TS Inter 1st Year

Here students can locate TS Inter 1st Year Physics Notes 9th Lesson Gravitation to prepare for their exam.

TS Inter 1st Year Physics Notes 9th Lesson Gravitation

→ Kepler’s Laws :
Law of orbits (1st law) ‘.All planets move in an elliptical orbit with the sun is at one of its foci.

→ Law of areas (2nd law) : The line joining the planet to the sun sweeps equal areas in equal intervals of time, i.e., \(\) = constant.
i. e., planets will appear to move slowly when they are away from sun, and they will move fast when they are nearer to the sun.

→ Law of periods (3rd law) : The square of time period of revolution of a planet is proportional to the cube of the semi major axis of the ellipse traced out by the planet.
i.e., T² ∝ R³ ⇒ \(\frac{\mathrm{T}^2}{\mathrm{R}^3}\) = constant

→ Newton’s law of gravitation (OR) Universal law of gravitation: Every body in universe attracts other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
F ∝ m¹m², F ∝ \(\frac{1}{\mathrm{r}^2}\) ⇒ F = G\(\frac{\mathrm{m}_1 \mathrm{~m}_2}{\mathrm{r}^2}\)

→ Central force : A central force is that force which acts along the line joining the sun and the planet or along the line joining the two mass particles.

→ Conservative force : For a conservative force work done is independent of the path. Work done depends only on initial and final positions only.

→ Gravitational potential energy : Potential energy arising out of gravitational force is called gravitational potential energy.
Since gravitational force is a conservative force gravitational potential depends on position of object.
V = \(-\frac{\mathrm{Gm}_1 \mathrm{~m}_2}{\mathrm{r}}\)

→ Gravitational potential : Gravitational potential due to gravitational force of earth is defined as the “potential energy of a particle of unit mass at that point”.
Gravitational potential V = \(\frac{G M}{r}\)
(r = distance from centre of earth)

→ Acceleration due to gravity (g) :
Acceleration due to gravity ‘g’ = \(\)

→ Acceleration due to gravity below and above surface of earth :
1) For points above earth total mass of earth seems to be concentrated at centre of earth.
For a height ‘h’ above earth
g(h) = \(\frac{\mathrm{GM}_{\mathrm{E}}}{\left(\mathrm{R}_{\mathrm{E}}+\mathrm{h}\right)^2}\)
where h << RE
g(h) = g\(\left(1+\frac{h}{R_E}\right)^{-2}\) = g\(\left(1-\frac{2 \mathrm{~h}}{\mathrm{R}_{\mathrm{E}}}\right)\)

2) For a point inside earth at a depth’d’ below the ground mass of earth (M_s) with radius (R_E – d) is considered. That mass seems to be at centre of earth.
g’ = g\(\left(1-\frac{d}{R}\right)\)

→ Escape speed (v₁)_min : The minimum initial velocity on surface of earth to overcome gravitational potential energy is defined as “escape speed v_e”
v_e = \(\sqrt{2 \mathrm{gR}}=\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}}\)

→ Orbital velocity: Velocity of a body revolving in the orbit is called orbital velocity.
Orbital velocity V₀ = \(\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}\) ⇒ V₀ = \(\sqrt{\mathrm{gR}}\)

Note:

• Relation between orbital velocity V and escape speed v_e = √2V₀
• If velocity of a satellite in the orbit is increased by √2 times or more it will go to infinite distance.

→ Time period of the orbit (T) : Time taken by a satellite to complete one rotation in the orbit is called “time period of rotation”.
T = 2π\(\frac{\left(\mathrm{R}_{\mathrm{E}}+\mathrm{h}\right)^{3 / 2}}{\sqrt{\mathrm{GM}_{\mathrm{E}}}}\)

→ Geostationary orbit : For a geostationary orbit in equatorial plane its time period of rotation is 24 hours, i.e., angular velocity of satellite in that orbit is equal to angular velocity of rotation of earth.
Geostationary orbit is at a height of 35800 km from earth.

→ Geostationary satellite : Geostationary satellite will revolve above earth in geostationary orbit along the direction of rotation of earth. So it always seems to be stationary w.r.t earth.
Time period of geostationary satellite is 24 hours. It rotates in equatorial plane in west to east direction.

→ Polar satellites: Polar satellites are low attitude satellites with an altitude of 500 km to 800 km. They will revolve in north-south direction of earth.
Time period of polar satellites is nearly 100 minutes.

→ Weightlessness: Fora freely falling body its weight seems to be zero. Weight of a body falling downwards with acceleration ‘a’ is w’ = mg’ = m(g – a). When a = g the body is said to be under free fall and it seems to be weightless.

→ Force between two mass particles, F = \(\frac{\mathrm{Gm}_1 \mathrm{~m}_2}{\mathrm{r}^2}\)

→ Universal gravitational constant, G = \(\frac{\mathrm{Fr}^2}{\mathrm{~m}_1 \mathrm{~m}_2}\)
G = 6.67 × 10^-11 Nm² / Kg² D.F.: M^-1 L³ T^-2

→ Relation between g and G is,
g = \(\frac{\mathrm{GM}}{\mathrm{R}^2}=\frac{4}{3}\)πρG.R

→ Variation ofg with depth, g_d = g(1 – \(\frac{d}{R}\))

→ Variation ofg with height, g_h = g(1 – \(\frac{2h}{R}\))
For small values of h i.e., h < < R then
g_h = g(1 – \(\frac{2h}{R}\))

(a) Gravitational potential, U = –\(\frac{\mathrm{GMm}}{\mathrm{R}}\)
(b) If a body is taken to a height h’ above the ground then
Gravitational potential, Uh = –\(\frac{\mathrm{GMm}}{(\mathrm{R}+\mathrm{h})}\)

→ Orbital velocity, V₀ = \(\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}=\sqrt{\mathrm{gR}}\)

→ Orbital angular velocity, ω₀ = \(\sqrt{\frac{\mathrm{GM}}{\mathrm{R}^3}}=\sqrt{\frac{\mathrm{g}}{\mathrm{R}}}\)

→ Escape velocity, V_e = \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}}=\sqrt{2 \mathrm{gR}}\)

→ Time period of geostationary orbit = 24 hours.

→ Angular velocity of earth’s rotation
(ω) = \(\frac{2 \pi}{24 \times 60 \times 60}\) = 0 072 × 10^-3 rad/sec

Here students can locate TS Inter 1st Year Physics Notes 10th Lesson Mechanical Properties of Solids to prepare for their exam.

TS Inter 1st Year Physics Notes 10th Lesson Mechanical Properties of Solids

→ Elasticity: The property of a body by virtue of which it tends to regain its original size (or) shape when applied force is removed.

→ Plasticity: It is the inability of a body not to regain its original size (or) shape when external force is removed.
Note : Substances which will exhibit plasticity are called plastic substances.

→ Stress (σ): The restoring force per unit area is called stress (σ).
Stress (σ) = Force/area
Unit Nm^-2 (or) pascal; D.F : ML^-1 T^-2

→ Tensile Stress: When applied force is normal to area of cross section of the body then restoring force per unit area is called tensile stress.

→ Tangential (or) shearing stress: The restor-ing force developed per unit area of cross section when a tangential force is applied is known as shear stress or tangential stress.

→ Hydraulic stress (Volumetric stress): For a body in a fluid force is applied on it in all directions perpendicular to its surface.
“The restoring force developed in the body per unit surface area under hydraulic compression is called hydraulic stress.”

→ Strain: Change produced per unit dimension is called strain. It is a ratio.
Types:

• Longitudinal strain : The ratio of increase in length to original length is called as longitudinal strain. Longitudinal strain = \(\frac{\Delta \mathrm{L}}{\mathrm{L}}\)
• Tangential (or) shear strain : The ratio of relative displacement of faces Ax to the perpendicular distance between the faces is called shear strain.
  Shear strain = \(\frac{\Delta \mathrm{L}}{\mathrm{L}}\) = tan θ
• Volume strain: The ratio of change in volume AV to the original volume (V) is called volume strain.
  Volume strain = \(\frac{\Delta \mathrm{V}}{\mathrm{V}}\)

→ Hooke’s Law : For small deformations the stress is proportional to strain.
Stress ∝ strain ⇒ stress/strain = constant.
This proportional constant is called modulus of elasticity.

→ Elastic constant: The ratio of stress to strain is called “elastic constant”. Unit: Newton/m². D.F: ML^-1T^-2
Note : Elastic constants are three types.

→ Young’s modulus (Y): The ratio of tensile stress (or) compressive stress to longitudinal strain or compressive strain is called Young’s modulus.
Y = \(\frac{\sigma}{\varepsilon}=\frac{\text { Tensile or compressive stress }(\sigma)}{\text { Tensile or compressive strain }(\varepsilon)}\)

→ Shear modulus (G) : The ratio of shearing stress to the corresponding shearing strain is called shear modulus.
Shear modulus (G) = \(\frac{\text { Shear stress }\left(\sigma_{\mathrm{s}}\right)}{\text { Shearstrain }(\theta)}\)

→ Bulk modulus (B): The ratio of hydraulic stress to the corresponding hydraulic strain is called Bulk modulus.
Bulk modulus (B) = \(=-\frac{\text { Hydraulic pressure (F/A) }}{\text { Hydraulic strain }(\Delta \mathrm{V} / \mathrm{V})}\)

→ Compressibility (K): The reciprocal of bulk modulus is called compressibility. Compressibility K = 1/B

→ Poisson’s ratio: In a stretched wire the ratio of lateral contraction strain to longitudinal elongation strain is called Poisson’s ratio.
Poisson’s ratio σ = \(\frac{\Delta \mathrm{d} / \mathrm{d}}{\Delta \mathrm{L} / \mathrm{L}}\)
Poissons ratio is a ratio of two strains so it has only numbers.
Note : For steel Poisson’s ratio is 0.28 to 0.30, for aluminium alloys it is upto 0.33.

→ Elastic potential energy (u) : When a wire is under tensile stress, work is done against the inter atomic forces. This work is stored in the wire in the form of elastic potential energy.
(or)
Work done to stretch a wire against inter atomic forces is termed as “elastic potential energy”.
Elastic potential energy (u) = \(\frac{1}{2} \frac{\mathrm{YA} l^2}{\mathrm{~L}}\) = \(\frac{1}{2}\)σs
or u = \(\frac{1}{2}\) stress × strain × volume of wire.

→ Ductile materials : If the stress difference between ultimate tensile strength and fracture point is high then it is called ductile material. Ex : Silver, Gold.

→ Brittle material : If the stress difference between ultimate tensile strength and fracture point is very less then that substance is called
brittle material. Ex : Cast iron.

→ Elastomers : Substances which can be stretched to cause large strains are called elastomers. Ex : Rubber, Tissues of aorta.

→ Stress = \(\frac{\text { Force }}{\text { Area }}=\frac{F}{A}\); Unit: N/m2 (or) Pascal.

→ Strain = \(\frac{\text { elongation }}{\text { original length }}=\frac{\Delta \mathrm{L}}{\mathrm{L}}\); No units

→ Hooke’s Law: Within elastic limit, stress
stress oc strain (or) \(\frac{\text { stress }}{\text { strain }}\) = constant (Elastic constant).

→ Young’s modulus (Y) = \(\frac{\text { longitudinal stress }}{\text { longitudinal strain }}\)
= \(\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{e} / \mathrm{L}}=\frac{\mathrm{FL}}{\mathrm{A} \Delta \mathrm{e}}\). In Searle’s apparatus Y = \(\frac{g L}{\pi r^2} \cdot \frac{M}{e}\)

• If two wires are made with same materials have lengths l₁ l₂ and radii r₁, r₂ then ratio of elongations \(\frac{\mathrm{e}_1}{\mathrm{e}_2}=\frac{l_1}{l_2} \cdot \frac{\mathrm{r}_2^2}{\mathrm{r}_1^2}\)\ (∵ e ∝ l / r)
• If two wires are made with same material and same volume has areas Aj and A2 are subjected to same force then ratio of elongations \(\frac{e_1}{e_2}\) = r₂⁴/r₁⁴ (∵ e ∝ \(\frac{1}{\mathrm{r}^4}\))
• If two wires of same length and area of cross section are subjected to same force then ratio of elongations e₁/e₂ = \(\frac{y_2}{y_1}\) (∵ e ∝ \(\frac{1}{y}\))
• If two wires are made with same material have lengths /j and 1% and masses m} and m2 are subjected to same force then ratio of elongations \(\frac{\mathrm{e}_1}{\mathrm{e}_2}=\frac{l_1^2}{l_2^2} \times \frac{\mathrm{m}_2}{\mathrm{~m}_1}\) (∵ e ∝ \(\frac{l^2}{\mathrm{~m}}\))

→ Rigidity modilus, G = \(=\frac{\text { shear stress }}{\text { shear strain }}=\frac{\mathrm{F}}{\mathrm{A} \theta}=\frac{\mathrm{FL}}{\mathrm{Al}}\)

→ Shear strain,
θ = \(\frac{\text { relative displacement of upper layer }}{\text { perpendicular distance between layers }}=\frac{\Delta x}{z}=\frac{l}{L}\)

→ Bulk modulus (B) = \(\frac{\text { volumetric stress }}{\text { volumetric strain }}=\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{V} / \mathrm{V}}=\frac{\mathrm{PV}}{\Delta \mathrm{V}}\)

→ \(\frac{1}{B}\) is called “coefficient of compressibility” (K).

→ Poisson’s ratio,
σ = \(\frac{\text { lateral contraction strain }}{\text { longitudinal elongation }}=-\frac{\Delta D / D}{e / L}=-\frac{L \Delta D}{D \cdot e}\)

→ Theoretical limits of poisson’s ratio σ is – 1 to 0.5
Practical limits of poisson’s ratio σ is 0 to 0.5

→ Relation between Y, G, B and σ are
B = \(\frac{Y}{3(1-2 \sigma)}\)
η = \(\frac{Y}{2(1+\sigma)}\)
σ = \(\frac{3 B-2 G}{2(3 B+G)}\)
Y = \(\frac{9 \mathrm{~GB}}{3 \mathrm{~B}+\mathrm{G}}\)

→ Relation between volume stress and linear stress is \(\frac{\Delta V}{V}=\frac{\Delta L}{L}\)(1 – 2σ)

→ Strain energy = \(\frac{1}{2}\) × load × extension = \(\frac{1}{2}\) × F × e

→ Strain energy per unit volume = \(\frac{1}{2}\) × stress × strain or \(\frac{\text { stress }^2}{2 \mathrm{Y}}\) or \(\frac{\left({strain}^2\right) Y}{2}\)

→ When a body is heated and expansion is prevented then thermal stress will develop in the body.

• Thermal stress Y ∝ Δt
• Thermal force = YA ∝ Δt

→ When a wire of natural length L is elongated by tensions say T₁ and T₂ has final lengths l₁ and l₂ then Natural length of wire = L
= \(\frac{l_2 T_1-l_1 T_2}{\left(T_1-T_2\right)}\)

→ When a wire is stretched by a load has an elongation ‘e’. If the load is completely immersed in water then decrease In elongation e¹ = Vρgl/(AY) where V is volume of load, ρ = density of water.

Here students can locate TS Inter 1st Year Physics Notes 11th Lesson Mechanical Properties of Fluids to prepare for their exam.

TS Inter 1st Year Physics Notes 11th Lesson Mechanical Properties of Fluids

→ Pressure : It is defined as the normal force acting per unit area.
Pressure (P) = Force/Area.
Unit: Nm^-2 (or) pascal
D.F: ML^-1 T^-2
It is a “scalar quantity”.
Note : Common unit of pressure is atmosphere. 1 atmosphere = 1.013 × 10⁵ pascal.

→ Density (ρ):
The ratio of mass to volume of a body or mass per unit volume.
Density (ρ) : \(\frac{\text { mass }}{\text { volume }} \frac{\mathrm{m}}{\mathrm{v}}\)
Unit: kg m^-3, D.F: ML^-3

→ Relative density: The ratio of the density of the body to the density of water at 4 °C is called Relative density.

→ Pascal’s law: The pressure in a fluid at rest is the same at all the points if they are at the same height in the liquid.
Note : If the pressures were not equal in different parts of the fluid then some resultant force must act on it and the fluid will flow. Since the fluid is assumed to be at rest then pressure must be same every where.

→ Gauge pressure : The excess of pressure P – Pa at a depth h’ is called gauge pressure where P’ is the pressure at given point and Pa’ is atmospheric pressure.

→ Hydrostatic paradox:

• In a liquid the pressure at all the points of same height is equal.
• The pressure difference depends on vertical height only. From these two concepts “whatever is the shape and mass of liquid in the container, it will have same pressure difference when h is same”. This is called “hydrostatic paradox”.

→ Mercury barometer : Mercury barometer is used to measure atmospheric pressure.
It consists of one metre long glass tube with one side closed. It is filled with mercury and inverted into a trough con-taining mercury. Height of mercury column above mean level gives atmospheric pressure.

→ Manometer: Manometer is useful to measure pressure difference. It consists of a ‘U’ tube. One arm of ‘U’ tube is connected to the vessel where pressure is to be measured. The other long arm filled with mercury is open to atmospheric pressure. Gauge pressure = P – P_a.
Note: To measure small pressure differences we will use Torr . 1 Torr = 1 m.m of Hg = 133 pascal (P_a)
1 Atmospheric pressure = 1 Bar = 10⁵ P_a
Note : Hydraulic lifts and hydraulic brakes will work on principle of Pascal’s law.

→ Hydraulic lift : It consists of two cylinders with different areas of cross section. Both are separated and connected through a liquid. Let force F( is applied on small piston of area A, then pressure P = F₁/A₁ will be transmitted to large piston of area
From Pascal’s law
P = \(\frac{F_2}{A_2}=\frac{F_1}{A_1}\) ⇒ F₂ = F₁\(\frac{\mathrm{A}_2}{\mathrm{~A}_1}\)
Where A₂/A₁ is called mechanical advantage. In hydraulic lifts force applied on small piston F₁, multiplied by a factor A₂/A₁, at large piston and it will lift heavy loads with small force.

→ Hydraulic brakes : In hydraulic brakes a small force is applied on a master cylinder filled with fluid through brake pedal. It is transmitted through brake oil to other cylinders attached to four wheels. As a result brake linings of all the wheels will expand with uniform force and braking is very smooth.

→ Fluid dynamics: It is a branch of physics which deals about the study of fluids in motion.

→ Steady flow: The flow of the fluid is said to be “steady” if at any given point the velocity of each passing fluid particle remains constant in time.

→ Stream line: The path taken by a fluid particle under a steady flow is called stream line.
Note: The tangent drawn at any point gives the direction of velocity of fluid particle at that point.

→ Equation of continuity: It states that the volume flux (or) flow rate remains constant throughout the pipe of flow.
Mathematical form of equation of continuity is AV = constant (or) A₁V₁ = A₂V₂
Note : Equation of continuity is a consequence of law of conservation of mass of an incompressible fluid in motion.

→ Bernoulli’s principle: For a fluid in stream line flow the sum of pressure energy (p), kinetic energy \(\left(\frac{\rho V^2}{2}\right)\) and potential energy (ρgh) per unit volume remains constant.
i. e., P + \(\frac{1}{2}\)ρV² + ρgh
or P₁ + \(\frac{1}{2}\)ρV₁² + ρgh₁ = P₂ + —ρV₂² + ρgh₂

→ Limitations of Bernoulli’s principle :

• The fluids must be non viscous, be-cause in the derivation we assumed that no energy is lost due to friction.
• The fluids must be uncompressible, because in the derivation elastic energy of the fluid is not taken into account.

→ Speed of efflux (Torricelli’s Law) :

• Speed of efflux from an open tank V = \(\sqrt{2 \mathrm{gh}}\)
• Speed of efflux from a closed tank V = \(\sqrt{2 g h+\frac{2\left(P-P_a\right)}{\rho}}\)
  Where P’ pressure on fluid in closed tank and P_a is atmosphere pressure.

→ Venturi meter: Venturimeter is used to measure the flow speed of uncompressible liquids.
It consists of a tube with broader diameter ‘A’ with a small constriction of area ‘a’ at middle. When liquids flow through this tube pressure difference (h) will develop due to high velocity at constriction.
Velocity of fluid V = \(\sqrt{\left(\frac{2 \rho_m g h}{\rho}\right)}\left[\left(\frac{A}{a}\right)^2-1\right]^{-1 / 2}\)
Where ρ = density of liquid and ρ_m = density of mercury.

→ Dynamic lift: Dynamic lift is the force that acts on a body moving through a fluid. Ex : Hydro foil, Aeroplane wing, spinning ball.
When a body is moving through a fluid due to its shape or motion some pressure difference (P₂ – P₁) will develop at top and bottom layers of the body.
Dynamic lift = (P₂ – P₁) × Area (A)

→ Magnus effect : The dynamic lift due to spinning of a body is called magnus effect.

→ Viscosity: The resistance to the flow of liquid between different layers is called viscosity.
(or)
Friction between liquid layers in motion is called viscosity.

→ Coefficient of viscosity (η): It is defined as the ratio of shearing stress to shearing strain rate.
S.I unit of viscosity is poiseiulle (PI) or NS m^-2.
Dimensional formula: ML^-1 T Note : Poise is C.G.S unit of viscosity. 1 poise = 10^-1 poiseiulle (PI)

→ Strain rate : For a flowing liquid strain increases continuously with time. So rate of
change of strain is called “strain rate”.
Strain rate = \(\frac{\text { strain }}{\text { time }}=\frac{\Delta \mathrm{x} / l}{\Delta \mathrm{t}}=\frac{\mathrm{V}}{l}\) (∵ \(\frac{\Delta x}{\Delta t}\) = V)

→ Variation of viscosity of fluids with temperature :
1. Viscosity of liquids will decrease with temperature.
In liquids when temperature increases separation between molecules increases and cohesive force between molecules decreases. So viscosity decreases.

2. In case of gases viscosity increases with temperature.
In gases when temperature increases velocity of gases increases and exchange of momentum due to collision of gas molecules increases. So resistance to flow of gas molecules (viscosity) increases.

→ Stoke’s Law: From Stoke’s law for a smooth spherical body falling through a fluid :

• Force due to viscosity is proportional to velocity of the body. It acts in opposite direction of velocity F ∝ – V.
• Force due to viscosity is proportional to viscosity of the fluid F ∝ η.
• Force due to viscosity is proportional to radius of the body (a) i.e., F ∝ a
  From Stoke s law force due to viscosity F ∝ – aηV, magnitude of force F = 6π a ηV

→ Terminal velocity : When a body is falling through a fluid after sometime the body will attain a constant speed V called “terminal velocity”.
At terminal velocity force due to viscosity (F_v) and force due to buoyancy (F_b) are equal to weight of the body (mg).

Terminal velocity of a body
V_Y = \(\frac{2}{9} \frac{r^2 g\left(\rho-\rho_0\right)}{\eta}\)

→ Reynolds number (R_e): Reynolds number represents the ratio of internal force to viscous force of a flowing liquid.
Internal force = ρAV²,
Force due to viscosity = \(\frac{\eta \mathrm{AV}}{\mathrm{d}}\)
Reynolds number = \(\frac{\text { Internal force }}{\text { Forcedue to viscosity }}\)
= \(\frac{\rho \mathrm{AV}^2}{\eta \mathrm{AV} / \mathrm{d}}=\frac{\rho \mathrm{Vd}}{\eta}\)
Note : For steady or laminor flow R_e < 1000
For turbulent flow R_e > 2000

→ Critical Reynolds number: For every fluid in motion the laminor flow will become turbulent flow for R = 1000 to 2000.
The velocity of the fluid where laminor flow becomes turbulent is called “critical velocity”. Reynolds number corresponding to critical velocity is called “critical Reynold’s number”.

→ Critical velocity (V): The maximum velocity of a fluid in a tube for which the flow remains streamlined is called critical velocity.
Critical velocity V_c = R_e η/pd.

→ Surface energy: In a liquid molecules near the surface are pulled down due to cohes’ive forces of liquid molecules. As a result these surface molecules will have some negative potential energy.
Energy possessed by the surface molecules of a liquid is known as “surface energy.”
Note : Due to surface energy liquid layers tends to have least surface area.

→ Surface tension (S) : Surface tension is the force per unit length acting in the plane of the interface between the plane of the liquid and any other substance.
Unit: newton/metre. D.F = MT^-2

→ Surface energy & surface tension: Surface energy per unit area of the liquid interface is equal to “surface tension.”
Surface tension (S) : \(\frac{\text { Surface energy }}{\text { Area of liquid surface }}\)

→ Angle of contact (θ): The angle between the tangent to the liquid surface at the point of contact and solid surface inside the liquid is called “angle of contact ‘θ’.”

→ Drops and bubbles:

• Rain drops and soap bubbles are sphe-rical because of negative surface energy. Liquid surface tries to occupy minimum possible area. Among all geometrical shapes sphere has least surface area. So drops and bubbles are spherical.
• Excess pressure inside a liquid drop is P = \(\frac{2 \mathrm{~S}_{\mathrm{T}}}{\mathrm{r}}\)
  r = radius of the drop
• Excess pressure inside a bubble is P = \(\frac{4 \mathrm{~S}_{\mathrm{T}}}{\mathrm{r}}\)

→ Capillary rise: The rise or fall of a liquid in a capillary tube above or below mean level is called capillary rise. Capillary rise is due to surface tension of liquids.

Note :

• When angle of contact θ < 90° then capillary rise takes place.
• When angle of contact θ > 90° capillary depression takes place.

→ Wetting agents: Substances which decrease angle of contact between the liquid and solid molecules are known as “wetting agents”.
Wetting agents are used in dyeing fabrics. Note : Due to wetting agents force of attraction between the solid and liquid molecules (S_SA) is more than force due to surface tension force of liquid (S_Li)

→ Water proof agents : Substances which increase the angle of contact between the liquid and solid molecules are known as “Water proof agents.”
Water proof agents are used in marine paints and in soaps and detergents.

Note : Due to water proof agents force of attraction between liquid and solid molecules (S_sa) is less than force of attraction between liquid molecules due to surface tension (S_Li).

→ Detergent action: Molecules of detergents are hair pin shaped. One end is attached to water molecule and the other end to dirt particles or grease particles etc. Detergent molecules will also reduce the surface tension of water drastically. As a result force between dirt particles and liquid molecules is reduced. Due to strong force between dirt molecules and detergent molecules thej* can be easily removed.

→ Average pressure, P_av = \(\frac{\mathrm{F}}{\mathrm{A}}\)

→ Hydrostatic pressure of a liquid, P = hρg
ρ = density of liquid; h = height of liquid

→ Pressure at any point in a liquid at a depth of ‘h’ is, P = P₀ + h dg
where P₀ = atmospheric pressure.

→ Pressure energy per unit volume = P = hydrostatic pressure

→ Upward force due to buoyancy = Vρg
= weight of liquid displaced by the body

→ Fraction of volume of a body submersed in a liquid = \(\frac{\text { density of object }}{\text { density of fluid }}\)

→ From equation of continuity A₁V₁ = A₂V₂ i.e., At steady flow volume flux is constant.

→ Critical Velocity, V_c = \(\frac{R \cdot \eta}{d \rho}\) or number, R = \(\frac{V_c d \rho}{\eta}\)
d = diameter of tube;
ρ = density of fluid,
η = coefficient of viscosity.

→ Bernoulli’s theorem states that total energy of a fluid at any point in that fluid is constant.
P + \(\frac{1}{2}\)ρv² + ρgh = a constant

→ At horizointal points or fluid is flowing horizontally then \(\frac{P_1}{\rho_1}+\frac{v_1^2}{2}=\frac{P_2}{\rho_2}+\frac{v_2^2}{2}\)

→ Aero – dynamic lift on wing of aeroplane p ressure difference × Area of wing
Dynamic lift = (P₂ – P₁) A

→ Velocity gradient, \(\frac{\Delta v}{\Delta x}\)
= \(\frac{\text { Change in velocity }}{\text { Perpendicular distance between layers }}\)

→ Coefficient of viscosity, η = \(\frac{\mathrm{F}}{\mathrm{A}} \frac{\Delta \mathrm{x}}{\Delta v}\) or
η = \(\frac{\pi \mathrm{Pa}^2}{8 \mathrm{Ql}}\) (Poiseuille’s equation)
(where Q = volume of liquid flowing per second, r = radius of tube, l = length of tube and P = pressure difference)

→ Torricelli’s formula : Velocity of liquid through a hole, v = \(\sqrt{2 g h}\)
(h = height of liquid above centre of hole.)

→ Stake’s Law : Viscous force on a smooth spherical body, F = 6π η av

→ Terminal velocity is the constant velocity with which the body is falling through the fluid.
Terminal velocity, v_t = \(\frac{2}{9} a^2 \frac{g\left(\rho-\rho_0\right)}{\eta}\)
ρ = density of the body,
ρ₀ = density of fluid,
η = coefficient of viscosity of fluid.

→ Surface tension = \(\frac{\text { Force }}{\text { Length }}\) ⇒ S = \(\frac{\mathrm{F}}{l}\)

→ Surface Tension, S = \(\frac{\mathrm{rdg}\left(\mathrm{h}+\frac{\mathrm{r}}{3}\right)}{2 \cos \theta}=\frac{\mathrm{rhdg}}{2 \cos \theta}\)
For water θ = 0°
S = \(\frac{\text { rhdg }}{2}\)

→ Height of liquid h ∝ S, h ∝ \(\frac{1}{r}\). This is called “Jurin’s law”.

→ When a capillary tube is tilted by an angle a’ with vertical then vertical height of liq¬uid is same. But length of liquid in capillary tube increases.
h₁ = \(\frac{\mathrm{h}}{\cos \alpha}\)
Ratio of lengths for two different angles is
\(\frac{\mathrm{h}_1}{\mathrm{~h}_2}=\frac{\cos \alpha_2}{\cos \alpha_1}\)

→ If a capillary tube of insufficient length is placed in a liquid, then liquid will rise upto top of capillary tube but angle of contact 0 will increase.
\(\frac{\mathrm{h}_1}{\cos \theta_1}=\frac{\mathrm{h}_2}{\cos \theta_2}\) or h₁cos θ₂ = h₂ cos θ₁

→ Excess of pressure inside a soap bubble,
P = \(\frac{4 \mathrm{~S}}{\mathrm{r}}\)

→ Excess of pressure inside a liquid drop,
P = \(\frac{2 \mathrm{~S}}{\mathrm{r}}\)

→ Work done to blow a soap bubble from soap solution, w = 8πr²S

→ Work done to increase the radius of a soap bubble from r₁ to r₂ is, w = 8πS (r₂² – r₁²)
(where r₂ > r₁)

→ Work done to divide a large drop of liquid into ‘n’ small droplets, w = 4πr²S (n^{\(\frac{1}{3}\)} – 1)

→ Energy released when ‘n’ small droplets are combined to form a large bubble is,
w = 4πr²S (n^{\(\frac{2}{3}\)} – 1)

Here students can locate TS Inter 1st Year Physics Notes 12th Lesson Thermal Properties of Matter to prepare for their exam.

TS Inter 1st Year Physics Notes 12th Lesson Thermal Properties of Matter

→ Temperature: Temperature is a relative measure of hotness or coldness of a body.

→ Heat: Heat is a form of energy transferred between two systems and its surroundings by virtue of temperature difference.

→ Measurement of temperature: Thermometers are used to measure temperature. Commonly used property in measuring temperature is variation of volume of a liquid with temperature.
Since temperature is a relative concept- two fixed points are used for standard scale. They are

• Freezing point of pure water is taken as “lower fixed point”.
• Boiling point of pure water at standard atmospheric pressure is taken as “upper fixed point”.

→ Celsius scale (°C): In Celsius scale freezing point of water at standard atmospheric pressure is taken as zero. Boiling point of water at standard atmospheric pressure is taken as 100. The temperature difference between these two limits is divided into 100 equal parts and each part is called as 1°C.

→ Fahrenheit scale (°F): In Fahrenheit scale, freezing point of water is taken as 32 °F and boiling point of water is taken as 212 °F. The temperature difference between these two limits is divided into 180 equal parts. Each part is called 1 °F.
Note: Relation between Celsius scale and
Fahrenheit scale is \(\frac{\mathrm{T}_{\mathrm{F}}-32}{180}=\frac{\mathrm{T}_{\mathrm{c}}}{100}\)

→ Boyle’s Law: When temperature is held constant, volume of a given mass of gas (V) is inversely proportional to its pressure (P)
i.e., V ∝ \(\frac{1}{P}\) ⇒ PV = constant.

→ Charles’ Law (I): When pressure is kept constant, the volume of a given mass of gas (V) is proportional to the temperature (T).
i.e., V ∝ T ⇒ \(\frac{P}{T}\) = constant

→ Charles’ Law (II): When volume is constant the pressure of a given mass of gas (P) is proportional to the temperature (T).
i.e., P ∝ T ⇒ \(\frac{P}{T}\) = constant

→ Ideal gas equation: From Boyle’s law for given quantity of gas PV = constant. From Charles’ law \(\frac{V}{T}\) is constant for given quantity of gas. By combining these two laws, \(\frac{PV}{T}\) = constant for given quantity of gas.
i.e., \(\frac{PV}{T}\) = nR or PV = nRT
Where n = Number of gram moles of gas taken,
R = Universal gas constant = 8.31 J mole^-1. K^-1.
Note: The temperature’T’ used in Charles’ law and in ideal gas equation is absolute scale or Kelvin scale of temperature.

→ Kelvin scale or Absolute scale of temperature: The temperature scale in which -273.15 °C is taken as the zero point (‘0’K) is called”Kelvin scale or absolute scale of temperature.”
The size of unit in Kelvin scale and in Celsius scale are equal. A relation between them is T = t_c + 273.15.
Note: The temperature scales of Kelvin and Celsius does not coincide numerically. Whereas the magnitudes of Celsius scale and Fahrenheit scales are numerically equal at – 40 i.e., – 40 °F = – 40 °C.

→ Thermal expansion: Most substances expand on heating and contract on cooling.
The increase in the dimensions of a body per unit dimension due to increase in temperature is called Thermal expansion.
Note:

• Expansion in length is called linear expansion.
• Expansion in area is called areal expansion.
• Expansion in volume is called volume expansion.

→ Linear expansion coefficient or coefficient of linear expansion (α): For a body in the form of a rod the fractional change in length \(\frac{\Delta l}{l}\) is proportional to change in temperature ΔT.
i.e., \(\frac{\Delta l}{l}\) ∝ ΔT ⇒ \(\frac{\Delta l}{l}\) = αΔT
α = \(\frac{\Delta l}{l \cdot \Delta \mathrm{T}}\) ⇒ α = \(\frac{l_2-l_1}{l_1\left(\mathrm{t}_2-\mathrm{t}_1\right)}\)/°C
Where α = coefficient of linear expansion.

→ Coefficient of areal expansion (α_a): For a body in the form of thin sheet fractional increase in area (Δa/a) is proportional to change in temperature (ΔT).
i.e., \(\frac{\Delta a}{a}\) ∝ ΔT ⇒ \(\frac{\Delta a}{a}\) = α_aΔT ⇒ α_a = \(\frac{a_2-a_1}{a_1 \Delta T}\)/°C
Where α_a = coefficient of areal expansion.

→ Coefficient of volume expansion (α_v): For a body the fractional change in volume ( ΔV/V) is proportional to change in temperature ΔT .
i.e., \(\frac{\Delta V}{V}\) ∝ ΔT (or) \(\frac{\Delta V}{V}\) = α_VΔT ⇒ α_v = \(\frac{\Delta \mathrm{V}}{\mathrm{V} \cdot \Delta \mathrm{T}}\)/°C
(or) coefficient of volume expansion
α_v = \(\frac{V_2-V_1}{V_1\left(t_2-t_1\right)}\)/°C

→ Relation between expansion coefficients α, α_a and α_v: Coefficient of areal expansion α_a = 2a
Coefficient of volume expansion α_v = 3α

→ Thermal stress: When a body is heated and expansion is prevented then stress will develop in the body. This is known as Thermal stress.
When a body is heated fractional increase in length Δl/l = αAT.
But from principles of elasticity Δl/l is strain.
Young’s modulus Y = \(\frac{\text { stress }}{\text { strain }}=\frac{F / A}{\Delta l / l}\)
Stress produced = Y . Δl/l. This is called thermal stress.

→ Anomalous expansion of water: Water when heated from 0°C to 4 °C its volume will decreases and4°-100°Cits volume increases. This phenomenon is called “Anomalous expansion of water.”
Note: Due to anomalous expansion water has maximum density at 4 °C.

→ Importance of anomalous expansion of water: Due to anomalous expansion of water in polar region and in cold countries water freezes from the top layers of lakes or rivers etc. But water will exist at 4 °C at bottom layers. Hence aquatic life such as plants, fishes etc., are able to survive at bottom layers even in winter.

→ Heat capacity (S): It is the quantity of heat required by a substance to change its temperature by one unit.
Heat capacity
S = \(\frac{\text { Heat supplied }}{\text { Change in temperature }}=\frac{\mathrm{dQ}}{\mathrm{dt}}\)
Unit: joule/kelvin ; D.F = ML²T^-2K^-1

→ Specific heat (s): The amount of heat absorbed or rejected to change the temperature of unit mass of a body by one unit temperature difference is called “specific heat.”
Specific heat s = \(\frac{\mathrm{S}}{\mathrm{m}}=\frac{1}{\mathrm{~m}} \cdot \frac{\mathrm{dQ}}{\mathrm{dt}}\)
Unit: joule kg^-1 k^-1; D.F = L²T^-2 K^-1

→ Molar specific heat: Heat capacity of substance per mole of substance is called “Molar specific heat.”
Molar specific heat C = \(\frac{S}{n}=\frac{1}{n} \frac{d Q}{d T}\)
Unit: joule/mole – kelvin (or) joule, mole^-1 kelvin^-1; D.F = ML²T^-2 mol^-1 K^-1
Note: Since volume and pressure of a gas depend much on temperature, we have two molar specific heats for gases.

• Molar specific heat at constant pressure (C_P) The amount of heat required to raise the temperature of one gram mole of gas through 1 °Cor T K at constant pressure.
• Molar specific heat at constant volume (C_v): The amount of heat required to raise the temperature of one mole of gas through 1 °C or 1 Kelvin at constant volume.

Note: In gases C_P > C_v. When a gas is heated at constant volume all the heat energy supplied is useful to rise the temperature of the gas only. Whereas when a gas is heated at constant pressure heat energy supplied is useful to increase the temperature of the gas and also to do external work.

→ Principle of calorimetry: In a thermally isolated system, when a hot body and cold body are mixed together then “Heat lost by hot body = Heat gained by cold bodyThis is called “principle of calorimetry”.

→ Change of state: The transition from one state to another state of matter is called “change of state”.
Ex: Conversion of a solid into liquid (or)
Conversion of a liquid into vapour or vice versa.

→ Melting point: Change of state from solid to liquid is called “melting point”.
At melting point temperature of solid and liquid forms are in thermal equilibrium. Temperature of the liquid does not increase until the whole solid is converted into liquid.

→ Normal melting point: Melting point at standard atmospheric pressure is called “normal melting point”.

→ Vaporisation: The change of state from liquid to vapour (or gas) is called vaporisation.

→ Boiling point: The temperature at which the liquid and vapour states of a substance coexist is called “boiling point”.
At boiling point temperature of the vapour does not increase until the whole liquid is converted into vapour.

→ Effect of pressure on boiling point:

• The boiling point of a liquid increase when pressure increases.
• The boiling point of a liquid decreases when pressure decreases.

→ Normal boiling point: Boiling point of a liquid at standard atmospheric pressure is called “normal boiling point”.

→ Sublimation: The change from solid state to vapour state without passing through liquid state is called “sublimation”.
Ex: Dry ice (solid CO₂), Iodine.
Note: In sublimation when solid is heated it will directly convert into vapour state.

→ Regelation: The process of conversion of a solid into liquid due to increased pressure and again refreezing into a solid when pressure is reduced is called “regelation”.

→ Triple point: The temperature and pressure where a substance can coexist in all its three states, i.e., The substance will exist as a solid, as liquid and as vapour at that particular temperature and pressure.
For water the triple point is at a temperature of 273.16K and at a pressure of 6.11 × 10^-3 atmospheres or nearly 610 pascals.

→ Latent heat (L): The amount of heat absorbed or rejected by unit mass of substance while converting from one state to another state at constant temperature.
Latent heat (L) = \(\frac{\text { Heat absorbed } / \text { rejected }}{\text { Mass of the substance }}=\frac{Q}{\mathrm{~m}}\)

→ Latent heat of fusion (L_f): The amount of heat absorbed or rejected by unit mass of substance while converting from solid to liquid or liquid to solid state.
L = 80 cal / gm

→ Latent heat of vaporisation (L_v): The amount of heat absorbed or rejected by unit mass of substance while converting from liquid to vapour or vapour to liquid states.
L = 540 cal / gm

→ Heat transfer: The transmission of heat energy from one system to another system or from one part of the system to another part is known as heat transfer.
Note: Heat transfer will take place in three different methods called

• Conduction,
• Convection,
• Radiation.

→ Convection: It is a process of heat transfer by actual motion of matter. Convection takes place in fluids (i.e., in liquids and gases).

→ Natural convection: In natural convection gravity plays an important part.
When a fluid is heated it will expand, its density will decrease. So it goes to upper layers. Fluid with high density will come down due to gravity. In this way heat energy is transmitted.

→ Forced convection: In forced convection the fluid is forced to move with the help of a pump or by some other mechanism.
Ex: Blood circulation in human body.

→ Trade winds: Trade wind is a result of natural convection.
Due to natural convection a steady surface wind blows on earth in North-east direction towards equator. This is called trade wind.
Note: At equator, temperature is high, so a is heated and goes up. At poles temperature is less, so air is cooled and it will blow towards equator. But due to rotation of earth this air cycle is not exactly between poles and equator. But hot air reaches earth at 30° north to poles and returns to equator.

→ Radiation: In radiation heat energy transfer takes place without any medium. Heat radiation takes place in the form of electromagnetic waves. They do not require any medium to travel.
Ex: Heat energy from Sun reaches Earth in the form of radiation.
Note: Amount of heat energy radiated depends on colour of the body, temperature of the body and its surface area.

→ Black body radiation: An ideal black body will radiate all the heat energy given to it. Thermal radiation of black body at any given temperature is a continuous spectrum which contains electromagnetic waves of different wave lengths.

→ Wein’s displacement law: From Wein’s law, the wave length λ_m corresponding to maximum energy radiated will decrease with increasing temperature of the body.
From Wein’s law λ_mT = constant (b)
Wein’s constant b = 2.9 × 10^-3 mK
Note: In the solar spectrum λ_m = 4753 Å. It indicates that surface temperature of Sun is T = 6060 K.

→ Stefan-Boltzmann’s law: The energy emitted per unit time from a hot bodv L ∝ AT⁴ Where A is area of the body and T is temperature in Kelvin scale, or H = σAT⁴. Where σ is a constant. It is called Stefan – Boltzmann’s constant, a = 5.67 x 10^-8m²K^-4.

Note:

• For a body other than black body from Stefan – Boltzmann’s law H = AeσT^-4
  Where e = emissivity of the body.
• If the body is surrounded by a medium of temperature Ts then H = Aeσ(T^-4 – T_s^-4)

→ Green house effect: Earth will absorb heat radiation and reradiate heat energy of longer wave length. This longer wave length heat radiation is reflected back to earth due to green house gases such as Carbondioxide [CO₂], Methane (CH₄), Chloroflurocarbons, Ozone (O₃) etc. As a result temperature of earth’s atmosphere is gradually increasing. This is known as “green house effect.”
Note: CO₂, CH₄, Chloroflurocarbons, Ozone (O₃) which reflect back longer radiation to earth are called “green house gases”.

→ Global warming: Earth receives heat energy during day time from sun. It reradiates heat energy in the form of longer electromagnetic waves.
But due to presence of green house gases the longer electromagnetic waves were reflected back to earth. As a result temperature of earth’s atmosphere is gradually increasing.
This process will increase with the increased content of green house gases in atmosphere. As a result temperature of earth’s atmosphere increases gradually.

→ Newton’s law of cooling: From Newton’s law of cooling the rate of loss of heat (-dQ/ dt) of a body is directly proportional to the difference of temperature (ΔT = T₂ – T₁) of the body and the surroundings.
This law holds good when ΔT = T₂ – T₁ is small.
\(-\frac{\mathrm{d} Q}{\mathrm{dt}}\) (T₂ – T₁) or \(\frac{\mathrm{d} Q}{\mathrm{dt}}\) = K(T₂ – T₁)
Where K = constant that depends on area and nature of the body.

→ Relation between Celsius and Fahrenheit scales is \(\frac{C-0}{100}=\frac{F-32}{180} \Rightarrow \frac{C}{5}=\frac{F-32}{9}\)
C = \(\frac{5}{9}\)(F – 32) or F = \(\frac{9}{5}\)C + 32

→ Linear expansion coefficient or Final length,
(α) = \(\frac{l_2-l_1}{l_1\left(t_2-t_1\right)}\)/°C
or Final length, l₂ = l₁[1 + α(t₂ – t₁)] or l₂ = l₁(1 + αΔt)
Increase in length, l₂ – l₁ = l = l₁α(t₂ – t₁) or l = l₁αΔt

→ Areal expansion coefficient
(α_a) = \(\frac{A_2-A_1}{A_1\left(t_2-t_1\right)}\)/°C
Final Area (A₂) = A₁ [(1 + β(t₂ – t₁)] or A₂ = A₁ (1 + βΔt)
Increase in area (A₂ – A₁) = ΔA
= A₁ α (t₂ – t₁) or ΔA = AβΔt

→ Volume expansion coefficient,
(γ) = \(\frac{V_2-V_1}{V_1\left(t_2-t_1\right)}\)/°C
Final volume (V₂) = V₁ [1 + γ(t₂ – t₁)] or V₂= V₁ (1 + γΔt)
Increase in volume = (V₂ – V₁) = ΔV
= V₁ γ (t₂ – t₁) or ΔV = V₁γ Δt

→ Relation between α, α_a and α_v is α_a = 2a, α_v = 3a (or) α: α_a: α_v = 1: 2: 3

→ Change of density with temperature,
ρ_t = ρ₁[1 + α_v (t₂ – t₁)] or ρ_t = ρ₁(1 + α_vΔt)

→ Two different metal rods will always keep the same length difference at all temperatures if l₁ α₂ = l₂ α₂

→ The volume of unoccupied portion of liquid in a vessel will always remain constant at all temperatures if V₁α_v1 = V₂α_v2

→ Volume expansion coefficient of a gas,
α = \(\frac{V_t-V_0}{V_0 t}\)/°C or α = \(\frac{V_2-V_1}{V_1 t_2-V_2 t_1}\)/°C

→ Pressure coefficient of a gas,
β = \(\frac{P_t-P_0}{P_0 t}\)/°C or β = \(\frac{\mathrm{P}_2-\mathrm{P}_1}{\mathrm{P}_1 \mathrm{t}_2-\mathrm{P}_2} \mathrm{t}_1\)/°C

→ In gases, volume coefficient (α) = pressure coefficient (β) = \(\frac{1}{273}\)

→ From Boyle’s law at constant temperature V ∝ \(\frac{1}{P}\) or PV = constant or P₁V₁ = P₂V₂

→ From Charle’s Law at constant pressure
V ∝ T or \(\frac{V}{T}\) = constant or = \(\frac{V_1}{T_1}=\frac{V_2}{T_2}\)

→ From Charle’s Law at constant volume
P ∝ T or \(\frac{P}{T}\) = constant or \(\frac{\mathrm{P}_1}{\mathrm{~T}_1}=\frac{\mathrm{P}_2}{\mathrm{~T}_2}\)

→ Ideal gas equation PV = RT for one mole PV = nRT for n moles of gas or
PV = \(\left(\frac{\mathrm{m}}{\mathrm{M}}\right)\)RT

→ Gas equation in terms of mass of gas is
PV = mrT where r = R/M.

→ Universal gas constant,
R = \(\frac{\mathrm{PV}}{\mathrm{T}}\) = 8.317 J / mole – k

→ Rate of flow of heat energy, \(\frac{Q}{t}\) ∝ A\(\frac{\left(\theta_2-\theta_1\right)}{\mathrm{d}}\)
or \(\frac{Q}{t}\) = KA\(\frac{\left(\theta_2-\theta_1\right)}{\mathrm{d}}\)

→ Coefficient of thermal conductivity,
K = \(\frac{\mathrm{Qd}}{{At}\left(\theta_2-\theta_1\right)}\) Unit of K is w/m-k

→ Temperature difference across the ends of a conductor (θ₂ – θ₁) = \(\left(\frac{\mathrm{Q}}{\mathrm{t}}\right) \frac{\mathrm{d}}{\mathrm{KA}}\) where \(\frac{\mathrm{d}}{\mathrm{KA}}\) = R is known as thermal resistance.

→ When two conductors are joined in end to end, total thermal resistance R = (R₁ + R₂)
∴ R = \(\left[\frac{\mathrm{d}_1}{\mathrm{~K}_1 \mathrm{~A}_1}+\frac{\mathrm{d}_2}{\mathrm{~K}_2 \mathrm{~A}_2}\right]\)
(a) Temperature gradient = \(\frac{d \theta}{t}=\frac{\left(\theta_2-\theta_1\right)}{l}\)
(b) Temperature at junction θ = \(\frac{\mathrm{K}_1 \theta_1 l_2+\mathrm{K}_2 \theta_2 l_1}{\mathrm{~K}_1 l_2+\mathrm{K}_2 l_1}\)

→ In thermal convection, rate of flow of heat energy, \(\frac{Q}{t}\) = hA Δθ where h = coefficient of convection.
A = Surface area over which fluid moves.
Δθ = (t₂ – t₁) = Temperature difference between the surface and that of fluid.

→ Emissive power, e_λ = \(\frac{\mathrm{d} \phi}{\mathrm{d} \lambda}\) Energy emitted per unit wavelength within the wavelength limit of λ and λ + dλ, at given temperature.

→ Absorptive power, a_λ = \(\frac{\mathrm{dQ}_\lambda}{\mathrm{d} \phi}\)
= \(=\frac{\text { Energy flux absorbed in certain time }}{\text { Total energy flux incident on the body in the same time }}\)

→ Wein’s Law states that λ_max × T = constant (b) where b = Wein’s constant = 2.9 × 10^-3 mk.

→ Stefan’s Law: Energy radiated by a body is proportional to fourth power of the absolute temperature.
∴ H ∝ T⁴ for black body of unit area where P is energy radiated per second per unit area.
(a) H = σ AT⁴ where a = 5.670 × 10^-8 W/m2 – k. A is Area of surface.
(b) For any body other than black body a Thermal power radiated, H = e_λ σ AT⁴ where e_λ is emissivity of the body.
(c) If a body at T kelvin is in an enclosure of temperature T₁ then heat energy radiated per second is H = eσ A (T⁴ – T₁⁴)

→ From Newton ‘s Law of cooling energy radiated per second,
\(-\frac{\mathrm{dQ}}{\mathrm{dt}}\) ∝ ( T – T_s) or \(\frac{\mathrm{dQ}}{\mathrm{dt}}\) = -K(T – T_s)
or Rate of cooling \(\frac{\mathrm{dT}}{\mathrm{dt}}\) = -K(T – T_s)
where T is temperature of the body and T_s is temperature of surroundings.

Here students can locate TS Inter 1st Year Physics Notes 13th Lesson Thermodynamics to prepare for their exam.

TS Inter 1st Year Physics Notes 13th Lesson Thermodynamics

→ Thermodynamics: It is a branch of physics in which we shall study the process where work is converted into heat and vice versa.

→ Thermodynamic variables: In thermodynamics the state of a gas is specified by macroscopic variables such as pressure, temperature, volume, mass and composition that are felt by our sense perceptions and are measurable.

→ Thermal equilibrium: In general at thermal equilibrium the temperatures of the two bodies or systems are equal.
In a thermally isolated system it is said to be in “thermal equilibrium” if the thermodynamic variables such as pressure, volume, temperature, mass and composition do not change with time and they have fixed values.

→ Zeroth law of thermodynamics: It states that if two systems say A & B are in thermal equilibrium with a third system ‘C’ separately then the two systems A and B are also in thermal equilibrium with each other.

→ Internal energy: It includes only the energy associated with random motion of molecules of the system
i. e., internal energy is simply the sum of kinetic and potential energies of these molecules. Internal energy is denoted by ‘U’.

→ First law of thermodynamics: The heat energy (dQ) supplied to a system is partly used to increase its internal energy (dU) and the rest is used to do work (dW)
i. e., dQ = dU + dW. (OR)
Heat energy supplied to a system (dQ) always equals to the sum of change in internal energy (dU) and workdone (dW).
This law is a consequence of ”law of conservation of energy.”

→ Isothermal expansion: If a system is taken through a thermodynamic process in which ΔU = 0 then it is called Isothermal process.
In isothermal process change in internal energy ΔU = 0 i.e., temperature of the system is constant. Isothermal process obeys gas equation PV = RT.

→ Adiabatic process: In an adiabatic process system is insulated from the surroundings. So energy absorbed or released is zero (ΔQ = 0). In adiabatic process temperature of the system may change. It follows the equation PV^γ = constant. Where γ = \(\frac{C_P}{C_V}\) ratio of specific heats of a gas.

→ Isobaric process: In isobaric process pressure P’ is kept constant, volume and temperature changes are permitted. Work done in isobaric process
W = P(V₂ – V₁) = µR(T₂ – T₁).

→ Isochoric process: In isochoric process volume (V) of the system is kept constant. Work done by isochoric process is zero. In this process heat energy absorbed is totally used to increase the internal energy of the system.

→ Cyclic process: In a cyclic process the system returns to initial state (P, V and T). Change in internal energy ΔU = 0. Heat absorbed during cyclic process is equal to work done.

→ Reversible process : A thermodynamic process is said to be reversible if the process can be turned back such that both the system and surroundings return to their original state, with no other change any where else in universe.

→ Irreversible process : If a thermodynamic process cannot be reversed exactly in opposite direction of direct process then it is called irreversible process.
All spontaneous process of nature are irreversible.

→ Quasi static process: In a quasi static process at every stage the difference on pressure and temperature of systems and surroundings is infinitesimally small.
i.e., P + ΔV ≈ P and T + ΔT = T .
In this process the thermodynamic variables (P,V,T) will change very slowly so that it remains in thermal and mechanical equilibrium with surroundings throughout that process.
Note: Quasi static process is an imaginary concept only.

→ Heat engines: A heat engine is a device by which a system is made to undergo a cyclic process. As a result heat is converted into work.
Work done by heat engine W= Q₁ – Q₂;

Work done by heat engine W= Q1 — Q2; efficiency η = 1 – \(\frac{\mathrm{Q}_2}{\mathrm{Q}_1}\) (or) η = 1 – \(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\)

Important parts of heat engine : every heat engine mainly consists of

• hot source,
• working substance,
• cold reservoir.

→ Refrigerators or heat pumps : A Refrigerator is a heat pump which is a reverse of heat engine. Here working substance extracts heat Q₂ from cold body at temperature T₂ and delivers it to hot reservoir at temperature T₁. Coefficient of performance of refrigerator
α = \(\frac{\mathrm{Q}_2}{\mathrm{~W}}=\frac{\mathrm{Q}_2}{\mathrm{Q}_1-\mathrm{Q}_2}\)

→ Second law of thermodynamics :
(a) Kelvin – Planck statement: No process is possible whose sole resultant is the absor-ption of heat from a reservoir and the complete conversion of heat into work.
(b) Clausius statement: No process is possible whose sole resultant is the transfer of heat from a colder object to a hotter object.

Second law of thermodynamics gives a fundamental limitation to the efficiency of heat engine i.e. heat released to a colder body will never become zero. So 100% efficiency of heat engine cannot be achieved.

→ Carnot engine : Carnot engine operates between a hot reservoir of temperature T₁ and a coldreservoir of temperature T₂ through a cyclic process.
This cyclic process consists of

• Iso-thermal expansion,
• Adiabatic expansion,
• Isothermal compression and
• Adiabatic compression.

In this cyclic process it absorbs heat energy from source and releases heat energy Q2 to cold reservoir efficiency of

Carnot engine η = 1 – \(\frac{\mathrm{T}_2}{\mathrm{~T}_1}\)

→ Carnot theorem: (a) Any heat engine working between two given temperatures T₁ and T₂ cannot have efficiency more than that of carnot engine, (b) The efficiency of a carnot engine is independent of nature of working substance.

→ Isotherm: The pressure (p) and volume (v) curve for a given temperature is called isotherm.

→ adiabatic wall: An insulating wall that does not allow heat energy to flow from one side to another side is called “adiabatic wall.”

→ diathermic wall : It is a conducting wall which transfers heat energy from one side to another side.

→ Heat mechanical equivalent (J): In M.K.S system heat and work are measured with same unit ‘joule’. But in C.G.S system heat is measured in calorie and work in erg (1 joule = 10⁷ erg).
So in C.G.S system a conversion factor heat mechanical equivalent (J) is used to convert work into heat or vice versa.
1 Calorie = 4.2 Joules ⇒ J = 4.2 Joule/ cal. or J = 4200 joule/kilocal.

→ Calorie: The amount of heat energy required to rise the temperature of 1 gram of water through 1°C or 1 K is defined as “calorie.” Note: Magnitude of calorie slightly changes with the initial temperature of water.

→ Mean 15 °C calorie : The amount of heat energy required to rise the temperature of 1 gram of water from 14.5 °C to 15.5 °C is called “mean 15 °C calorie.”

→ Joule’s Law, work W ∝ Q ⇒ W = JQ where J = mechanical of heat equivalent
J = \(\frac{\text { Work }}{\text { Heat }}\) = 4.18 J/Cal.

→ From 1st Law of thermodynamics, dQ = dU + dW

→ Heat capacity of a body = \(\frac{\Delta \mathrm{Q}}{\Delta \mathrm{t}}\) = me (i.e., mass × specific heat)

→ Specific heat S or C = \(\frac{\Delta Q}{m \Delta t}\) = \(\frac{\text { Heat energy supplied }}{\text { mass } \times \text { temperature difference }}\)

→ From method of mixtures, Heat lost by hot body = Heat gained by cold body

→ When two spheres of radii r₁ : r₂ and ratio of specific heats S₁: S₂ and densities p₁: p₂ then their thermal capacities ratio
= \(\frac{m_1 S_1}{m_2 S_2}=\left(\frac{r_1}{r_2}\right)^3\left(\frac{\rho_1}{\rho_2}\right)\left(\frac{S_1}{S_2}\right)\)

(a) Specific heat of a gas, C_p = ΔQ / mΔT
(b) Molar specific heat, C_p = \(\frac{\Delta \mathrm{Q}}{\mathrm{n} \Delta \mathrm{t}}\) (n = Number of moles)
(c) Ratio of specific heats, y = C^p/ C^v;
C^v = \(\frac{\mathrm{R}}{\gamma-1}\), C^p = \(\frac{\gamma \mathrm{R}}{\gamma-1}\)
C^v = \(\frac{C_V}{M}=\frac{1}{M} \frac{R}{(\gamma-1)}\)
= \(\frac{\mathrm{PV}}{\mathrm{M}(\gamma-1) \mathrm{T}} \frac{\mathrm{P}}{\rho \mathrm{T}(\gamma-1)}\)J/kg-K
or C^v = \(\frac{P}{\mathrm{~J} \rho \mathrm{T}(\gamma-1)}\)k.cal/kg.K

→ Work done in expanding a gas against constant pressure (P) is W = P dV.

→ Work done during ideal expansion
W = P (V₂ – V₁) or W = nR (T₂ – T₁)
n = number of moles of gas;
R = universal gas constant.

→ Relation between C^p and C^v ⇒ C^P – C^v = R

→ Isothermal relation between P, V & T is PV = RT or PV = nRT.

→ Adiabatic relation between P, V & T

• PV^γ = constant
• TV^γ-1 = constant
• PV^1-γT^γ = constant.

→ Work done in Isothermal process
(a) W = RT log_e \(\frac{V_2}{V_1}\)
(b) W = 2.303 RT log₁₀ \(\frac{V_2}{V_1}\)

→ Work done in adiabatic process,
(a) W = \(\frac{1}{\gamma-1}\) (P₁V₁ – P₂V₂) per mole (OR)
(b) W = \(\frac{\mathrm{nR}}{\gamma-1}\) (T₁ – T₂);
n = number of moles.

→ Efficiency of heat engine, η = 1 – \(\frac{\mathrm{Q}_2}{\mathrm{Q}_1}\) or η = 1 – \(\frac{\mathrm{T}_2}{\mathrm{T}_1}\)
T₁ = Temperature of source,
T₂ = Temperature of sink.

→ Heat energy supplied to heat a body within the same state is Q = mct.

→ Heat energy supplied during change of state is Q = mL.

Here students can locate TS Inter 1st Year Physics Notes 14th Lesson Kinetic Theory to prepare for their exam.

TS Inter 1st Year Physics Notes 14th Lesson Kinetic Theory

→ Avogadro’s law: Equal volumes of all gases at equal temperature and pressure have the same number of molecules.

→ Avogadro number (N_A) : At S. T.P 22.4 liters of any gas contains 6.02 × 10²³ atoms. This is known as “Avogadro’s number (N_A)”.

→ Mean free path : The average distance that a gas molecule can travel without colliding is called “mean free path”.

→ Dalton’s law of partial pressures : For a mixture of non interacting ideal gases at same temperature and volume total pressure in the vessel is the sum of partial pressures of individual gases.
i. e., P = P₁ + P₂ +…. where P is total pressure
P₁, P₂, …… etc. are individual pressures of each gas.

→ Assumptions of kinetic theory :

• Gas is a collection of large number of mole-cules.
• Gas molecules are always in random motion.
• The interaction between gas molecules is negligible.
• They will always move in straight lines.
• Molecules collide with each other and also with walls of the container.
• These collisions are considered as totally elastic collisions.
• During collisions kinetic energy and momentum are totally conserved.

Note: From kinetic theory pressure of ideal gas P = \(\frac{1}{3}\) nmV̄²
Where V̄² denotes the mean of the squared speed.

→ Average kinetic energy of gas molecules: Internal energy E’ of an ideal gas is purely kinetic.
∴ E = N(\(\frac{1}{2}\) nmV̄²) = \(\frac{3}{2}\) K_BNT
or Average kinetic energy of gas molecule
\(\frac{E}{N}=\frac{3}{2}\) K_BT

Note: Average kinetic energy of gas mole cule is proportional to absolute temperature
\(\frac{E}{N}\) ∝ T

→ Law of equipartition of energy : The total energy of a gas is equally distributed in all possible energy modes, with each mode having an average energy equal to \(\frac{1}{2}\) K_BT.
This is known as “law of equipartition of energy.”

Explanation: A gas molecule is free to move in space in all t he three directions (x, y & z). At a given temperature T the average kinetic energy
< Er > = \(\frac{1}{2}\)mV²_x + \(\frac{1}{2}\)mV²_y + \(\frac{1}{2}\)mV²_z = \(\frac{3}{2}\)K_BT
But we assume that molecule is free to move equally in all possible directions
∴ \(\frac{1}{2}\)mV²_x = \(\frac{1}{2}\)mV²_y = \(\frac{1}{2}\)mV²_z = \(\frac{1}{2}\)K_BT
∴ Average kinetic energy for each translational degree of freedom is \(\frac{1}{2}\)K_BT .

→ Specific heat predictions (From law of equipartition of energy)

→ Solids : In a solid the atoms are free to vibrate in all three dimensions. Energy for each degree of freedom of vibration is K_BT.
∴ U = 3K_BT × N_A = 3RT
∴ Specific heat C = \(\frac{\mathrm{dU}}{\mathrm{dT}}\) = 3R

→ Specific heat of water: Water (H20) contains three atoms. So specific heat of water
U = 3 × 3RT = 9RT
∴ Specific heat of water ^ ^
= 9 × 8.31 = 75 Jmol^-1 K^-1

Note: Specific heat predictions from law of equipartition of energy are not applicable at low temperature because at nearly ‘O’ kelvin the degrees of freedom gets frozen.

→ Ratio of specific heats of gas (γ ): In gases the ratio of molar specific heats of a gas \(\frac{C_P}{C_V}\) is called “ratio of specific heats” (γ).
∴ γ = \(\frac{C_P}{C_V}\)

• For monoatomic gas γ = \(\frac{5}{3}\) = 1.66̄ =1.67
• For diatomic gas γ = \(\frac{7}{5}\) = 1.4
• For tri or polyatomic gas γ =1.33.

→ The ideal gas equation connecting pressure (P), volume (V) and absolute temperature (T) is
PV = µRT = K_BNT
Where µ is the number of moles and N is the number of molecules. R and K_B are universal constants.
R = 8.314 J mol^-1 K^-1; K_B = \(\frac{\mathrm{R}}{\mathrm{N}_{\mathrm{A}}}\)
= 1.38 × 1o^-23 JK^-1

→ Kinetic theory of an ideal gas gives the relation, P = \(\frac{1}{3}\) nmv² where n is number density of molecules, m is the mass of the molecule and v² is the mean of squared speed.

→ Kinetic interpretation of temperature is,
\(\frac{1}{3}\) nmv² = \(\frac{3}{2}\) k_BT
v_rms = (v²)^{\(\frac{1}{2}\)} = \(\sqrt{\frac{3 K_B T}{m}}\)

→ Translational kinetic energy, E = \(\frac{3}{2}\) k_BNT

→ Mean free path, l = \(\frac{1}{\sqrt{2} \mathrm{n} \pi \mathrm{d}^2}\)
Where n is the number density and d is the diameter of the molecule.

→ Root mean square (rms) speed of a gas at temperature ‘T’ is, c_rms = \(\sqrt{\frac{3 \mathrm{RT}}{\mathrm{M}}}\)
Where ‘M’ is the molecular weight of molar mass of the gas.

→ If n’ molecules of a gas have speeds c₁, c₂, c₃ …… c_n respectively then rms speed is given by,
c_rms = \(\sqrt{\frac{c_1^2+c_2^2+c_3^2+\ldots \ldots \ldots \ldots+c_n^2}{n}}\)

→ If a gas has ‘f degrees of freedom then,
γ = \(\frac{c_p}{c_v}\) = 1 + \(\frac{2}{f}\)

→ The relation between rms velocity and absolute temperature of a gas is c ∝ √T .

Here students can locate TS Inter 1st Year Physics Notes 5th Lesson Laws of Motion to prepare for their exam.

TS Inter 1st Year Physics Notes 5th Lesson Laws of Motion

→ Force: Force is that which changes or tries to change the state of a body. Force is a vector.
D.F = MLT^-2, Unit: Newton (N)

→ Newton’s Laws of Motion :
1st Law : Every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force.

→ Inertia : It is the property of the body to oppose any change in its state.
Simply inertia means resistance to change. Mass of a body m’ is a measure for the inertia of a body.

→ 2nd Law: The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts.
Note : Internal forces cannot change the momentum of the body or system,
F ∝ \(\frac{\mathrm{d} \overline{\mathrm{p}}}{\mathrm{dt}}, \quad \frac{\mathrm{d} \overline{\mathrm{p}}}{\mathrm{dt}}=\mathrm{m} \frac{\mathrm{dv}}{\mathrm{dt}}\) or F = k.ma

→ Momentum (p) : It is the product of mass (m) and velocity (v) of a body.
Momentum (p) = mass x velocity = m v It is a vector, unit: Kg – m/sec. D.F = MLT^-1

→ Impulse : When force acts between two bodies in contact for a very small time then product of force and time is defined as Impulse.

Impulse = Force × time = F.t,
Impulse = change in momentum
Impulse is a vector. Unit: Kg-m/sec,
D.F. = MLT^-1

→ Some observations of momentum:
1) If equal force is applied on two bodies of different masses the body with less mass will gain more velocity and body with more mass will gain less velocity. But change in momentum is same for both bodies.

2) To stop a fast moving cricket ball abruptly we require a large force. Whereas if we move our hands along the direction of motion of the ball we require less force to stop it.
I = Ft; F ∝ \(\frac{1}{t}\)
Ex : When a horse pulls a cart, horse applies force on the cart. Whereas cart applies the reaction on the ground so motion is possible.

3) In some cases action and reaction app-lies on the same system then the body is in equilibrium. In this case motion is not possible.
Ex : When you sit on a bench or chair force (F = ma) equal to your weight is applied on the bench or chair called action. At the same time the chair or bench will apply equal amount of force on you as reaction. In this case the person and bench or chair are in equilibrium and motion is not possible.

→ Law of conservation of momentum: Under the absence of external force, “The total momentum of an isolated system of interacting particles is conserved” i.e., total momentum of system is constant.

→ Friction : It is a contact force parallel to the surfaces in contact. Friction will always oppose relative motion between the bodies.

→ Normal reaction (N): When two bodies are one over the other, force applied by the lower body on the bottom layers of upper body is called normal reaction.
On a horizontal surface normal reaction N = mg weight of upper body.

On an inclined surface normal reaction N = mg cos θ

→ Motion of a car on a horizontal road : On a horizontal road when a car is in circular motion three forces will act on it. They are

• weight of car (mg)
• normal reaction (N)
• Frictional force (f)

In this type of motion friction between road and tyres gives necessary centripetal force.
For safe journey centripetal force must be equal to Frictional force i.e., \(\frac{\mathrm{mv}^2}{\mathrm{R}}\) = µmg
Safe velocity of car v = \(\sqrt{\mu \mathrm{gR}}\)

→ Static friction: Friction between two bodies at rest is called static friction.
Static friction does not exist by itself. It will come into account when a force tries to develop motion between the bodies.

→ Laws of static friction :

• Static Friction does not exist indepen-dently i.e. when external force is zero static friction is zero.
• The magnitude of static friction gradually increases with applied force to a maximum value called limiting static friction (f_s)_max
• Static friction opposes impending motion.
• Static friction is independent of area of contact.
• Static friction is proportional to normal reaction.
  (f_s)_maxµN (or) (f_s)_max = µ_sN

→ Kinetic friction (f_k) : Frictional force that opposes relative motion between moving bodies is called kinetic friction.

→ Laws of kinetic friction: When a body begins to slide on the other surface static friction abruptly decreases and reaches to a constant value called kinetic friction.

• Kinetic friction is independent of area of contact.
• Kinetic friction is independent of velocities of moving bodies.
• Kinetic friction is proportional to normal v reaction N.
  f_k µN (or) f_k = µ_kN

→ Rolling friction (f_r): When a body is rolling on a plane without slip then contact forces between the bodies is called rolling friction.
It opposes rolling motion between the surfaces.

→ Laws of rolling friction :

• Rolling friction will develop a point contact between the surface and the rolling sphere. For objects like wheels line of contact will develop.
• Rolling friction (f_r) has least value for given normal reaction when compared with static friction (f_s) or kinetic friction
• Rolling friction is directly proportional to normal reaction, f_r = µN.
• In rolling friction the surfaces in contact will get momentarily deformed a little.
• Rolling friction depends on area of contact. Due to this reason friction increases when air pressure is less in tyres (Flattened tyres).

→ Advantages of friction :

• We are able to walk because of friction.
• It is impossible for a car to move on a slippery road.
• Breaking system of vehicles works with the help of friction.
• Friction between roads and tyres provides the necessary external force to accelerate the car. Transmission of power to various parts of a machine through belts is possible by friction.

→ Disadvantages of friction :

• In many cases we will try to reduce friction because it dissipates energy into heat.
• It causes wear and tear to machine parts.

→ Methods to reduce friction :

• Lubricants are used to reduce friction.
• Ball bearings are used between moving parts of machine to reduce friction.
• A thin cushion of air maintained between solid surfaces reduces friction.
  Ex: Air pressure in tyres.

→ Ball bearings : Ball bearings will convert sliding motion into rolling motion due to their special construction. So sliding friction is converted into rolling friction. Hence friction decreases.

→ Banking of roads : In a curved path the outer edge of road is elevated with some angle ‘θ’ to the horizontal. Due to this arrangement centripetal force necessary for circular motion is provided by gravitational force on vehicle.
Angle of banking θ = tan^-1\(\left(\frac{\mathrm{v}^2}{\mathrm{rg}}\right)\)
Safe velocity on a banked road
V_max = \(\sqrt{g R \tan \theta}\)

→ Motion of a car on a banked road: When a road is banked driving will become safe and safe velocity of vehicles will also increase. Safe velocity of vehicle o,n a banked road
v = \(\sqrt{\mathrm{gR} \tan \theta}\)
Due to baking wear and tear of tyres will decrease. Driving is also easy.

→ Momentum, P = mass × velocity

→ From Newton’s second law,
F ∝ \(\frac{\mathrm{dP}}{\mathrm{dt}}=\mathrm{m} \frac{\mathrm{dv}}{\mathrm{dt}}\) ⇒ F = ma = m\(\frac{(\mathrm{v}-\mathrm{u})}{\mathrm{t}}\)

→ When a body of mass m’ is taken in a lift move with acceleration a’
Moving in upwards apparent weight, W₁ = m (g + a) ⇒ W₁ = W\(\)

→ Motion of lawn roller:
(i) When pulling the lawn roller of mass m with a force F
(a) Horizontal component useful for motion, F_x = F cos θ
(b) Normal reaction, N = mg – F sin θ.

(ii) When lawn roller is pushed with a force F
(c) Horizontal component of force,
F_x = F cos θ
(d) Normal reaction, N = mg + F sin θ

Here students can locate TS Inter 1st Year Physics Notes 4th Lesson Motion in a Plane to prepare for their exam.

TS Inter 1st Year Physics Notes 4th Lesson Motion in a Plane

→ Vector: A physical quantity which has both magnitude and direction is called as vector. Ex: Displacement, Velocity, Force, etc.

→ Scalar: A physical quantity which has only magnitude is called as scalar.
Ex: Distance, Speed, Work, etc.

→ Equality of vectors: If two vectors are equal both in magnitude and direction are called ns-equal vectors.

→ Resultant vector : If the effect of many vectors is represented by a single vector then that single vector is called resultant vector.

→ Triangle law: If the magnitude and direction of two vectors are represented by two sides of a triangle taken in order then the third side of the triangle taken in reverse order will give the resultant both in magnitude and direction.

From the above figure, R = P + Q represents vectorial addition of P and Q.

→ Parallelogram law : If two vectors are represented by the two adjacent sides of a parallelogram then the diagonal passing through the intersection of those two vectors will represent the resultant both in direction and magnitude.

In the above figure, diagonal OB is sum of vectors i.e. R = A + B, diagonal AC is subtraction of vectors i.e. R = A – B

→ Laws of vector addition :

• Vector addition is commutative i.e.,
  A̅ + B̅ = B̅ + A̅
• Vector addition obeys associative law i.e.,
  (A̅ + B̅) + C̅ = A̅ + (B̅ + C̅)

→ Unit Vector: If the magnitude of any vector is unity then it is called unit vector.
Ex: unit vector A̅ = \(\frac{\overline{\mathrm{A}}}{|\overline{\mathrm{A}}|}\) = 1
Note: Unit vectors along X and Y directions are represented by i and j. In space unit vectors along X, Y and Z directions are represented by i, j and k.

→ Null vector: If the magnitude of a vector is zero then it is called null vector.
Null vector has only direction.
Ex: A̅ – A̅ = 0 If has only direction, magni¬tude is zero.
A̅ × 0̅ = 0̅ It has only direction, magnitude is zero.

→ Position vector: Any vector in a plane can be represented as A̅ = A_x i̅ + A_y j̅
Any vector in space can be represented as A̅ = A_xi̅ + A_yj̅ + A_zk̅
Where Ax, Ay and Az are magnitudes along X, Y and Z directions.

→ Resolution of vectors : Every vector can be resolved into two mutually perpendicular components. This division is with funda-mental principles of trigonometry.

A̅ = A̅_x + A̅_y
A̅ = A̅_xî + A̅_y ĵ
Ex Let A̅ makes an angle ‘O’ with X – axis then
X – component of \(\overline{\mathrm{A}}_{\mathrm{x}}=\overline{\mathrm{OB}}\) = A̅ cosO
Y-component of \(\overline{\mathrm{A}}_{\mathrm{y}}=\overline{\mathrm{OC}}\)=XsinO
Note: If values of A, and A are given then
Resultant A̅ = \(\sqrt{\mathrm{A}_{\mathrm{X}}^2+\mathrm{A}_{\mathrm{Y}}^2}\)
Angle made by vector A̅ with X-axis
θ = tan^-1\(\left[\frac{A_Y}{A_X}\right]\)

→ Projectile : When a body is thrown into the space with some angle 0 (θ ≠ 90) to the horizontal it moves under the influence of gravity then it is known as projectile.
Note: The path of a projectile can be represented by the equation
y = ax – bx². It represents a parabola.
Time taken to reach maximum height
t = \(\frac{v_0 \sin \theta}{g}\)
Maximum height reached hmax = \(\frac{v_0^2 \sin ^2 \theta}{2 g}\)

→ Time of flight (T) : The time interval from the instant of projection to the instant where it crosses the same plane or it touches the ground is defined as time of flight.
Time of flight T = \(\frac{v_0 \sin \theta}{g}\)

Note: For horizontally projected projectiles
T = \(\frac{v_0 \sin \theta}{g}\)

→ Range (or) horizontal range (R) : It is the horizontal distance from the point of projection to the point where it touches the ground.
Range R = \(\frac{2 \mathrm{v}_0^2 \operatorname{Sin} 2 \theta}{\mathrm{g}}\)
For horizontal projection R = v₀\(\sqrt{2 h / g}\)

→ Uniform circular motion : If a body moves with a constant speed on the periphery of a circle then it is called uniform circular motion.

→ Time period: In circular motion time taken to complete one rotation is defined as time period (T).
Time period (T) = 2π/ω
Note: Frequency υ = \(\frac{1}{T}\) is equal to number of rotations completed in one second. Relation between ω and υ is ω = 2πυ or
v = 2πυ R.

→ Relative velocity in two-dimensional motion: Let two bodies A and B are moving with velocities V̅_A and V̅_B then relative velocity of A w.r.t B is V̅_AB = V̅_A – V̅_B
Relative velocity of B w.r.t. A is
V̅_BA = V̅_B – V̅_A

→ For (like) parallel vectors say P̅ and Q̅ resultant R̅ = P̅ + Q̅

→ For antiparallel vectors say P̅ and Q̅ resultant R̅ = P̅ – Q̅

→ Rectangular components of a vector R̅ are R_x = R cos θ and R_y = R sin θ

→ Resultant of vectors is given by parallelogram law.
(a) Resultant, (R) = \(\sqrt{\mathrm{P}^2+\mathrm{Q}^2+2 \mathrm{PQ} \cos \theta}\)
(b) Angle made by Resultant
α = tan^-1\(\sqrt{\mathrm{P}^2+\mathrm{Q}^2+2 \mathrm{PQ} \cos \theta}\)

(c) Difference of vectors
= \(\sqrt{\mathrm{P}^2+\mathrm{Q}^2-2 \mathrm{PQ} \cos \theta}\)
where θ is the angle between P̅ and Q̅.

→ If two vectors a̅ and b̅ are an ordered pair then from triangle law, resultant R̅ = a̅ + b̅
(a) When two bodies A, B are travelling in the same direction ⇒ relative velocity, V_R = V_A – V_B.
(b) Two bodies travelling in opposite direction ⇒ relative velocity, V_R = V_A + V_B

→ Crossing of a river in shortest path :
(a) To cross the river in shortest path, it must be rowed with an angle, θ = sin^-1(V_WE/ V_BW) perpendicular to flow of water.
(b) Velocity of boat with respect to earth,
V_BE = \(\sqrt{\mathrm{V}_{\mathrm{BW}}^2-\mathrm{V}_{\mathrm{WE}}^2}\)
(c) Time taken to cross,
t = \(\frac{\text { width of river }(l)}{\text { velocity of boat w.r.t earth }}=\frac{l}{\mathrm{~V}_{\mathrm{BE}}}\)

→ Crossing the river in shortest time :
(a) Time taken to cross the river,
t = \(\frac{\text { width of river } l \text { l }}{\text { velocity of boat w.r.t water } \mathrm{V}_{\mathrm{BW}}}\)
(b) Resultant velocity of boat,
VR = \(\sqrt{\mathrm{V}_{\mathrm{BW}}^2+\mathrm{V}_{\mathrm{WE}}^2}\)
(c) Angle of resultant motion with
θ = tan^-1(V_WE/ V_BE) down the stream

→ Dot product:
A̅ . B̅ = |A̅| . |B̅| cos θ

→ Let A̅ = x₁ i̅ + y₁ j̅ + z₁ k̅ and
B̅ = x₂ i̅ + y₂ j̅ + z₂ k̅ then
(a) A + B = (x₁ + x₂)i̅ + (y₁ + y₂)j̅ + (z₁ + z₂)k̅
(b) |A| = \(\sqrt{\mathrm{x}_1^2+\mathrm{y}_1^2+\mathrm{z}_1^2}\)
|B| = \(\sqrt{\mathbf{x}_2^2+\mathbf{y}_2^2+\mathbf{z}_2^2}\)
(c) A̅.B̅ = x₁x₂ + y₁y₂ + z₁z₂

→ In dot product i̅ i̅ = j̅ j̅ = k̅ k̅ = 1 i.e., dot product of heterogeneous vectors is unity.

→ In dot product i̅ . j̅ = j̅ . k̅ = k̅ . i̅ = 0

→ Projectiles thrown into the space with some angle ‘θ’ to the horizontal: Horizontal component (u_x) = u cos θ. Which does not change.

→ Vertical component, U_y = u sin θ (This component changes with time)

→ Time of flight, (T) = \(\frac{2 u \sin \theta}{g}\)
H = \(\frac{u^2 \sin ^2 \theta}{2 g}\)
Range (R) = \(\frac{\mathrm{u}^2 \sin 2 \theta}{\mathrm{g}}\)

→ Velocity of projectile, v = \(\sqrt{v_x^2+v_y^2}\) where v_x = u_x = u cos θ and v_y = u sin θ – gt

→ Angle of resultant velocity with horizontal, v.
α = tan^-1\(\left[\frac{v_{\mathrm{y}}}{\mathrm{v}_{\mathrm{x}}}\right]\) where v_y = u sin θ – gt and v_x = u cos θ

→ In projectiles range R is same for complementary angles (θ and 90 – θ).
For θ = 45°, Range is maximum.
R_max = \(\frac{\mathrm{u}^2}{\mathrm{~g}}\) corresponding to h_max = \(\frac{u^2}{4 \mathrm{~g}}\)

→ Relation between R_max and h_max is
R_max = 4h_max

→ For complimentary angles of projection,
h₁ + h₂ = \(\frac{u^2}{2 g}\);
Range, R = 4\(\sqrt{\mathrm{h}_1 \mathrm{~h}_2}\); R_max = 2 (h₁ + h₂)

→ Horizontally projected projectiles: Time of flight, t = \(\sqrt{\frac{2 h}{g}}\)

→ Range, R = u × t = \(\sqrt{\frac{2 h}{g}}\)

→ Velocity of projectile after a time t is, v = \(\sqrt{v_{\mathrm{x}}^2+\mathrm{v}_{\mathrm{y}}^2}\) where v_x = u_x = u and v_y = gt
∴ v = \(\sqrt{\mathrm{u}^2+\mathrm{g}^2 \mathrm{t}^2}\)

→ Angle of resultant with x – axis,
α = tan^-1\(\left[\frac{v_y}{v_x}\right]\) where v_x = u and v_y = gt
α = tan^-1\(\left[\frac{\mathrm{gt}}{\mathrm{u}}\right]\)

Here students can locate TS Inter 1st Year Physics Notes 3rd Lesson Motion in a Straight Line to prepare for their exam.

TS Inter 1st Year Physics Notes 3rd Lesson Motion in a Straight Line

→ Motion: A body is said to be in motion if the position changes with respect to time.

→ Kinematics : It is a branch of physics dealing with the laws of motion of a body without referring to reasons of motion.

→ Motion in a straight line (or) linear motion : If the motion of a body is restricted to a straight line then it is said to be in linear motion.

→ Displacement (X) :
Change in the position of a body in particular direction is called displacement. It is a vector.
Displacement consists of magnitude and direction.

→ Path length: The total distance travelled by a body during its journey is called “path length”.
It is a scalar.
Note : The displacement of a body may or may not equal to path length. In some cases even though displacement is zero its path length is not equal to zero.

→ Displacement – time curves (x -1 graphs or x -1 curves) : When a graph is plotted with time ’t’on x- axis and displacement ‘x’ on y – axis then it is known as displacement – time graph.

→ Various types of displacement time graphs:
(a) For a body at rest ⇒ x – t graph is a straight line parallel to x – axis.
(b) For a body starting from rest and moving with uniform velocity ⇒ x -1 graph is a straight line passing through origin.
Slope of the line \(\frac{d y}{d x}\) will give uniform velocity of the body.
(c) For a body moving with uniform acceleration x – t graph is an upshooting curve. Slope of the curve \(\frac{d y}{d x}\) at any point gives velocity of the body at that point.

→ Velocity (v): The rate of change in displacement is known as velocity. Unit m/s. It is a vector.

→ Average Velocity: The ratio of total displacement of a body to the total time taken is called as average velocity. Unit: m/s, it is a vector.
Average velocity v̅ = \(\frac{x_2-x_1}{t_2-t_1}=\frac{\Delta x}{\Delta t}\)
Where x₁ and x₂ are initial and final positions of the body.

→ Average Speed: The ratio of total path length travelled to the total time taken is known as average speed.
Speed and average speed are scalar quantities so no direction for these quantities.
Average speed = \(\frac{\text { Total path length }}{\text { Total time interval }}\)

→ Instantaneous Velocity (v): The velocity of a body at a given instant of time is known as instantaneous velocity.
Mathematically v = \({Lt}_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{d x}{d t}\)
When time interval is extremely small then \(\frac{d x}{d t}\) is velocity at that instant.
Note : Instantaneous velocity may be or may not be equal to average velocity of the body.

→ Velocity – time graphs (v – t graphs): When a graph is plotted with time ‘t’ on x- axis and velocity ‘v’ on y – axis then that graph is known as velocity – time graph.

→ Different types of velocity – time graphs
(a) For uniform velocity ⇒ v – t graph is a straight line parallel to x – axis.
(b) Starting from rest and moving with uniform acceleration ⇒ v-1 graph is a straight line passing through origin. It has positive slope with t – axis.
(c) For a body moves with some initial velocity ‘u’ and with some uniform acceleration ⇒ v – t graph is a straight line with some is a positive quantity).
(d) For a body moving with uniform retardation ⇒ graph is a straight line with some negative slope (i.e., \(\frac{d y}{d x}\) is a negative quantity).

→ Uses of velocity – time graphs :

• Slope (dy/dx) of velocity – time graph gives acceleration of the body.
• Area under velocity – time graph gives total displacement of the body.
• Basic equations of motion v = u0 + at,
  x = u₀t + \(\frac{1}{2}\)at² and v² – u² = 2 ax can be de-rived from v – t graphs.

→ Acceleration : Rate of change in velocity is defined as “acceleration”. It is a vector. Unit: m/s²

→ Instantaneous acceleration : It is defined as the acceleration of the body at any given instant of time.
When time interval Δt is extremely small then \(\frac{d v}{d x}\) is called instantaneous acceleration,

→ Freely falling body: When a body drops from certain height it is called freely felling body.
For a freely falling body u = 0, accelera-tion a = g

→ Relative velocity : Let two bodies A and B are in motion. The velocity of B with respect to A or velocity of A with respect to (w. r. t) B’ is called relative velocity.
Ex: Let velocity of A is v_A and that of B is v_B. When both are moving in
(a) same direction
Velocity of A w.r.t B is v_AB = v_A – v_B
Velocity of B w.r.t A is v_BA = v_B – v_A

(b) opposite direction
Velocity of A w.r.t B is v_AB = v_A + v_B
Velocity of B w.r.t A is v_BA = v_B + v_A

→ For a body moving with uniform velocity, x = vt

→ Average Speed, V = \(\frac{\text { total distance }}{\text { total time }}\)

→ Instantaneous velocity, V = \({Lt}_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{\mathrm{dx}}{\mathrm{dt}}\)

→ Slope of displacement (x), time (t) graph gives velocity.

→ Average acceleration,
a = \(\frac{\text { change in velocity }}{\text { change in time }}=\frac{v_2-v_1}{t_2-t_1}\)

→ Instantaneous acceleration,
a = \(\mathrm{Lt}_{\Delta t \rightarrow 0} \frac{\Delta \mathrm{v}}{\Delta \mathrm{t}}=\frac{\mathrm{dv}}{\mathrm{dt}}\)

→ Slope of velocity (v), time (t) graph gives acceleration.

→ Area under velocity (v), time (t) graph gives total displacement.

→ For a body moving with uniform velocity (v) its v – t graph is a straight line parallel to x – axis.

→ For a body moving with some acceleration (a) its v – t graph is a straight line inclined to x-axis.

→ Equations of motion

• v = u + at
• s = ut + \(\frac{1}{2}\) at²
• v² – u² = 2as

→ Distance travelled by a body during nth second, S_n = u + a (n – \(\frac{1}{2}\))

→ Equations of a freely falling body are

• v = gt
• s = h = \(\frac{1}{2}\)gt²
• v² = 2gh

→ For a body thrown up vertically use

• v = u – gt
• s = h = ut – \(\frac{1}{2}\)gt²
• v² – u² = -2gh
• time of ascent, t = \(\frac{u}{g}\), time of descent t = \(\frac{u}{g}\)
• time spent in air, T = 2t = \(\frac{2u}{g}\)
• Maximum height reached h_max = \(\frac{\mathrm{u}^2}{2 \mathrm{~g}}\)

Here students can locate TS Inter 1st Year Physics Notes 2nd Lesson Units and Measurements to prepare for their exam.

TS Inter 1st Year Physics Notes 2nd Lesson Units and Measurements

→ Fundamental Quantity : A fundamental quantity is one which is unique and freely existing. It does not depend on any other physical quantity. Ex: Length (L), Time (T), Mass (M) etc.

→ Fundamental quantities in SI System : In SI system length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity are taken as fundamental quantities.

→ Derived quantity: A derived quantity is pro-duced by the combination of fundamental quantities (i.e., by division or by multiplica-tion of fundamental quantities).
Ex: Velocity = \(\frac{\text { displacement }}{\text { time }}=\frac{\mathrm{L}}{\mathrm{T}}\) or LT^-1
Acceleration = \(\frac{\text { change in velocity }}{\text { time }}\)
= \(\frac{\mathrm{LT}^{-1}}{\mathrm{~T}}\) = LT²

→ Unit: The standard which is used to measure the physical quantity is called the Unit’.

→ Fundamental unit: The units of the funda-mental quantities are called the “fundamental units”.
Ex : Length → Meter (m), Mass → Kilogram (kg), Time Second (sec) etc.

→ Basic units or fundamental units of SI system : The basic units in S.I. system are Length → metre (L), Mass → kilogram (kg), Time second (s); electric current → ampere (amp), Thermodynamic temperature → Kelvin (K); Amount of substance → mole (mol); Luminous intensity → candela (cd); Auxilliary units : Plane angle → Radian (rad); Solid angle → steradian (sr)

→ Derived units: The units of derived quantities are known as “derived units”.
Ex: Area → square meter (m²),
Velocity → meter/sec (m/s) etc.

→ International system of units (S.I. units) :
S.I. system consists of seven fundamental quantities and two supplementary quantities. To measure these quantities S.I. system consists of seven fundamental or basic units and two auxiliary units.

→ Accuracy: Accuracy indicates the closeness of a measured value to the true value of the quantity. If we are very close to the true value then our accuracy is high.

→ Precision : Precision depends on the least measurable value of the instrument. If the least measurable value is too less, then precision of that instrument is high.
Ex : Least measured value of vernier callipers is 0.1 mm
Least count of screw gauge is 0.01 mm.
Among these two, the precision of the screw gauge is high.

→ Error: The uncertainty of measurement of a physical quantity is called “error”.
→ Systematic errors : Systematic errors always tend to be in one direction i.e., positive or negative. For systematic errors, we know the reasons for the error. They can be reduced by proper correction or by proper care. Ex:

• Zero error in screw gauge and
• A faulty calibrated thermometer

Note :
Systematic errors are classified as

• Instrumental errors
• Imperfection of experimental technique
• Personal errors.

1) Instrumental errors: These errors arise due to the imperfect design or faulty calibration of instruments.
Ex : Zero error in screw gauge.

2) Imperfection of experimental technique: These errors are due to the procedure followed during experiment or measurements. Ex : 1) Measurement of body temperature at armpit 2) Simple pendulum oscillations with high amplitude.

3) Personal errors: These errors arise due to an individual’s approach or due to lack of proper setting of apparatus.
Ex : Parallax error is a personal error.

→ Methods To Reduce Systematic Errors :
Systematic errors can be minimized by improving experimental techniques, by selecting better instruments and by removing personal errors.

→ Random errors :
These errors will occur irregularly. They may be positive (or) negative in sign. We cannot predict the presence of these errors.
Ex:

• Voltage fluctuations of power supply
• Mechanical vibrations in experimental set up.

→ Least count error: This is a systematic error. It depends on the smallest value that can be measured by the instrument.
Least count error can be minimized by using instruments of highest precision.

→ Arithmetic mean: The average value of all the measurements is taken as arithmetic mean.
Let the number of observations be a₁, a₂, a₃ ……….. a_n

Then the arithmetic mean
a_mean = \(\frac{\mathbf{a}_1+\mathbf{a}_2+\mathbf{a}_3+\ldots \ldots \ldots .+\mathbf{a}_{\mathbf{n}}}{\mathbf{n}}\)
or a_mean = \(\sum_{i=1}^n \frac{a_i}{n}\)

→ Absolute error (|Δa|): The magnitude of the difference between the individual measurement and true value of the quantity is called absolute error of the measurement. It is denoted by |Δa|
Absolute error
|Δa| = |a_mean – a_i|
= |True value – measured value|

→ Mean absolute error ( Aa[nrnnl: The arithmetic mean value of all absolute errors is known as mean absolute error.
Let ‘n’ measurements are taken, then their absolute errors are, say |Δa₁|, |Δa₂|, |Δa₃| …….. ||Δa_n|, then
|Δa_mean| = \(\frac{\left|\Delta a_1\right|+\left|\Delta a_2\right|+\left|\Delta a_3\right|+\ldots \ldots \ldots+\left|\Delta a_n\right|}{n}\)
or
Δa_mean = \(\frac{1}{n} \sum_{i=1}^n \Delta a_i\)

→ Relative error: Relative error is the ratio of the mean absolute error A amean to the mean value a mean of the quantity measure.
Relative error = \(\frac{\Delta \mathbf{a}_{\text {mean }}}{\mathbf{a}_{\text {mean }}}\)

→ Percentage error (δa): When relative error is expressed in percent then it is called percentage error.
Percentage error (δa) = \(\frac{\Delta \mathbf{a}_{\text {mean }}}{\mathbf{a}_{\text {mean }}}\) × 100

→ Significant figures: The scientific way to report a result must always have all the reliably known (measured) values plus one uncertain digit (first digit). These are known as “significant figures”.
This additional digit indicates the uncertainty of measurement.
Ex: In a measurement, the length of a body is reported as 287.5 cm. Then, in that measu-rement, the length is believable up to 287 cm
i. e., the digits 2, 8 and 7 are certain. The first digit (5) is uncertain. Its value may change.

→ Rules in determining significant numbers

• All the non-zero digits are significant.
• All the zeros in between two non-zero digits are significant.
• If the number is less than one, the zeros on the right of decimal point to the first nonzero digit are not significant.
  Ex : In a result 0.002308 the zeros before the digit ‘2’ are non significant.
• The terminal or trailing zeros in a number without decimal point are not significant. Ex: In the result 123 m = 12300 cm = 123000 mm the zeros after the digit ‘3’ are not significant.
• The trailing zeros in a number with a decimal point are significant.
  Ex : In the result 3.500 or 0.06900 the last zeros are significant. So number of significant figures are four in each case.

→ Rules for arithmetic operation with sig-nificant figures
1. In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
Ex : In the division \(\frac{4.237}{2.51}\) the significant figures are 4 and 3, so least significant figures are ‘3’.
\(\frac{4.237}{2.51}\) = 1.69 i.e., final answer must have only ‘3’ significant digits.

2. In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.
Ex: 436.26g + 227.2 g Here least number of significant figures after decimal point is one.
436.26 + 272.2 = 708.46 must be expressed as 708.5 (after rounding off the last digit).

→ Rounding off the uncertain digits Rules for rounding off procedure : In rounding off the numbers to the required number of significant digits the following rules are followed.

• The preceding significant digit is raised by one if the first non-significant digit is more than 5.
• The preceding significant digit is left unchanged if the first non-significant digit is less than 5.
• If the first non-significant figure is 5 then
  (a) If the preceding significant figure is an odd number then add one to it.
  (b) If the preceding significant figure is an even number then it is unchanged and 5 is discarded.

→ Dimension: The power of a fundamental quantity in the given derived quantity is called
dimension.
Ex: Force dimensional formula MLT^-2 Here dimensions of Mass → 1, Length → 1, Time → 2

→ Dimensional formula: It is a mathematical expression giving relation between various fundamental quantities of a derived physical quantity.
Ex : Momentum (P),MLT^-1,
Energy ML²T^-2 etc.

→ Uses of dimensional methods :

• To convert units from one system to another system.
• To check the validity of given physical equations. For this purpose, we will use homogeneity of dimensions on L.H.S and on R.H.S.
• To derive new relations between various physical quantities.

→ Dimensional formulae of physical quantities:

Here you will find Telangana TSBIE State Board Syllabus TS Inter 1st Year Physics Study Material Pdf free download, TS Intermediate 1st Year Physics Textbook Solutions Questions and Answers in English Medium and Telugu Medium according to the latest exam curriculum. The chapter-wise TS Inter 1st Year Study Material will help the students in understanding the concept behind each question in a detailed way.

Students can check the TS Inter 1st Year Physics Syllabus & TS Inter 1st Year Physics Important Questions for strong academic preparation. Students can use TS Inter 1st Year Physics Notes as a quick revision before the exam.

TS Intermediate 1st Year Physics Study Material Pdf Download | TS Inter 1st Year Physics Textbook Solutions Telangana

TS Inter 1st Year Physics Study Material in Telugu Medium

• Chapter 1 భౌతిక ప్రపంచం
• Chapter 2 ప్రమాణాలు, కొలత
• Chapter 3 సరళరేఖాత్మక గమనం
• Chapter 4 సమతలంలో చలనం
• Chapter 5 గమన నియమాలు
• Chapter 6 పని, శక్తి, సామర్ధ్యం
• Chapter 7 కణాల వ్యవస్థలు, భ్రమణ గమనం
• Chapter 8 డోలనాలు
• Chapter 9 గురుత్వాకర్షణ
• Chapter 10 ఘనపదార్ధాల యాంత్రిక ధర్మాలు
• Chapter 11 ప్రవాహుల యాంత్రిక ధర్మాలు
• Chapter 12 పదార్ధ ఉష్ణ ధర్మాలు
• Chapter 13 ఉష్ణోగతిక శాస్త్రం
• Chapter 14 అణుచలన సిద్ధాంతం

TS Inter 1st Year Physics Study Material in English Medium

• Chapter 1 Physical World
• Chapter 2 Units and Measurements
• Chapter 3 Motion in a Straight Line
• Chapter 4 Motion in a Plane
• Chapter 5 Laws of Motion
• Chapter 6 Work, Energy and Power
• Chapter 7 Systems of Particles and Rotational Motion
• Chapter 8 Oscillations
• Chapter 9 Gravitation
• Chapter 10 Mechanical Properties of Solids
• Chapter 11 Mechanical Properties of Fluids
• Chapter 12 Thermal Properties of Matter
• Chapter 13 Thermodynamics
• Chapter 14 Kinetic Theory

TS Inter 1st Year Physics Weightage Blue Print 2022-2023

TS Inter 1st Year Physics Syllabus

Telangana TS Intermediate 1st Year Physics Syllabus

TELANGANA STATE BOARD OF INTERMEDIATE EDUCATION, HYDERABAD
Physics-I
Syllabus (w.e.f. 2012-13)

Chapter 1 PHYSICAL WORLD
1.1. What is Physics ? 1.2. Scope and excitement of physics 1.3. Physics, technology and society 1.4. Fundamental forces in nature 1.5. Nature of physical laws.

Chapter 2 UNITS AND MEASUREMENTS
2.1 Introduction 2.2 The International system of units 2.3 Measurement of length, Measurement of Large Distances, Estimation of Very Small Distances: Size of Molecule, Range of Lengths 2.4 Measurement of Mass, Range of Mass 2.5 Measurement of Time
2.6 Accuracy, precision of instruments and errors in measurement, Systematic errors, random errors, least count error, Absolute Error, Relative Error and Percentage Error, Combination of Errors 2.7 Significant Figures, Rules for Arithmetic Operations with Significant Figures, Rounding off the Uncertain Digits, Rules for Etermining the Uncertainly in the Results of Arithmatic Calculations 2.8 Dimensions of Physical Quantities 2.9 Dimensional Formulae and dimensional equations 2.10 Dimensional Analysis and its Applications, Checking the Dimensional Consistency of Equations, Deducting Ration among the Physical Quantities.

Chapter 3 MOTION IN A STRAIGHT LINE
3.1 Introduction 3.2 Position, Path Length and Displacement 3.3 Average Velocity and Average Speed 3.4 Instantaneous Velocity and Speed 3.5 Acceleration 3.6 Kinematic equations for uniformly accelerated motion 3.7 Relative velocity – Elements of Calculus.

Chapter 4 MOTION IN A PLANE
4.1 Introduction 4.2 Scalars and Vectors, Position and Displacement Vectors, Equality of Vectors 4.3 Multiplication of Vectors by real members 4.4 Addition and Subtraction of Vectors – graphical method 4.5 Resolution of vectors 4.6 Vector addition Analytical method 4.7 Motion in a plane, Position Vector and Displacement, Velocity, Acceleration 4.8 Motion in a plane with constant acceleration 4.9 Relative velocity in two dimensions 4.10 Projectile Motion, Equation of path of a projectile, Time of Maximum height, Maximum height of a projectile, Horizontal range of projectile 4.11 Uniform circular motion.

Chapter-5: LAWS OF MOTION
5.1 Introduction 5.2 Aristotle’s fallacy 5.3 The law of inertia 5.4 Newton’s first law of Motion 5.5 Newton’s second law of Motion 5.6 Newton’s third law of Motion, Impulse 5.7 Conservation of momentum 5.8 Equilibrium of a particle 5.9 Common forces in Mechanics, Friction 5.10 Circular Motion, Motion of a car on a level road, Motion of a car on a banked road 5.11 Solving problems in Mechanics.

Chapter 6 WORK, ENERGY AND POWER
6.1 Introduction 6.2 Notions of Work and Kinetic Energy: The work-energy theorem. 6.3 Work 6.4 Kinetic Energy 6.5 Work done by a variable force 6.6 The work-energy theorem for a variable force 6.7 The concept of Potential Energy 6.8 The conservation of Mechanical Energy 6.9 The Potential Energy of a spring 6.10 Various forms of energy: the law of conservation of Energy. Heat, Chemical Energy, Electrical Energy, The Equivalence of a Mass and Energy, Nuclear Energy, The Principle of Conservation of Energy. 6.11 Power 6.12 Collisions, Elastic and Inelastic Collisions, Collisions in one dimension, Coefficent – Power consumption in walking

Chapter 7 SYSTEM OF PARTICLES AND ROTATIONAL MOTION
7.1 Introduction, What kind of motion can a rigid body have? 7.2 Centre of mass. Centre of gravity 7.3 Motion of Centre of Mass 7.4 Linear momentum of a System of particles 7.5 Vector product of Two Vectors 7.6 Angular Velocity and its relation with linear velocity, Angular acceleration, kinematics of Rotational motion about a fixed axis. 7.7 Torque and angular Momentum, Moment of force (Torque), Angular momentum of a particle, Torque and angular momentum for a system of a particles, conservation of angular momentum 7.8 Equilibrium of a Rigid Body, Principle of moments 7.9 Moment of Inertia 7.10 Theorems of perpendicular and parallel axis, Theorem of perpendicular axes, Theorem of parallel axes 7.11 Dynamics of Rotational Motion about a Fixed Axis. 7.12 Angular momentum in case of rotations about a fixed axis, Conservation of angular momentum 7.13 Rolling Motion, Kinetic Energy of Rolling Motion.

Chapter 8 OSCILLATIONS
8.1 Introduction 8.2 Periodic and Oscillatory Motions, Period and frequency, Displacement 8.3 Simple Harmonic Motions (SHM) 8.4 Simple Harmonic Motion and Uniform Circular Motion 8.5 Velocity and Acceleration in Simple Harmonic Motion 8.6 Force Law for Simple Harmonic Motion 8.7 Energy in Simple Harmonic Motion 8.8 Some systems executing Simple Harmonic Motion, Oscillations due to a Spring, The Simple Pendulum 8.9 Damped Simple Harmonic Motion 8.10 Forced Oscillations and Resonance.

Chapter 9 GRAVITATION
9.1 Introduction 9.2 Kepler’s Laws 9.3 Universal Law of Gravitation 9.4 The Gravitational Constant 9.5 Acceleration due to Gravity of the Earth 9.6 Acceleration due to gravity below and above the surface of Earth 9.7 Gravitational Potential Energy 9.8 Escape Speed 9.9 Earth Satellite 9.10 Energy of an orbiting satellite 9.11 Geostationary and Polar satellites 9.12 Weightlessness.

Chapter 10 MECHANICAL PROPERTIES OF SOLIDS
10.1 Introduction 10.2 Elastic behavior of Solids 10.3 Stress and Strain 10.4 Hook’s law 10.5 Stress – strain curve 10.6 Elastic Moduli, Young’s Modulus, Determination of Yong’s Modulus of the Material of a Wire, Shear Modulus Bulk Modulus, Poisson’s Ratio. 10.7 Applications of elastic behaviour of Materials.

Chapter 11 MECHANICAL PROPERTIES OF FLUIDS
11.1 Introduction 11.2 Pressure, Pascal’s Law, Variation of Pressure with Depth, Atmospheric Pressure and Gauge Pressure, Hydraulic Machines 11.3 Streamline flow 11.4 Bernoulli’s principle, Speed of Efflux, Torricelli’s Law, Venturi-meter, Blood Flow and Heart Attack, Dynamic Lift 11.5 Viscosity, Variation of Viscosity of fluids with temperature, Stoke’s Law 11.6 Reynolds number 11.7 Surface Tension, Surface Energy, Surface Energy and Surface Tension, Angle of Contact, Drops and Bubbles, Capillary Rise, Detergents and Surface Tension; What is blood pressure.

Chapter 12 THERMAL PROPERTIES OF MATTER
12.1 Introduction 12.2 Temperature and Heat 12.3 Measurement of Temperature 12.4 Ideal – Gas Equation and Absolute Temperature 12.5 Thermal Expansion 12.6 Specific Heat Capacity 12.7 Calorimetry 12.8 Charge of State, Regelation, Latent Heat 12.9 Heat transfer, Conduction, thermal conductivity, Convection, Radiation, Blackbody Radiation, Greenhouse Effect 12.10 Newton’s Law of Cooling.

Chapter 13 THERMODYNAMICS:
13.1 Introduction 13.2 Thermal Equilibrium 13.3 Zeroth Law of Thermodynamics 13.4 Heat, Internal Energy and Work 13.5 First Law of Thermodynamics 13.6 Specific Heat Capacity 13.7 Thermodynamic State Variables and Equation of State 13.8 Thermodynamic Processes, Quasi-static Isothermal Process, Adiabatic Process, Irochoric Process, Cyclic Process. 13.9 Heat Engines 13.10 Refrigerators and Heat Pumps 13.11 Second Law of Thermodynamics 13.12 Reversible and Irreversible Processes 13.13 Carrot Engine, Carnot’s Theorem.

Chapter 14 KINETIC THEORY
14.1 Introduction 14.2 Molecular Nature of Matter 14.3 Behaviour of Gases 14.4 Kinetic Theory of an Ideal Gas, Pressure of an Ideal Gas 14.5 Laws of equipartition of energy 14.6 Specific Heat Capacity, Monatomic Gases, Diatomic Gases, Polyatomic Gases, Specific Heat Capacity of Solids, Specific Heat Capacity of Water 14.7 Mean Free Path.

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Here students can locate TS Inter 1st Year Physics Notes 1st Lesson Physical World to prepare for their exam.

TS Inter 1st Year Physics Notes 1st Lesson Physical World

→ Physics: Physics is the study of nature and natural phenomena.

→ Fundamental forces in nature : In physics

• Gravitational force
• Electromagnetic force
• Strong nuclear force
• Weak nuclear force.

→ Gravitational force : It is the force of attraction between any two objects by virtue of their masses.
These are very weak forces. They are very long distance forces. For heavy bodies like planets and stars etc. the magnitude of these forces is high. These forces are very important in planetary motion, and in formation of Stars and Galaxies.

→ Electromagnetic forces : It is the force between two charged particles.
Between like charges they are “repulsive forces” and between unlike charges they are “attractive forces”. These forces are very strong forces. These are long distance forces.

→ Strong nuclear forces: Strong nuclear forces bind protons and neutrons in a nucleus.
These are very strong attractive forces. They are 100 times stronger than electromagnetic forces. They are short range forces. Their effect is upto very few fermi.

→ Weak nuclear forces: Weak nuclear forces will appear only in certain nuclear processes such as β – decay of nucleus where nucleus emits electron and neutrino. These are weak forces, their range is upto few fermi.

→ Conserved quantities: In physics any physical phenomenon is governed by certain forces. Si.oeral physical quantities will change with time but some special physical quantities will remain constant with time. Such physical quantities are called conserved quantities of nature.
Ex : For motion under an external conser-vative force such as gravitational field the total mechanical energy (i.e., P.E + K.E) is constant or energy is conserved.

→ Some physicists and their major contributions

→ Fundamental forces of nature

→ Fundamental constants of Physics

→ Conversion factors:

→ Important Prefixes:

→ The Greek Alphabet:

→ Formulae of geometry :

• Area of triangle = \(\frac{1}{2}\) × base × height
• Area of parallelogram = base × height
• Area of square = (length of one side)²
• Area of rectangle = length × breadth
• Area of circle = πr² (r = radius of circle)
• Surface area of sphere = πr² (r = radius of sphere)
• Volume of cube = (length of one side of cube)³
• Volume of parallelopiped = length x breadth x height
• Volume of cylinder = πr²l
• Volume of sphere = \(\frac{4}{3}\)πr³
• Circumference of square = 41
• Volume of cone = \(\frac{1}{3}\)πr²h
• Circumference of circle = 2πr

→ Formulae of algebra:

• (a + b)² = a² + b² + 2ab
• (a – b)² = a² + b² – 2ab
• (a² – b²) = (a + b) (a – b)
• (a + b)³ = a³ + b³ + 3ab (a + b)
• (a – b)³ = a³ – b³ – 3ab (a – b)
• (a + b)² – (a – b)² = 4ab
• (a + b)² + (a – b)² = 2(a² + b²)

→ Formulae of differentiation:

• \(\frac{d}{d x}\) (constant) = 0 differentiation with respect to x = \(\frac{d}{d x}\)
• \(\frac{d}{d x}\) (xⁿ) = n x^n-1
• \(\frac{d}{d x}\) (sin x) = cos x
• \(\frac{d}{d x}\)(cos x) = – sin x dx

→ Formulae of Integration:
Integration with respect to x = ∫dx

• ∫dx = x
• ∫xⁿ dx = \(\frac{x^{n+1}}{n+1}\)
• ∫sin x dx = cos x + c
• ∫cos x dx = sin x + c

→ Formulae of logarithm :

• log mn = (log m + log n)
• log(\(\frac{m}{n}\)) = (log m – log n)
• log mⁿ = n log m

→ Value of trigonometric functions :

→ Signs of trigonometrical ratios :

• sin (90° – θ) = cos θ ; sin (180° – θ) = sin θ
• cos (90° – θ) = sin θ ; cos (180° – θ) = – cos θ
• tan (90° – θ) = cot θ ; tan (180° – θ) = – tan θ

→ According to Binomial theorem :
(1 + x)ⁿ ≈ (1 + nx) if x < < 1

→ Quadratic equation:
ax² + bx + c = 0
x = \(\left(\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}\right)\)