Mahesh

Here students can locate TS Inter 1st Year Chemistry Notes 13th Lesson Organic Chemistry: Some Basic Principles and Techniques to prepare for their exam.

TS Inter 1st Year Chemistry Notes 13th Lesson Organic Chemistry: Some Basic Principles and Techniques

→ Compounds containing only carbon and hydrogen atoms are called hydro-carbons.

→ A single atom or group of atoms which is responsible for the characteristic properties of an organic compound is called a functional group.

→ Series of compounds in which adjacent members differ by a CH₂ group are called homologous series,

→ Compounds with the same molecular formula but having different properties are called isomers and the phenomenon is called isomerism.

→ Isomerism due to the difference in the carbon chain is called chain isomerism.

→ Isomerism due to the difference in the position of a substituent, a functional group or a multiple bond is called position isomerism.

→ Isomerism due to the difference in the nature of the alkyl groups attached to the same functional group is called functional isomerism.

→ Alkanes are saturated hydrocarbons having carbon – carbon single bonds and their general formula is C_nH_2n-2

→ Isomerism due to the difference in the nature of the alkyl groups attached to the same functional group is called, metamerism.

→ A reaction in which an atom or a group of atoms attached to carbon atom is replaced by a new atom or group of atoms is called substitution reaction.

→ Alkenes are unsaturated hydrocarbons having carbon – carbon double bond and their general formula is C_nH_2n.

→ Alkynes are also unsaturated hydrocarbons having carbon – carbon triple bond and their general formula is C_nH_{2n -2}.

→ The process of breaking a pi bond and adding two atoms is called addition.

→ The decomposition of an organic compound into smaller products by heating in the absence Of air is called pyrolysis or cracking.

→ Alkyl magnesium halide is called Grignard reagent.

→ Alkaline KMnO₄ solution is called Bayer’s reagent.

→ Alkyl halides on heating with sodium metal in presence of dry ether gives an alkane with twice the number of carbon atoms. This reaction is known as Wurtz reaction.

→ Electrolysis of a concentrated aqueous solution of sodium or potassium salt of carboxylic acid gives a hydrocarbon at anode. This reaction is called Kolbe’s electrolysis.

→ Aromatic compounds are cyclic, planar and obey Huckle’s rule.

→ Benzene undergoes electrophilic substitut-ion reactions rather than addition reactions.

→ A mixture of sodium hydroxide and calcium oxide is called sodalime.

→ Chromatography is a method of separation of components of a mixture between a stationary phase and a mobile phase.

→ Inductive effect is defined as the polarization of a bond caused by the polarization of adjacent o bond.

→ Electromeric effect is defined as the complete transfer of a shared pair of n electrons to one of the atoms joined by a multiple bond on the demand of an attacking reagent.

→ The electron pair displacement caused by an atom or group along a chain by a conjugative mechanism is called the mesomeric effect of that atom or group.

→ Resonance energy is the difference in energy between the actual energy of the molecule and that of the most stable canonical structure of the molecule.

→ Hyperconjugation is also called, no-bond resonance.

→ Electrophiles are the reagents that attack a point of high electron density or negative centres.

→ Nucleophiles are the reagents that attack a site of low electron density or positive centres.

→ Molecular rearrangements are those in which a less stable molecule rearranges into a more stable molecule.

→ Conformational isomers of an alkane are obtained by rotation about C – C bond.

→ Any intermediate conformation between staggered and eclipsed is called a skew conformation.

→ NORBORNANE is Bicyclo (2, 2, 1) heptane.

→ The existence of more than one compound having identical structures but differing in spatial arrangements of atoms or groups is called geometrical isomerism or cis-trans isomerism.

→ Geometrical isomers are diastereomers.

→ Markownikoff’s rule : When an unsymmetrical reagent adds to a double bond, the positive part of the adding reagent attaches itself to a carbon of the double bond so as to give the more stable carbocation as the intermediate.

→ In presence of a peroxide, anti-Markownikoff s addition takes place. It is called, Kharsch effect.

→ Carcinogenic (cancer-producing) substances are generally formed due to incomplete combustion of organic substances like coal, petroleum, tobacco etc.

→ A substance which rotates the plane polarised light is called an ‘optically active substance.

→ Inorganic substances like quartz, some rock crystals, crystals of KClO₃, KBrO₃, NaIO₄ etc. are optically active.

→ Organic compound exhibits optical activity when it is chiral.

→ Most chiral compounds have a chiral centre, which is a carbon atom, bonded to four different atoms or groups.

→ The chiral molecule and its mirror image are not superimposable. They are called enantiomers.

Here students can locate TS Inter 1st Year Physics Notes 7th Lesson Systems of Particles and Rotational Motion to prepare for their exam.

TS Inter 1st Year Physics Notes 7th Lesson Systems of Particles and Rotational Motion

→ Rigid body: A rigid body is a body with perfectly definite and unchanging shape. The distance between all the pair of particles of such body do not change.
Note: There is no real body that is truely rigid. All real bodies deform under the influence of forces. But in many cases this deformation is negligible.

→ Translational motion: In translational motion the body will move as a whole from one place to another place.
In pure translational motion all the particles of the body will have same velocity at any instant of time.

→ Axis of rotation: To prevent translational motion a rigid body has to be fixed along a straight line. Then the only possible motion is rotation about that fixed line. This fixed line is called axis of rotation.

→ Rotation: In rotation every particle of a rigid body moves in a circle which lies in a plane perpendicular to the axis of rotation. The rotating body has a centre on the axis.

→ Centre of mass: For a rigid body or system of particles the total mass seems to be concentrated at a particular point is called centre of mass.
Such point will behave as if it is the representative of whole translational motion of that body.

→ Centre of gravity: It is a point in the body where the total weight of the body seems to be concentrated.
If we apply an equal and opposite force to weight of the body (W = mg) the body will be in mechanical equilibrium (i.e. both in translational and rotational equilibrium).

→ Co-ordinates of centre of mass:
1. For symmetric bodies when origin is taken at geometric centre then centre of mass is also at geometric centre. Ex: Thin rod, disc, sphere etc.
2. Let two bodies of masses say m₁ and m₂ are at distances say x₁ and x₂, from origin then centre of mass = x_c = \(\frac{m_1 x_1+m_2 x_2}{m_1+m_2}\)
So we can assume that position centre of mass is the ratio of sum of moment of masses and total mass of the body.

Note:
1. Let two bodies of equal masses m and m are separated by a distance x then centre of mass of that svstem is at \(\frac{x}{2}\).
2. If three equal masses are at the corners of a triangle then centre of mass is at centroid of that triangle.
3. Let a system of particles say m₁, m₂, m₃ ………….. m_n are in a plane or in space then co-ordinates of centre of mass will have x, y for plane and X, Y and Z for space where

→ Characteristics of centre of mass:

• Total mass of the body seems to be concentrated at centre of mass.
• The total external force (F_ext) applied on a body seems to be applied at centre of mass. F_ext = M a_c where ‘a’ is acceleration of centre of mass.
• Internal forces cannot change the motion of centre of mass.
• A complex motion is a combination of translational and rotational motions. In complex motion centre of mass represents the entire translational motion of the whole body.
• The momentum of a body is the product of mass of the body and velocity of centre of mass P̅ = MV_c
• Co-ordinates of centre of mass do not depend on the co-ordinate system chosen.

→ Motion of centre of mass:
1. Motion of centre of mass represents the translational motion of the whole body.

2. Velocity of centre of mass
V_c = \(\frac{m_1 v_1+m_2 v_2+\ldots \ldots \ldots+m_n v_n}{\Sigma m_i}\)
i.e., velocity of centre of mass is the ratio of sum of momentum of all particles to total mass of the body.

3. Momentum of centre of mass P̅_c is the sum of momentum of all the particles of the body.
P̅_c = MV̅_c = m₁v₁ + m₂v₂ + ……………… + m_nv_n

4. External force acting on centre of mass ^
F = M A_c = m₁a₁ + m₂a₂ ………………+ m_na_n or
F = MA_c = F₁ + F₂ + …………….. + F_n
Where a₁, a₂,…….. a_n are accelerations of individual particles of masses m₁, m₂ …. m_n

→ Explosion of a shell in mid air: Let a shell moves along a parabolic trajectory explodes in mid air and divided into number of fragments. Still then centre of mass of that system of fragments will follow “the same parabolic path”.
Explanation: Explosion is due to internal forces. Internal forces cannot change the momentum of a body. So algebraic sum of momentum of all fragments is constant. So velocity of centre of mass is constant.
Hence centre of mass will follow the same parabolic path.

→ Cross product or vector product of vectors:
If the multiplication of two vectors generates a vector then that vector multiplication is called cross product. Mathematically
A̅ × B̅ = |A̅||B̅|sin θ n̂
Where n is a unit vector perpendicular to the plane of A̅ and B̅.
Note: The new vector generated is always perpendicular to both A̅ and B̅ i.e., perpendicular to the plane containing A̅ and B̅.

Properties of cross product:

• Cross product is not commutative i.e., A̅ × B̅ ≠ B̅ × A̅ But A̅ × B̅ = -B̅ × A̅
• Cross product obeys distributive law i.e., A̅ × (B̅ + C̅) = (A̅ × B̅) + (A̅ × C̅)
• If any vector is represented by the combination of i̅, j̅ and k̅ then cross product will obey right hand screw rule.
• The product of two coplanar perpendicular unit vectors will generate a unit vector perpendicular to that plane i.e. i̅ × j̅ = k̅, j̅ × k̅ = i̅ and k̅ × i̅ = j̅
• Cross product of parallel vectors is zero
  i. e. i̅ × i̅ = j̅ × j̅ = k̅ × k̅ =0

→ Angular displacement (θ): The angle subtended by a body at the centre when it is in angular motion is called angular dis-placement. Unit: Radian.

→ Angular velocity (ω): Rate of change in angular displacement is called angular velocity.
Let angular displacement is Δθ over a time interval Δt then average angular velocity
ω = \(\frac{\Delta \theta}{\Delta t}\), Unit = Radian / sec

Note:

• Relation between angular velocity and linear velocity is v = rω
• For a body in rotatory motion all the particles will have same angular velocity ‘ω’. But linear velocity ‘v’ changes.

→ Angular acceleration (α): The rate of change of angular velocity is defined as angular acceleration.
Angular acceleration α = \(\frac{\mathrm{d} \omega}{\mathrm{dt}}\)
Unit: Radian /sec²

→ Moment of force couple or torque (τ): The moment of force is called torque or moment of force couple.
Let a force F̅ is applied on a point P’.
The position vector of P from origin is r then
Torque τ = r̅ × F̅ = |r̅ | × |F̅ |sin θ . n̂
It is a vector. Its direction is perpendicular to both r̅ and F̅ (or) it is perpendicular to the plane containing r̅ and F̅.
Unit: Newton metre (N – m) ; D.F: ML² T^-2

Note:
1. Torque is the product of force F and perpendicular distance between force (F) and point of application (i.e. r sin θ).
2. Torque represents the energy with which a body is rotated. So units of torque and energy are same.

→ Angular momentum (L): Let a particle of mass m’ has a linear momentum P and its position vector is r from origin then angular momentum of that particle is defined as
L = r̅ × p̅ = |r̅||p̅|sin θ n̂
Angular momentum is a vector. L is perpendicular to the plane containing r and P .
Unit: Kg – metre² and D.F.: ML²
Note: Angular momentum is the product of momentum P̅ and perpendicular distance (r sin θ) from origin. It is also written as L = Iω

→ Torque and angular momentum: The time rate of change of the angular momentum of a particle is equal to torque acting on it.
Torque τ = \(\frac{\mathrm{d} \overline{\mathrm{L}}}{\mathrm{dt}}=\overline{\mathrm{r}} \cdot \frac{\mathrm{d} \overline{\mathrm{p}}}{\mathrm{dt}}\)

Note:
The time rate of change of the angular momentum of a system of particles about a point is equal to the sum of external torque i.e.,
\(\frac{\mathrm{d} \overline{\mathrm{L}}}{\mathrm{dt}}\) = τ_ext lust like \(\frac{\mathrm{d} \overline{\mathrm{p}}}{\mathrm{dt}}\) = F_ext

→ Law of conservation of angular momentum:
When external torque (τ_ext) is zero, the total angular momentum of a system is conserved i. e., it remains constant.
When τ_ext, = 0 then \(\frac{\mathrm{d} \overline{\mathrm{L}}}{\mathrm{dt}}\) = 0 i.e. angular momentum L̅ is constant, i.e. I₁ω₁ + I₂ω₂ = constant

Note:
1. For a system of particles when τ_ext = 0 then \(\mathrm{d} \overline{\mathrm{L}}_1+\mathrm{d} \overline{\mathrm{L}}_2+\ldots \ldots+\mathrm{d} \overline{\mathrm{L}}_{\mathrm{n}}=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{d} \overline{\mathrm{L}}_{\mathrm{i}}\) = 0.
2. Law of conservation of angular momentum is similar to law of conservation of linear momentum in linear motion.

→ Equilibrium of a rigid body: A rigid body is said to be in mechanical equilibrium if both its linear momentum and angular momentum are not changing with time. Then the body has neither linear acceleration nor angular acceleration.
(or)
1. The vector sum of all the forces acting on a rigid body must be zero.
i.e. F₁ + F₂ + ……………. + F_n = \(\sum_{i=1}^n F_i\) = 0
This condition provides translational equilibrium of the body.

2. The vector sum of all the torques acting on a rigid body must be zero i.e.,
τ₁ + τ₂ + ……………. + τ_n = \(\sum_{i=1}^n \tau_i\) = 0
This condition provides rotational equilibrium of the body.

→ Couple (or) Force couple: A pair of equal and opposite forces with different lines of action is known as couple.
A couple produces rotation without translation.

→ Moments or moment of force: It is defined as the product of force and perpendicular distance between force and its point of application.

→ Principles of moments: For a lever to be in mechanical equilibrium let R is the reaction of the support at fulcrum. It is directed upwards and F₁, and F₂ are the forces then

1. For translational equilibrium R – F₁ – F₂ = 0 i.e. algebraic sum of forces must be zero.
2. For rotational equilibrium d₁F₁ = d₂F₂ = 0. i.e. algebraic sum of moments must be zero.

→ Lever: An ideal lever is a light rod pivoted at a point along its length. This point is called “fulcrum”. In levers d₁F₁ = d₂F₂ i.e .
Load arm × load = Force arm × Force
Mechanical advantage (M.A) = \(\frac{\mathrm{F}_1}{\mathrm{~F}_2}=\frac{\mathrm{d}_2}{\mathrm{~d}_2}\)
If effort arm d₂ is larger than load arm then M.A > 1 i.e. we can lift greater loads with less effort.

→ Moment of Inertia (I): The inertia of a rotating body is called moment of inertia.
Mathematically, Moment of Inertia, I = \(\sum_{i=1}^n\) m₁r₁² = MR²

→ Radius of gyration (k): Radium of gyration of a body about an axis may be defined as the distance from the axis of a mass point whose mass is equal to whole mass of the body and whose moment of inertia is equal to moment of inertia of the whole body about that axis.
Fly wheel: Fly wheel is a metalic body with large moment of inertia.
It is used in rotational motion of engines like automobiles. It allows a gradual change in speed and prevents jerky motion.

→ Perpendicular axis theorem: The moment of inertia of a plane body (lamina) about an axis perpendicular to its plane is equal to the sum of its moment of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body, i.e., I_z = I_x + I_y

→ Parallel axis theorem: The moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through the centre of mass and the product of its mass and square of the distance between the two parallel axes, i.e., I = I_G + MR²

→ Rolling motion: Rolling motion is a combination of translational motion and rotatory motion.

→ Kinetic energy of a rolling body: When a body is rolling on a body without slipping then it will have translational kinetic energy (\(\frac{1}{2}\)mv²) and rotational kinetic energy (\(\frac{1}{2}\)Iω²).
Total kinetic energy of rolling body
K.E_R = \(\frac{1}{2}\)mv² + \(\frac{1}{2}\)Iω>²

→ For two particle system of masses m₁ and m₂ with positions x₁ and x₂.
(a) Coordinates of centre of mass,
x_c = \(\frac{m_1 x_1+m_2 x_2}{m_1+m_2}\)
(b) If coordinate system coincides with m1 then x_c = \(\frac{\mathrm{m}_2 \mathrm{x}_2}{\mathrm{~m}_1+\mathrm{m}_2}\) or x_c = \(\frac{\mathrm{m}_2 \mathrm{~d}}{\mathrm{~m}_1+\mathrm{m}_2}\) where d is distance between m1 and m2.
(c) Ratio of distances from centre of mass is = \(\frac{\mathrm{d}_1}{\mathrm{~d}_2}=\frac{\mathrm{m}_2}{\mathrm{~m}_1}\)

→ For many particle system

→ Cross product: A̅ × B̅ is defined as |A̅||B̅| sin θ. n̅ where n̅ is a unit vector perpendicular to the plane of A̅ and B̅.

→ In cross product i̅ × i̅ = j̅ × j̅ = k̅ × k̅ = 0 i.e., cross product of parallel vectors is zero.

→ In cross product i̅ × j̅ = k̅, j̅ × k̅ = i̅ and k̅ × i̅ = j̅ cross product of two heterogeneous unit vectors will generate third unit vector taken in clockwise direction.

→ If A̅ = x₁ i̅ + y₁ j̅ + z₁ k̅ and B̅ = x₂ i̅ + y₂ j̅ + z₂ k̅ then

→ Angular velocity
ω = \(\frac{\text { angular displacement }}{\text { time }}=\frac{\theta}{t}\)
ω = \(\frac{\theta}{t}\)
For small quantities, ω = \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\)
Unit : Radian/sec
ω = \(\frac{\theta}{t}\) or ω = \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\) or ω = \(\frac{2 \pi \mathrm{n}}{\mathrm{t}}\) (n = number of rotations)

→ Angular acceleration,
α = \(\frac{\text { change in angular velocity }}{\text { time }}\)
α = \(\frac{\omega_2-\omega_1}{t}\) or α = \(\frac{\mathrm{d} \omega}{\mathrm{dt}}\)
Unit: Radian/sec²

→ Relation between v and ω is v = rω

→ Relation between a and α is a = rα

→ Centripetal acceleration, a_c = rω = vω = \(\frac{v^2}{r}\)

→ Centrifugal force = \(\frac{\mathrm{mv}^2}{\mathrm{r}}\) = mrω²

→ When a coin is placed on a gramphone disc or on a circular turn table or for a vehicle is moving in a curved path limiting friction,
μ_s = \(\frac{f_s}{N}=\frac{r \omega^2}{g}\)

→ When a vertically hanging body M’ is balanced by a rotating body m in horizontal planp then at equilibrium
Mg = mrω² or Angular velocity required for balance is ω = \(\sqrt{\frac{\mathrm{Mg}}{\mathrm{mr}}}\)

→ Torque, τ = r̅ × F̅ = |r̅| |F̅| sin θ. It represents the energy with which a body is turned.

→ Moment of force couple = one of the force in couple × distance between the directions of forces.

→ Moment of inertia, I = \(\sum_{i=1}^n\)m₁r₁² or I = MR²

→ If moment of inertia, I = MR² = MK² then K is called radius of gyration.

→ From parallel axis theorem moment of inertia about any parallel axis to the axis passing through centre of mass is, I = I_G + MR².

→ From perpendicular axis theorem M.O.I. about a perpendicular axis to the plane
I_z = I_x + I_y

→ Moment of inertia of a thin rod of length l.
(a) M.O. I of thin rod about its axis and perpendicular to length, I = \(\frac{\mathrm{M} l^2}{12}\); K = \(\frac{l}{\sqrt{12}}\)

(b) M.O.I. of thin rod about one end of the rod and perpendicular to length,
l = \(\frac{\mathrm{m} l^2}{3}\); K = \(\frac{l}{\sqrt{3}}\)

→ Moment of Inertia of a circular ring of radius R
a) M. O.O of a circular ring about an axis pass¬ing through the centre and perpendicular to its plane I = MR², K = R

b) M.O.I. of a circular ring about any diameter,

c) M.O.I. about any tangent and parallel to the diameter

→ Moment of Inertia of a disc of radius R
(a) M.O.I about an axis passing through centre and perpendicular to the plane,
I = \(\frac{\mathrm{MR}^2}{2}\); K = \(\frac{\mathrm{R}}{\sqrt{2}}\)

(b) MOI. about any diameter,
I = \(\frac{\mathrm{MR}^2}{4}\); K = \(\frac{\mathrm{R}}{2}\)
(c) MOI about any rangent,
I = \(\frac{5}{4}\)MR²; K = \(\frac{\sqrt{5}}{2}\)R

→ Moment of inertia of a rectangular plane lamina

(a) about the centre and perpendicular to the plane
I = M\(\frac{\left(l^2+b^2\right)}{12}\); K = \(\sqrt{\frac{l^2+b^2}{12}}\)

(b) about the axis parallel to length,
I = \(\frac{\mathrm{Mb}^2}{12}\); K = \(\frac{\mathrm{b}}{\sqrt{12}}\)

(c) about any axis parallel to breadth,
I = \(\frac{\mathrm{b}}{\sqrt{12}}\); K = \(\frac{l}{\sqrt{12}}\)

→ Moment of inertia oía solid sphere
(a) about an axis passing through diameter
I = \(\frac{2}{3}\)MR²; K = \(\sqrt{\frac{2}{5}}\)R

(b) about any tangent, I = \(\frac{7}{5}\)MR²; K = \(\sqrt{\frac{7}{5}}\)R

→ Moment of inertia of a hollow sphere
(a) about any diameter, I = \(\frac{2}{3}\)MR²; K = \(\sqrt{\frac{2}{3}}\)R

(b) about any tangent I = \(\frac{5}{3}\)MR²; K = \(\sqrt{\frac{5}{3}}\)R

→ Moment of inertia of a solid cylinder of length l and radius R
a) M.O.I. about natural axis of cylinder,
I = \(\frac{\mathrm{MR}^2}{2}\); K = \(\frac{\mathrm{R}}{\sqrt{2}}\)
b) M.O.I. about an axis perpendicular to length and passing through centre,
I = M\(\left(\frac{l^2}{12}+\frac{\mathrm{R}^2}{4}\right)\); K = \(\sqrt{\frac{l^2}{12}+\frac{\mathrm{R}^2}{4}}\)

→ M. O.I. of a hollow cylinder of length / and radius R

a) about the natural axis, I = MR²; K = R
b) about an axis perpendicular to length and passing through centre
I = \(\left(\frac{l^2}{12}+\frac{\mathrm{R}^2}{2}\right)\); K = \(\sqrt{\frac{l^2}{12}+\frac{\mathrm{R}^2}{2}}\)

→ Angular momentum, L = Iω

→ Relation between angular momentum (L) and torque (τ) is, τ = \(\frac{\mathrm{dL}}{\mathrm{dt}}=\frac{\mathrm{L}_2-\mathrm{L}_1}{\mathrm{t}}\)

→ Relation between τ and α is τ = Iα

→ From law of conservation of angular momentum
I₁ω₁ + I₂ω₂ = constant (When no external torque acts on the body)

Here students can locate TS Inter 1st Year Physics Notes 8th Lesson Oscillations to prepare for their exam.

TS Inter 1st Year Physics Notes 8th Lesson Oscillations

→ Periodic motion: A motion that repeats itself at regular intervals of time is called periodic motion.

→ Oscillations or vibrations: When a small displacement is given to a body at rest position (i. e. its mean position) then a force will come into play and tries to bring back the body to its rest position or equilibrium position by executing to and fro motion about mean position. These are called vibrations or oscillations.

→ Time period (T): In periodic motion the smallest time interval after which the motion is repeated is called its time period.

→ Displacement: For a body in periodic motion the displacement changes with time, so displacement is given by x(t) (or)
f(t) = A cos ωt

→ Frequency: The reciprocal of time period (T) gives the number of repetitions that occur for unit time. It is called frequency (o).
υ = \(\frac{1}{T}\) = 1/Time period

→ Fourier theory: According to Fourier “any periodic function can be expressed as a super position of sine and cosine functions of different time periods with suitable coefficients.”

→ Simple harmonic motion (explanation of equation): In simple harmonic motion displacement is a sinusoidal function of time. i.e. x(t) = A cos(ωt ± Φ)
The particle will oscillate back and forth about the origin on X-axis within the limits + A and -A where A is amplitude, ω and Φ are constants.
(a) Amplitude A of simple harmonic motion (S.H.M.) is the magnitude of maximum displacement of the particle. Displacement of particle varies from +A to -A.
(b) Phase of motion (or) Argument: For a body in S.H.M the position of a particle or state of motion of a particle at any time t’ is determined by the argument (wt + Φ). The term (wt + Φ) is called argument or phase of motion.
(c) Phase constant (or) phase angle: The value of phase of motion at time t = 0 is called”phase constant Φ” or”phase angle”.
(d) Simple harmonic motion is represented with a cosine function. It has a periodicity of 2n. The function repeats after a time period T.

→ Reference circle: Let a particle p’ moves uniformly on a circle. The projection of p on any diameter of the circle will execute S.H.M.

The particle ‘p’ is called “reference parti-cle” and the circle on which the particle moves is called “reference circle”.

→ Velocity of a body on S.H.M.: For a body in uniform circular motion speed v = coA. The direction of velocity v̅, at any time is along the tangent to the circle at that instantaneous point. Mathematically velocity of the body v̅(t) = -ωAsin(ωt + Φ) or v(t) = \(\frac{d}{d t}\)x(t) = \(\frac{d}{d t}\)Acos(ωt + Φ)
= -ωAsin(ωt + Φ)

→ Acceleration (a): For a body in S.H.M the instantaneous acceleration of the particle is a(t) = -ω² A cos(wt + Φ) = -ω² x(t)
or a(t) = \(\frac{d}{d t}\)v(t) = \(\frac{d}{d t}\)[-A ω sin(ωt + Φ)]
= -A(ω²cos(ωt + Φ))

Note:
a(t) implies the acceleration of the body is a function of time.
Maximum acceleration of the body amM = – co2A. Note: -ve sign indicates that acceleration and displacement are in opposite direction.

→ Force on a body in S.H.M:
Acceleration a(t) = – ω²x(t)
Force F = m a
∴ Force on a body in S.H.M
F(t) = ma = m(-ω²xt) = – K x(t)
Where K = mω² and m = mass of the body

→ Energy of a body in S.H.M: For a body in S.H.M both kinetic energy and potential energy will change with time. These values will vary between zero and their maximum value.
Kinetic energy K = \(\frac{1}{2}\)mv
= \(\frac{1}{2}\)mω² A² sin² (ωt + Φ)
= \(\frac{1}{2}\)kA² sin² (ωt + Φ)

Potential energy U = \(\frac{1}{2}\)kx
= \(\frac{1}{2}\)kA² cos² (ωt + Φ)

→ Springs:

• In case of springs for smaller displace-ments when compared with length of the spring Hooke’s law will hold good.
• The small oscillations of a block of mass m’ connected to a spring can be taken as simple harmonic.
• In case of springs the restoring force acting on the block of mass m is F(x) = – k (x)
• Spring constant k is defined as the force required for unit elongation.
  Unit: newton/metre, K= force/displacement ⇒ K = \(\frac{-F}{x}\) = ve sign indicates that force and displacement are opposite in direction. For a stiff spring k is high. For a soft spring k is less.
• Angular frequency of a loaded spring ω = \(\sqrt{\frac{K}{m}}\)
  Time period of oscillation T = 2π\(\sqrt{\frac{\mathrm{m}}{\mathrm{K}}}\)
  Where m is load attached and k is constant of spring.

→ Simple pendulum:
Time period of oscillation T = 2π\(\sqrt{l / \mathrm{g}}\)

→ Damped oscillations: In damped oscillations the energy of the system is dissipated continuously
When damping is very small the oscillations will remain approximately periodic. Eg: Oscillations of simple pendulum.
Damping force depends on nature of medium.

→ Free oscillations: When a system is displaced from its equilibrium position and allowed free to oscillate then oscillations made by die body are known as free oscillations.
Frequency of vibration is known as natural frequency of that body.

→ Forced or driven oscillations: If a body is made to oscillate under the influence of an external periodic force (say ω₀) then those oscillations are called forced oscillations.

→ Resonance: The phenomenon of increase in amplitude of vibrating body when driving force frequency ‘ω₀‘ is equal to or very close to natural frequency ‘ω’ of the oscillator is called resonance.
Note: When driving frequency is very close to natural frequency of vibrating body then also resonance will occur. Due to this reason damage to building is caused in earthquake effected area.

→ Displacement of a body in S.H.M
Y = A sin (ωt ± Φ)

→ Velocity of a body in S.H.M is
V = \(\frac{d y}{d t}=\frac{d}{d t}\) (A sin ωt)
V = Aω cos ωt ( where Y = A sin ωt)
or V = ω\(\sqrt{A^2-Y^2}\)
Maximum velocity, V_max = Aω

→ Acceleration of a body in S.H.M
a = -ω² A sin ωt or a = -ω²Y .
(where Y = A sin ωt)
(- ve sign shows that acceleration and displacement are opposite to each other) Maximum acceleration, a_max = ω²A

→ Angular velocity ω of a body in S.H.M:
∝ – Y or a = – ω²Y ( – ve sign for opposite direction only )
∴ ω = \(\frac{a}{Y}\) or ω = \(\sqrt{\frac{a}{Y}}\)
∴ Angular velocity of a body in S.H.M,
ω = \(\sqrt{\frac{\text { acceleration }}{\text { displacement }}}\)

→ Time period of a body in S.H.M: The time taken for one complete oscillation is called “time period”.
Time Period,
(T) = \(\frac{\text { Angular displacement for one rotation }}{\text { Angular velocity }}=\frac{2 \pi}{\omega}\)
∴ T = \(\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{Y}{\mathrm{a}}}=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}}\)

→ Frequency (υ) is the number of vibrations (or) rotations per second.
Frequency, υ = \(\frac{1}{\mathrm{~T}}=\frac{1}{2 \pi} \sqrt{\frac{\mathrm{a}}{\mathrm{Y}}}\),
⇒ υ = \(\frac{\omega}{2 \pi}\) or ω = /7io 2n

→ Springs:

• In springs force constant is defined as the ratio of Force to displacement.
  Spring constant, (K) = \(\frac{F}{Y}\)
• Time period, T = \(\sqrt{\frac{\mathrm{m}}{\mathrm{K}}}\) = 2π.\(\sqrt{\frac{\mathrm{m}}{\mathrm{K}}}\)
• If mass of the spring mj is also taken into account then time period,
  T₁ = 2π\(\frac{\sqrt{\left(m+\frac{m_1}{3}\right)}}{K}\)(For real spring )
• Frequency of vibration, n = \(\frac{1}{2 \pi} \sqrt{\frac{K}{m}}\)
  For Real spring, n = \(\frac{1}{2 \pi} \sqrt{\frac{K}{\left(m+\frac{m_1}{3}\right)}}\)
• When a spring is divided into ‘n’ parts then its force constant (k) will increase.
  New force constant k₁ = nk
  where n number of parts and k = original spring constant.

→ Simple Pendulum:

• In simple pendulum component of weight of the bob useful for to and fro motion is F = mg sin θ
• Time period, T = 2π\(\sqrt{\frac{l}{g}}\) ⇒ g = 4π\(\frac{l}{\mathrm{~T}^2}\)
• When a simple pendulum is placed in a lift moving with some acceleration then its time period changes.
• When lift moves up with an acceleration ‘a’ its time period decreases, T = 2π\(\sqrt{\frac{l}{g+a}}\)
• When lift moves down with an acceleration ‘a’ its time period increases, T = 2π\(\sqrt{\frac{l}{g-a}}\)
• For seconds pendulum time period, T = 2 sec
  Length of seconds pendulum on earth = 100 cm = 1 m ( nearly )
• In simple pendulum L – T² graph is a straight line passing through origin.

Here students can locate TS Inter 1st Year Physics Notes 6th Lesson Work, Energy and Power to prepare for their exam.

TS Inter 1st Year Physics Notes 6th Lesson Work, Energy and Power

→ Dot product (or) Scalar product: The scalar product (or) dot product of any two vectors
A̅ and B̅ is A̅ . B̅ = |A̅||B̅| cos θ
where θ is angle between them.
Ex : Work W = F̅ . S̅

→ Properties of dot product:

• Scalar product obeys “commutative law” i-e., A̅ . B̅ = B̅ . A̅
• Scalar product obeys “distributive law” A̅.(B̅ + C̅) = A̅.B̅ + A̅.C̅
• In unit vector i, j, k system
  i̅.i̅ = j̅.j̅ = k̅ k̅ = 1 i.e., dot product of like unit vectors is unity.
  i̅.j̅ = j̅.k̅ = k̅.i̅ = 0 i.e., dot product of perpendicular vectors is zero.

→ Work: Work done by a force is defined as the product of component of force in the direction of displacement and the magnitude of displacement.

W = F̅ . S̅ (or) W = F̅.S̅ cos θ
Work is a scalar, unit kg-m²/sec² called joule (J), D.F = ML²T^-2

→ Energy: The ability (or) capacity of a body to do work is called energy.
Note : Energy can be termed as stored work in the body.

→ Kinetic energy : Energy possessed by a moving body is called kinetic energy (k).
KE = \(\frac{1}{2}\)mv²
Ex : All moving bodies contain kinetic energy.

→ Relation between kinetic energy and momentum : Kinetic energy KE = \(\frac{1}{2}\)mv ; momentum P̅ = mv
∴ E = \(\frac{\mathrm{P}^2}{2 \mathrm{~m}}\)

→ Work energy theorem (For variable force):
Work done by a variable force is always equal to the change in kinetic energy of the body.
Work done W = \(\frac{1}{2}\)mV² – \(\frac{1}{2}\)mV₀² = K_f -K_i

→ Potential energy (V): It is the energy posses by virtue of the position (or) configuration of a body.

→ Law of conservation of mechanical energy:
The total mechanical energy of a system is conserved if the forces doing work on it are conservative.

→ Spring constant (k): It is defined as the ratio of force applied to the displacement produced in the spring.
k = –\(\) unit: Newton/metre, D.F = MT^-2
Note: A spring is said to be stiff if k is large. A spring is said to be soft if k is small.

→ Law of conservation of energy: When forces doing work on a system are conservative then total energy of the system is constant i. e. energy can neither be created nor destroyed.

→ Explanation: When conservative forces are applied on a system then the total mechanical energy is

• as kinetic energy which depends on motion
  OR
• as potential energy which depends on position of the body.

→ Collisions : In collisions a moving body collides with another body. During collision the two colliding bodies are in contact for a very small period. During time of contact the two bodies will exchange the momentum and kinetic energy.

→ Collisions are two types :

• Elastic collision
• Inelastic collision.

→ Elastic collisions: Elastic collisions will obey

• Law of conservation of momentum and
• Law of conservation of energy.

→ Inelastic collisions: Inelastic collisions will follow, law of conservation of momentum only.
In these collisions a part of energy is lost in the form of heat, sound etc.

→ Coefficient of restitution (e): The coefficient of restitution is the ratio or relative velocity of separation (v₂ – v₁) to the relative velocity of approach (u₁ – u₂).
Coefficient of restitution e = \(\frac{\mathrm{v}_2-\mathrm{v}_1}{\mathrm{u}_1-\mathrm{u}_2}\)

Note :

• For Perfect elastic collisions e = 1
• For Perfect inelastic collisions e = 0
• For collisions ‘e’ lies between ‘0’ and 1.

→ One dimensional collisions (or) head on collisions : If the two colliding bodies are moving along the same straight line they are called one dimensional collision or head on collisions.
For these collisions initial and final velocities of the two colliding bodies are along the same straight line.

→ Two dimensional collisions : If the two bodies moving in a plane collided and even after collision they are moving in the same plane then such collisions are called two dimensional collisions.

→ Power (P) : It is the rate of doing work.
Power (P) = \(\frac{\text { work }}{\text { time }}\) , unit: Watt
Dimensional formula ML²T^-3
Note : Power can also be expressed as
P = \(\frac{\mathrm{dw}}{\mathrm{dt}}\) = F̅.i\(\frac{\mathrm{dr}}{\mathrm{dt}}\) = F̅.V̅

→ Kilo Watt Hour (K.W.H) : If work done is at a rate of 1000joules/sec continuously for a period of one hour then that amount of work is defined as K.W.H.
K.W.H is taken as 1 unit for supplying electrical energy.

→ Horse power (H.P): 746 watts is called one horse power.
∴ 1 H.P = 746 Watt.

→ Work is the product of force and displacement in the same direction.

• When force and displacement are in same direction work, W = F.S.;
  Unit: joule; D.F. = ML²T^-2
• When force (F̅) is applied with some angle ‘θ’ with displacement vector (S̅) then work W = F̅ . S̅ cos0 or W = F̅ . S̅
• When a variable force is applied on a body, W = ∫F .dx

→ Power (P) is defined as rate of doing work. It is a scalar.
Power, (P) = \(\frac{\text { work }}{\text { time }}=\frac{W}{t}\); Unit: watt;
D.F. = ML²T^-3.

→ Work and energy can be interchanged.
In machine gun problems work done = kinetic energy stored in bullets
i.e. W = n.\(\frac{1}{2}\)mv²

→ In motor and lift problems work done = change in potential energy (mgh)

→ Potential energy, (PE) = mgh ;
Kinetic energy, KE = \(\frac{1}{2}\)mv²
Note : Work, P.E and K.E are scalars. Their units and Dimensional formulae are same.

→ Relation between KE and momentum are :

• Kinetic energy, KE = \(\frac{\mathrm{p}^2}{2 \mathrm{~m}}\)
• momentum, p = \(\sqrt{2 \mathrm{~m} \cdot \mathrm{KE}}\)

→ For a conservative force total work done in a closed path is zero. Ex: Gravitational force.
For a non – conservative force work done in a closed path is not equals to zero.
E : Frictional force.

→ Work energy theorem : Work done by an unbalanced force is equal to the difference of kinetic energy.
W = \(\frac{1}{2}\)mv² – \(\frac{1}{2}\)mu²

→ From Law of conservation of energy change . in potential energy is equal to work done.
∴ W = mgh₂ – mgh₁
1. In elastic collisions Relative Velocity of approach = relative velocity of separation
⇒ u₁ – u₂ = v₂ – v₁
2. In case of elastic collision, Law of conserva-tion of momentum and Law of conservation of Energy are conserved.
⇒ m₁ u₁ + m₂ u₂ = m₁V₁ + m₂ v₂
\(\frac{1}{2}\)m₁u₁² + \(\frac{1}{2}\)m₂u₂ ²=-m₁v₁² + \(\frac{1}{2}\)m₂v₂²

→ In elastic collisions final velocities after collisions are
v₁ = \(\left[\frac{m_1-m_2}{m_1+m_2}\right]\)u₁ + \(\left[\frac{2 m_2}{m_1+m_2}\right]\)u₂
v₂ = \(\left[\frac{2 m_1}{m_1+m_2}\right]\)u₁ + \(\left[\frac{m_2-m_1}{m_1+m_2}\right]\)u₂

→ In perfectly inelastic collision common velocity of the bodies, (v) = \(\frac{\mathrm{m}_1 \mathrm{u}_1+\mathrm{m}_2 \mathrm{u}_2}{\mathrm{~m}_1+\mathrm{m}_2}\)

→ Coefficient of restitution,
e = \(\frac{v_2-v_1}{u_1-u_2}=\frac{\text { Velocity of sepaıation }}{\text { Velocity of approach }}\)

→ For a body dropped from a height ‘h’
a) Velocity of approach, u = \(\sqrt{2 g^{\prime}}\)
b) Coefficient of restitution, e = \(\sqrt{\frac{{\mathrm{h}_2}}{\mathrm{~h}_1}}\)
c) Velocity of separation, v₁ = – e \(\sqrt{2 g h}\)
d) Height attained after nth bounce, h_n = e²ⁿh.

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.