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 = Kf -Ki
→ 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, hn = e2nh.