Work, Energy, Power |

 Work

Definition:

Work is said to be done on a body when an external force displaces a body in the direction of applied force.

S.I. Unit: Joule

1 joule = (1newton) (1 meter) or 1 J = 1 N . m

C.G.S unit = erg

1 J = $10^7$ erg

Dimension: $ML^2T^{-2}$

Mathematical Definition

Work (W) = $\overrightarrow{F} \cdot \overrightarrow{s}$

$\to$ Mathematically Work is the result of a dot product of two vectors i.e. force and displacement.

$\to$ Work is a scalar quantity.

Conditions

Work (W)= $FS \cos \theta$

$\to$ $\theta$ is the angle between force and displacement.

$\to$ $\theta= 0^{\circ}, W= FS$ displacement is in direction of force, the work done is maximum.

$\to$ $\theta= 90^{\circ} W=0$ displacement is perpendicular direction of force, the work done is zero.

$\textbf{For example}$

I . Work done by the centripetal force in displacing a particle alone a circular path is zero.

ii . Work done by centripetal force ( i.e gravitational pull ) in revolving satellite around the earth is zero.

iii. When a person carrying a load on the head moves over a horizontal work done against the gravitational force is zero.

iv. When a car moves with a uniform speed over a frictionless road, work done is zero.

$\to 180^{\circ} >\theta>90^{\circ} W= -ve$ work done is negative.

$\to$ Work done by friction and when the body is thrown up the work by gravitational pull is negative.

Work Done by Variable Force

If the force is the function of displacement, then, the work done can be found using integration

$W = \displaystyle{\int_{x=a}^{x=b}} F_xdx$

Energy

Definition:

The capacity of a body to do work is called energy.

S.I. Unit: Joule

C.G.S Unit: erg

Dimension: $ML^2T^{-2}$

Mechanical Energy

The energy possessed by the body due to its motion or position is called mechanical energy. They are kinetic energy and mechanical energy.

Kinetic Energy

The energy possesed by the body due to its motion is called kinetic energy.

$E = \dfrac{1}{2} mv^2 = \dfrac{p^2}{2m}$

$m$ is the mass of the body, $v$ is the velocity of the body and $p$ is the momentum of the body.

Hence For constant momentum, K.E is inversely proportional to the mass of the body.

Potential Energy

The energy possessed by a body due to its position or configuration is called potential energy.

$\to$ Gravitational Potential Energy

For the height h above the surface of the earth: $P.E. = - \dfrac{GMm}{R+h} = - \dfrac{mgR^2}{R+h}$, Where M = Mass of the earth, r = Radius of the earth, h= height above the earth

The change in potential energy when a body is taken from surface to height h is $\Delta PE = -\dfrac{mgR^2}{R+h} + \dfrac{mgR^2}{R} = \dfrac{mgRh}{R+h}$

If $h<< R$ $\Delta PE = mgh$

If $h>> R$ $\Delta PE = mgR$

$\to$ Elastic Potential Energy

The energy stored in the spring when it is stretched by a force (F) producing an extension of (x) is given as

$U = \dfrac{1}{2} Fx = \dfrac{1}{2} kx^2$

$\to$ KE is never negative but PE can be positive, negative and zero.

Work-Energy Theorem

$\to$ The work and energy are related and are equivalent.

$\to$ Work = Change in KE = $\dfrac{1}{2} mv^2 - \dfrac{1}{2} mu^2$

Conservation of energy

$\to$ The work-energy theorem is based on the conservation of energy.

$\to$ It states that energy can neither be created nor be destroyed but can be transferred from one form to another.

K.E. + P.E. = Constant

Work Done by Conservative and Non-Conservative Forces

$\to$ Conservative Forces

The work done by a conservative force is independent of path.

It depends on the final and initial state.

The work done in the closed path is zero.

Total mechanical energy is conserved

eg. Gravitational force, elastic force etc.

$\to$ Conservative Forces

The work done by a nonconservative force is dependent on the path taken.

The work done in the closed path is not zero.

Total mechanical energy is not conserved

eg. frictional force, viscous force etc.

Work and Energy

$\to$ Whenever work is done by a body, the work is + ve and energy decreases.

$\to$ Whenever work is done on a body, work is - ve, and its energy increases.

Power

Definition

The work done per unit time is known as power.

Power = $\dfrac{\text{Work Done}}{\text{Time Taken}}$

It is a scalar quantity.

S.I. Unit is Watt (W)

CGS Unit is ergs/s

Practical Unit: Horse Power (HP), 1 HP = 746 Watt

Dimension: $ML^2T^{-3}$

1 hp = 746 W = 0.746 kW hp= Horse power

Mathematical Definition

Work (W) = $\overrightarrow{F} \cdot \overrightarrow{v} = Fv \cos \theta$

$\to$ Mathematically Power is the result of a dot product of two vectors i.e. force and velocity.

$\to$ Power is a scalar quantity.

$\to$ It is also defined as rate of change of energy $\dfrac{\Delta E}{t}$

Collision

The interaction between two or more bodies for a short time after which their kinetic energy and momentum are changed is called collision.

Elastic Collision:

$\to$ Kinetic energy and linear momentum are conserved.

$\to$ The force creating elastic collision are conservative in nature.

$\to$ Mechanical Energy is conserved.

Inelastic Collision

A collision in which the total kinetic energy after the collision is less than before the collision is called an inelastic collision i.e. K.E is not conserved.

Linear momentum is conserved.

• Are there any examples of perfectly inelastic collisions?

$\to$The ballistic pendulum is a practical device in Qwhich an inelastic collision takes place. Until the advent of modern instrumentation, the ballistic pendulum was widely used to measure the speed of projectiles.

(Two bodies stick after perfect inelastic collision)

Perfectly inelastic collision: If the entire K.E. of bodies is converted to another form of energy after collision.

Laws of Collision

The velocity of separation between particles after the collision is directly proportional to the velocity of approach of these particles before the collision.

$u_1-u_2$ = velocity of approach

$v_2 – v_1$ = velocity of separation

According to the law of collision

$v_2 - v_1 \propto ( u_1 - u_2 )$

$v_2 - v_1 = e ( u_1 - u_2 )$

Where e is proportionality constant depending on the nature of the collision and is called the coefficient of restitution.

It is unitless and dimensionless.

Coefficient of restitution

The ratio of the velocity of separation after a collision to the velocity of approach before the collision is known as the coefficient of restitution.

Coefficient of restitution ( e ) = $\dfrac{\text{velocity of separation}}{\text{velocity of approach}} = \dfrac{(v_2 - v_1 )}{(u_1 - u_2 )}$

$\to$ For elastic collision, $e = 1$

$\to$