A system can do work provided it has energy for example, in lifting a body through a height h, work is done against the conservative force (gravity) by an external agent and thus work is stored in the form of energy. Similarly, in stretching a spring, work done is (1/2)kx^{2} , which is also stored in the spring.

Thus, energy (or stored work) is that physical quantity which enables a system to do work. With reference to sources and forms, energies have the form like heat, light, nuclear, mechanical etc.

In this section, we restrict ourselves to mechanical energy which comprises two forms :

(i) kinetic energy (ii) potential energy

__Kinetic energy & Work energy theorem :__

To get an expression for Kinetic energy, let us take an example shown in figure, in which a block of mass m kept on a rough horizontal surface acted upon by a constant force parallel to the surface. The corresponding F.B.D. is shown in the figure below, which gives

$ \displaystyle \vec{F}-\vec{f_k} = \vec{m a} $ ………(i)

and , N = mg ……….(ii)

Initially while the force is just applied, the block is at the position A and has a velocity v_{0}. The force acted on it for some interval of time ‘ t ‘ so that the block reaches to a position B at a distance x from A.

Now, the work done by the net external force along the surface is

$ \displaystyle W = (\vec{F}-\vec{f_k}) . \vec{x} = \vec{m a}.\vec{x} $

W = m a x

since cosθ = cos0° = 1 , θ being the angle between a^{→} and x^{→}

Therefore, W = ma x ………………….(iii)

Again from kinematical equation for the velocities at A and B, we have

v^{2} = v_{0}^{2} + 2 a x

where v is the velocity of the block at the position B.

$ \displaystyle ax = \frac{v^2 – v_0^2}{2} $

Putting the value of ‘ a x ‘ from above equation , we have

$ \displaystyle W = m (\frac{v^2 – v_0^2}{2}) $ …..(iv)

The work done by the other two forces in F.B.D. for the displacement x^{->} are zero because N^{→} .x^{→} = 0 and also mg^{→} . x^{→} = 0.

The equation (iv) has two important consequences.

(a) It establishes an important theorem related to work and energy, and

(b) It gives the concept of kinetic energy.

Firstly if v_{0} = 0 i.e. initially the block is at rest, then

$ \displaystyle W = \frac{1}{2}m v^2 $ …(v)

which implies kinetic energy is the energy possessed by the body in motion. If speed is zero, then kinetic energy is also zero.

Secondly , $ \displaystyle \frac{1}{2}m v_0^2 = K_i $ (let), initial kinetic energy,

and $ \displaystyle \frac{1}{2}m v^2 = K_f $ , called final kinetic energy for the interval of time ‘ t ‘ under consideration in the example.

Therefore equation (iv)can be written as

W = K_{f} − K_{i} …(vi)

Thus equation (vi) can be explained as the net work done by the external forces on a system gives the change in kinetic energy of the system.This itself is the Work Energy Theorem.

Exercise : Two masses M and m are connected by a light inextensible string which passes over a small pulley as shown in the diagram. If the mass m is moving downward with a velocity v when the string makes an angle of 45° with the horizontal, find the total K.E. of the two masses. Assume that the mass M moves horizontally.

Exercise : Shown in the figure is an inextensible light string that connects two bodies of masses M and m. The string passes over the pulleys P_{1}, P_{2} and P_{3} so that the body M moves down on a smooth horizontal surface and the body m moves down. If the instantaneous speed of M is v, find the Kinetic Energy of the system.

__Kinetic Energy in Terms of Momentum__

With reference to the adjacent figure, a body of mass m moving on a surface with a velocity v has a momentum ,

P = m v ……………(i)

The kinetic energy of the same body is,

$ \displaystyle K = \frac{1}{2}m v^2 $

$ \displaystyle K = \frac{1}{2 m}m^2 v^2 $ …….(ii)

Therefore from equations (i) and (ii), we have

$ \displaystyle K = \frac{p^2}{2 m} $ …….(iii)

### Potential energy :

Potential energy of any body is the energy possessed by the body by virtue of its position or the state of deformation. With every potential energy there is an associated conservative force.

The potential energy is measured as the magnitude of work done against the associated conservative force.

For example:

(i) To place an object at any point in gravitational field work is to be done against gravitational field force.

The magnitude of this work done against the gravitational force gives the measure of gravitational potential energy of the body at that position, which is U = mgh. Here h is the height of object from the reference level .

(ii) The magnitude of work done against the spring force to compress it gives the measure of elastic potential energy, which is

$ \displaystyle U = \frac{1}{2}k x^2 $

(iii) A charged body in any electrostatic field will have electrostatic potential energy.