Principle of Conservation of Energy

October 19, 2020

The energy is a conserved quantity. There won’t be any change in the energy of an isolated system.
The energy can neither be created nor can it be destroyed. It can only be changed from its one form to another form.
For example: if we throw a stone upward with certain velocity, initially, it has maximum velocity and hence maximum kinetic energy.
As it moves up, its velocity goes on At a certain height, its velocity becomes zero, and hence its kinetic energy becomes zero.
At the same time, the stone gains height as it moves up.
As the stone gains height, its potential energy increases. Hence, it is clear that when a stone is thrown up, there is continuous decrease in kinetic energy and increase in the potential energy.
However, the total energy of the stone remains the same. This is the principle of conservation of energy.
According to principle of conservation of energy, the energy can neither be created nor be destroyed but can be changed from one form to another form.

Image Source: nuclearpower Image Source: Braincart

Energy conservation in free fall

Consider a body of mass m is initially at a point A which is at a height h from the ground as in the given figure. Let the body fall freely under gravity so that the acceleration of the body is g– acceleration due to gravity. After a certain time, the body reaches a point B which is at height (h-x) from the ground.

As the body falls down, its velocity increases. Finally, the body strikes the ground C with velocity v.

For the point A

When the body is at A, velocity (u) = 0 and the height from the ground (h) = h.

Hence, the kinetic energy of the body (K.E)_a = ½ mu² = 0

And the potential energy of the body is (PE)_a = mgh

The total mechanical energy of body at point A is given by

E_a= (K.E)_a + (P.E)_a = 0 + mgh = mgh

∴ E_a = mgh …………………………………………………………. (i)

For the point B

When the body is at point B on its way down, we have initial velocity (u) = 0

Final velocity (v) = v_b

Distance travelled = AB=AC – BC = h – (h-x) = x

We have, v² = u² + 2gh (formula)

Or, v_b² = 0 + 2gx = 2gx

∴ v_b² = 2gx

Now, the kinetic energy of the body at point B is

(K.E)_b= ½ mv_b²= ½ m(2gx) = mgx

Also, the potential energy of the body at point B is

(P.E)_b = mgh = mg(h-x)

Therefore, the total mechanical energy of the body when it is at the point B is given by

E_b = (K.E)_b + (P.E)_b = mgx + mg(h-x) = mgx + mgh – mgx = mgh

∴ E_b= mgh…………………………………………………………… (ii)

For the point C

When the body is at point C, we have, for the motion of the body

Initial velocity (u) = 0

Final velocity (v) = v_c

Distance travelled = h

We have, v²= u² + 2gh

or, v_c²= 0 + 2gh =2gh

∴v_c²= 2gh

Hence, when the body is at point C, the kinetic energy is given by,

(K.E)_c = ½ mv_c² = ½ m(2gh) = mgh

Similarly, the potential energy of the body at the point C is given by,

(P.E)_c = mg(height of point C) = mg × 0 = 0

Hence, the total mechanical energy of the body when it is at the point C is given by,

E_c = (K.E)_c + (P.E)_c = mgh + 0 = mgh

∴ Ec = mgh………………………………………………………… (iii)

From equation (i), (ii) and (iii), it is seen that the total mechanical energy of a freely falling body when it is at point A is equal to the energy when it is at B or C.

This implies that the total mechanical energy of a freely falling body is conserved.

In other words, the principle of conservation of energy holds true in this case of a freely falling body.

If a graph is plotted between the total mechanical energy of a freely falling body and its height from the ground, a curve is obtained as in figure below.