Total Mechanical Energy of the System

Energy is a scalar quantity. The amount of energy stored in an object decides its capacity to work. Therefore, we have different types of energies in the form of potential energy, kinetic energy, chemical energy, thermal energy, mechanical energy, solar energy, etc. Speaking of mechanical energy, it is a sum of the potential as well as kinetic energies of an object. The type of energy working in an object is decided based on it being stationary or in motion. The amount of potential and kinetic energy in objects may vary, depending on their position in motion in the system. 

Mechanical Energy

In terms of an equation, total mechanical energy can be written as,

Emechanical = U + K

where U is the potential energy, and K is the kinetic energy. The potential energy of an object is calculated based on the mass of the object (m), gravitational acceleration acting on it (g), and the height at which the object is placed above the ground (h). 

Therefore, the formula of potential energy is,

U = m gh

Kinetic energy (K) of an object deals with the energy of an object in motion. It is the half of the mass of the object (m) and the square of the velocity of the object in motion (v)

K = 1/2 m v2

If conservative forces are acting on a body, the mechanical energy of the body remains constant. The difference between conservative and non-conservative forces is that conservative forces are independent of the path taken for an object to go from one point to another, while non-conservative forces are dependent on the path. 

Conservation of Mechanical Energy

We know that energy cannot be created nor destroyed but can only be converted from one form to another. This is exactly what happens in conservation of mechanical energy. Any system free of non-conservative forces like friction or air resistance has constant energy. In real life, non-conservative forces acting on the object have smaller effects; therefore, it is negligible. So, this law holds approximately true in all cases. 

Mechanical energy is conserved because the potential energy and kinetic energy of the object in the initial phase are almost equal to the potential energy and kinetic energy of the object in the final phase. 

For understanding the conservation of energy in detail, let us consider a ball in motion placed at height ‘h’ at point A. The ball has potential energy due to the height it is placed at and the gravitational pull acting on it towards the ground.  

When this ball is released from point A, we know that the ball will fall and its potential energy will be converted into kinetic energy. For understanding the total mechanical energy, we need to consider points B and C. Remember that during this process, we did not consider air resistance as it is a non-conservative force, and we are dealing with an isolated system, assuming that energy is conservative at all points. 

At a height, the ball has greater potential energy because the velocity of the ball increases due to the action of gravity. When the ball is released from point A, it reaches point B, where it falls to a point where its height from the ground becomes ‘x’. Therefore, the height from A to B becomes h – x.

Let us consider each point individually to understand conservation of mechanical energy in detail,

At point A,

Potential Energy,

U = m gh

This is the same as P.E = mgh, where

m = mass of the ball

g = gravitational acceleration

h = height from the ground

The Kinetic Energy is written as,

K.E = 1/2 m v2

But the ball is stationary at point A, thus

K.E = 0

Thus total mechanical energy,

Emechanical = P.E + K.E

Emechanical = P.E + 0

Thus, 

Emechanical = P.E 

Emechanical = mgh

At point B,

The ball is at a height of ‘x’ above the ground, therefore its potential energy,

P.E = mgx

The Kinetic Energy is written as,

K.E = 1/2 m v2 .  .  . (1)

For finding the velocity at this point, let us consider second equation of motion,

vf2 = vi2 + 2gS,

where 

vf = velocity at point B

vi = initial velocity

g = gravitational acceleration

S = distance the ball travelled from points A to B

We know that vi = 0, thus

vf2 = 0 + 2gS

vf2 = 0 + 2g(h – x) 

vf2 = 2g(h – x) .  .  . (2)

Substituting the formula of derived velocity from equation (2) in (1)

K.E = 1/2 m 2g(h – x)

K.E = mgh – mgx

The total mechanical energy at point B,

Emechanical = P.E + K.E

Substituting the derived values of P.E and K.E,

Emechanical = mgx + (mgh – mgx)

Emechanical = mgh

At point C,

The ball at point C has not touched the ground, it is just above the ground,

Here, P.E = 0

The Kinetic Energy is written as,

K.E = 1/2 m v2 .  .  . (1)

For finding the velocity at this point, let us consider second equation of motion,

vf2 = vi2 + 2gS,

where, 

vf = velocity at point C

vi = initial velocity

g = gravitational acceleration

S = distance the ball travelled from points A to C

We know that vi = 0, thus,

vf2 = 0 + 2gh

vf2 = 2gh  .  .  . (2)

Substituting the formula of derived velocity from equation (2) in (1)

K.E = 1/2 m 2gh

K.E = mgh

The total mechanical energy at point C,

Emechanical = P.E + K.E

Substituting the derived values of P.E and K.E,

Emechanical = 0 + mgh

Emechanical = mgh

Observe that at all points A,B, and C, total mechanical energy is,

Emechanical = mgh

Therefore, it is said that total mechanical energy is conserved.

Conclusion

The total mechanical energy in an isolated system is constant at any given point of motion. Without non-conservative forces, the mechanical energy is constant. The energy doesn’t leave the system. When non-conservative forces are involved like friction, and air drag, the system loses mechanical energy with each swing or motion. An ideal motion is very rare in everyday life. There is a bit of loss of mechanical energy, which is negligible. Therefore, the conservation of mechanical energy is applicable in most cases of motion. The loss in mechanical energy is often obtained as a gain in heat energy. So, the energy is not lost, just converted from one form to another.