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What is the law of conservation of mechanical energy? Prove that mechanical energy is conserved for a freely falling body.

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Law of Conservation of Mechanical Energy

‘The mechanical energy of any body or a system i. e., the sum of its kinetic energy and potential energy is constant in the presence of conservative forces.’ This is called the law of conservation of mechanical energy.

We clearly know that when conservative forces one applied on a system, its configuration changes and the kinetic energy of this system changes by AK. According to the definition of conservative force there should be a change equal in magnitude in the potential energy of the system but in opposite direction. Due to which the sum of both the changes becomes zero, i. e.,
\(\begin{aligned} \Delta U &=-\Delta K \\ \text { or } \quad \Delta U+\Delta K &=0 \\ \text { or } \quad U+K &=\text { constant } \end{aligned}\)
= constant ………………………(1)

Hence, in the presence of the conservative forces, the change in the kinetic energy (K) is equal and opposite to the change in the potential energy (U). Due to this the sum of the kinetic energy and potential energy is always constant.

This law is not true for non-conservative forces like-frictional force because some part of the non-conservative forces changes into the sound, heat, light and other types of energies.

Some Examples of Law of Conservation of Mechanical Energy:
Example: Free Falling Body : Fig shows a free falling body of mass m in the effect of gravitational force (conservative force). The height of point A from the earth’s surface is h; and the velocity of the object at A is zero (u = 0). Calculate the total mechanical energy of the system at A

When the body falls freely for a height × and reaches B then

∴ Total mechanical energy B

It is clear from the eqs (2), (3) and (4) that;
EA = EB = EC = mgh

Clearly, as the body falls, it’s potential energy decrease uniformly such that it’s total mechanical energy remains constant (mgh) at all point. Thus total mechanical energy is conserved during free fall of a body. Fig. 5.8 shows the variation of KE and PE and the total mechanical energy (TME) with height

Example 2:
A showing simple pendulum, is an also example of conservation of energy:

This is because a swinging simple pendulum is a body whose energy can either be potential or kinetic, or a mixture of potential and kinetic but it’s total energy at any instant of time remains the same.

When the pendulum bob is at position B it has only potential energy (but no kinetic energy) As the bob starts moving down from position B to position Ai it’s potential energy goes on decreasing but it’s kinetic energy goes on increasing when the bob reaches the centre position A it has only kinetic energy (but no potential energy)

As the bob goes from position A towards position C, it’s kinetic energy goes on decreasing but its potential energy on increase.
On reaching the extreme position C, the bob stops for a very small instant of time, so at position C, the bob has only potential energy.

+2 votes
by (1.7k points)
It may be shown that in the absence of external frictional force the total mechanical energy of a body remains constant.
Let a body of mass m falls from a point A, which is at a height h from the ground as shown in fig.

At A,

Kinetic energy kE = 0
Potential energy Ep = mgh
Total energy E = Ep + Ek = mgh + 0= mgh

During the fall, the body is at a position B. The body has moved a distance x from A.

At B,

velocity v2 = u2 + 2as

applying, v2 = 0 + 2ax = 2ax

Kinetic energy Ek = 1/2 mv2 = 1/2 m x 2gx = mgx
Potential energy Ep = mg (h – x)
Total energy E = Ep + Ek = mg (h-x) + mgx = mgh – mgx + mgx= mgh

If the body reaches the position C.

At C,

Potential energy Ep = 0
Velocity of the body C is
v2 = u2 + 2as
u = 0, a = g, s = h
applying v2 = 0 + 2gh = 2gh

kinetic energy Ek =1/2 mv2=1/2 m x 2gh= mgh

Total energy at C
                      E = Ep + Ek
                     E = 0 + mgh
                     E = mgh
Thus we have seen that sum of potential and kinetic energy of freely falling body at all points remains same. Under the force of gravity, the mechanical energy of a body remains constant.

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