Question

Prove that the motion of the simple pendulum obeys the conservation of energy law (the sum of the kinetic and potential energies is a constant of the motion). Hint: The kinetic energy is (m / 2)(ds / dt)² , where s = LX is the length of arc through which the pendulum swings and the potential energy is mgL[1 - cos(X)].

Answer 1

We know that the motion of the bob of a simple pendulum is simple harmonic motion. Here we have to prove that the energy is conversed during the motion of pendulum.
Proof: Consider a simple pendulum as shown in the diagram.

Energy Conservation At Point ‘A’

At point A’ velocity of the bob of simple pendulum is zero. Therefore, K.E. at point ‘A’ = 0. Since the bob is at a height (h), Therefore, P.E. of the bob will be maximum. i.e.
P.E. = mgh.
Energy total = K.E. + P.E
Energy total = 0 + mgh
Energy total = mgh
This shows that at point A total energy is potential energy.

Energy Conservation At Point ‘M’

If we release the bob of pendulum from point ‘A’, velocity of bob gradually increases, but the height of bob will decreases from point to the point. At point ‘M’ velocity will become maximum and the height will be nearly equal to zero.
Thus ,
K.E. = maximum = 1/2mV² but P.E. = 0.
Energy total = K.E. + P.E
Energy total = 1/2mV² + 0
Energy total = 1/2mV²
This shows that the P.E. at point is completely converted into K.E. at point ‘M’.

Energy Conservation At Point ‘B’

At point M the bob of Pendulum will not stop but due to inertia, the bob will moves toward the point ‘B’. As the bob moves from ‘M’ to ‘B’, its velocity gradually decreases but the height increases. At point ‘B’ velocity of the bob will become zero.
Thus K.E. at point ‘B’ = 0 but P.E. = max.
P.E. = mgh.
Energy total = K.E. + P.E.
Energy total = 0 + mgh
Energy total = mgh
This shows that at point B total energy is again potential energy.

CONCLUSION: Above analysis indicates that the total energy during the motion does not change. I.e. the motion of the bob of simple pendulum is according to the law of conservation of energy.