Conservation of Energy in the motion of simple pendulum
When a simple pendulum oscillates with simple harmonic motion, it gains some kinetic energy because of this type of motion. The motion of a pendulum is a classic example of mechanical energy conservation. A pendulum consists of a mass (known as a bob) attached by a string to a pivot point. As the pendulum moves it sweeps out a circular arc, moving back and forth in a periodic fashion. Besides, a restoring force always acts on the bob of the pendulum in the opposite direction of displacement. As a result, work is done during displacement. For this reason, the particle has also some potential energy as well. If there is no frictional force or similar type of dissipating force acting, total energy of the simple pendulum will remain constant.
Suppose, mass of the bob is in and at any moment its displacement and velocity are respectively x and v. So, at that moment kinetic energy of the bob i.e., simple pendulum is,
Ek = ½ MV2
Since, x = A sin (ωt + δ)
v = dx/dt = d/dt (A sin ωt + δ) = Aω cos (ωt + δ)
Ek = ½ mA2ω2 cos2 (ωt + δ)
Again, ω2 = K/m
So, Ek = ½ mA2 K/m cos2 (ωt + δ)
then, Kinetic energy, KE = ½ KA2 cos2 (ωt + δ) … … … (1)
From equation (1) it is seen that as the maximum value of cos2 (ωt + δ) is 1, hence the maximum kinetic energy is ½ KA2. During motion kinetic energy of the particle may change from zero to this maximum value.
Taking δ = 0, change of kinetic energy has been shown in figures 1 and 2.