Question

The angular velocity and the amplitude of a simple pendulum is '' and a respectively. At a displacement x from the mean position if its kinetic energy is k and potential energy is p then the ratio of potential energy to kinetic energy is

Answer 1

Energy in Simple Harmonic Motion

The total energy that a particle possesses while performing simple harmonic motion is energy in simple harmonic motion. Take a pendulum for example. When it is at its mean position, it is at rest. When it moves towards its extreme position, it is in motion and as soon as it reaches its extreme position, it comes to rest again. Therefore, in order to calculate the energy in simple harmonic motion, we need to calculate the kinetic and potential energy that the particle possesses.

Kinetic Energy (K.E.) in S.H.M

Kinetic energy is the energy possessed by an object when it is in motion. Let’s learn how to calculate the kinetic energy of an object. Consider a particle with mass m performing simple harmonic motion along a path AB. Let O be its mean position. Therefore, OA = OB = a.

The instantaneous velocity of the particle performing S.H.M. at a distance x from the mean position is given by

v= ±ω √a2 – x2

∴ v2 = ω2 ( a2 – x2)

∴ Kinetic energy= 1/2 mv2 = 1/2 m ω2 ( a2 – x2)

As, k/m = ω2

∴ k = m ω2

Kinetic energy= 1/2 k ( a2 – x2) . The equations Ia and Ib can both be used for calculating the kinetic energy of the particle.