Question

Give the expression for P.E &K.E in S.H.M

Answer 1

Hey!! Here is your Answer!!

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.

Potential energy is the energy possessed by the particle when it is at rest. Let’s learn how to calculate the potential energy of a particle performing S.H.M. Consider a particle of mass m performing simple harmonic motion at a distance x from its mean position. You know the restoring force acting on the particle is F= -kx where k is the force constant.

Now, the particle is given further infinitesimal displacement dx against the restoring force F. Let the work done to displace the particle be dw. Therefore, The work done dw during the displacement is

dw = – fdx = – (- kx)dx = kxdx

Therefore, the total work done to displace the particle now from 0 to x is

∫dw= ∫kxdx = k ∫x dx

Hence Total work done = 1/2 K x2 = 1/2 m ω2x2

The total work done here is stored in the form of potential energy.

Therefore Potential energy = 1/2 kx2 = 1/2 m ω2x2

Hope it helps you!!