Question

11VVI
3. Derive the equation for the kinetic energy and P.E of a simple harmonic oscillation
and show that the total energy of particle in simple harmonic motion is constant at any
point of a path?​

Answer 1

Answer:

Explanation:

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.

Learn how to calculate Velocity and Acceleration in Simple Harmonic Motion.

Potential Energy(P.E.) of Particle Performing S.H.M.

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

Equations IIa and IIb are equations of potential energy of the particle. Thus, potential energy is directly proportional to the square of the displacement, that is P.E. α x2.

Learn the Difference between Periodic and Oscillatory Motion.

Total Energy in Simple Harmonic Motion (T.E.)

The total energy in simple harmonic motion is the sum of its potential energy and kinetic energy.

Thus, T.E. = K.E. + P.E.  = 1/2  k ( a2  – x2) + 1/2 K x2 = 1/2 k a2

Hence, T.E.= E = 1/2 m ω2a2

Equation III is the equation of total energy in a simple harmonic motion of a particle performing the simple harmonic motion. As ω2 , a2 are constants, the total energy in the simple harmonic motion of a particle performing simple harmonic motion remains constant. Therefore, it is independent of displacement x.

As ω= 2πf , E= 1/2 m ( 2πf )2a2  

∴ E=  2mπ2f 2a2

As 2 and π2 constants, we have T.E. ∼ m, T.E. ∼ f 2, and T.E. ∼ a2

Thus, the total energy in the simple harmonic motion of a particle is:

Directly proportional to its mass

Directly proportional to the square of the frequency of oscillations and

Directly proportional to the square of the amplitude of oscillation.

The law of conservation of energy states that energy can neither be created nor destroyed. Therefore,  the total energy in simple harmonic motion will always be constant. However, kinetic energy and potential energy are interchangeable. Given below is the graph of kinetic and potential energy vs instantaneous displacement.

energy in simple harmonic motion