Frequency of Kinetic Energy in Simple Harmonic Motion

In simple harmonic motion (SHM), the kinetic energy (KE) of a particle varies periodically as the particle oscillates. If the frequency of the particle is f, what is the frequency of the kinetic energy?

Understanding Simple Harmonic Motion

When a particle oscillates in SHM, both its position and velocity vary sinusoidally with time. The velocity of the particle can be expressed as:

v(t) Aωcos(ωt φ)

where:

A is the amplitude of the motion, ω 2πf is the angular frequency, φ is the phase constant.

The kinetic energy (KE) of the particle is given by:

KE (1/2)mv^2

Substituting the velocity expression into the kinetic energy formula, we get:

KE(t) (1/2) m A^2 ω^2 cos^2(ωt φ)

The term cos^2(ωt φ) varies with time, and it has a frequency of 2f because the cosine function squared oscillates twice as fast as the original cosine function.

Frequency of Kinetic Energy Derivation

Therefore, the frequency of the kinetic energy of the particle in simple harmonic motion is:

Frequency of KE 2f

Here’s a step-by-step derivation of this result using a simple pendulum as an example:

Assume a simple pendulum with zero friction loss and zero air resistance. The string is light and inextensible.

The kinetic energy (KE) of the bob is zero at the extreme positions (E1 and E2) and maximum at the centre position (C).

Starting from the extreme position E1 and released with zero initial speed, the pendulum moves to C, then to E2, back to C, then again to E1, and so on. This represents one cycle (E1-C-E2-C-E1).

During this cycle, the KE goes from zero at E1 to maximum at C, then again to zero at E2, and back to maximum at C. Conversely, the potential energy (PE) goes from maximum at E1 to minimum at C, then to maximum at E2.

As you can see, the KE goes through two cycles in the time it takes to complete one cycle of the pendulum.

If the pendulum frequency is f, the KE frequency is 2f, since KE completes two cycles for every cycle of the pendulum.

To visualize, if we define the position displacement as x(t) ACos(ωt) where ω 2πf, the velocity at time t is given by v(t) -ωASin(ωt).

The kinetic energy expression then becomes:

KE (1/2)mv^2 (1/2)mω^2A^2 Sin^2(ωt)

Using the trigonometry identity sin^2(θ) (1 - cos(2θ))/2, we can write:

KE (1/2)mω^2A^2 [1 - cos(2ωt)]/2 (1/4)mw^2A^2 - (1/4)mw^2A^2 cos(4πft)

Here, only the cos(4πft) term is time-dependent and determines the frequency, showing clearly that the frequency of KE is 2f.

Conclusion

The frequency of the kinetic energy in simple harmonic motion is twice the frequency of the oscillator. This is a fundamental property of SHM and can be explained both visually and mathematically.

References

Understanding SHM and the relationship between KE, velocity, and position. Using trigonometric identities and calculus for rigorous derivation.