Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

As an experienced tutor registered on UrbanPro, I'm here to shed light on a fundamental principle in classical mechanics. Let me demonstrate why, in the case of a particle undergoing linear Simple Harmonic Motion (SHM), the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

Firstly, let's delve into the nature of SHM. In linear SHM, a particle oscillates back and forth along a straight line, with its acceleration proportional and opposite to its displacement from a fixed equilibrium point. This leads to a sinusoidal motion characterized by a restoring force.

Now, to prove the equality of average kinetic and potential energies over a period, we must understand the expressions for kinetic and potential energies in SHM.

The kinetic energy (KE) of a particle is given by KE=12mv2KE=21mv2, where mm is the mass of the particle and vv is its velocity.

In SHM, velocity varies sinusoidally with displacement, reaching maximum at the equilibrium point and minimum at the extremities. Thus, the average kinetic energy over a period can be represented by 12mvmax221mvmax2, where vmaxvmax is the maximum velocity.

On the other hand, the potential energy (PE) of a particle undergoing SHM is given by PE=12kx2PE=21kx2, where kk is the spring constant and xx is the displacement from equilibrium.

In SHM, potential energy also varies sinusoidally, reaching maximum at the extremities and minimum at the equilibrium point. Hence, the average potential energy over a period can be represented by 12kxmax221kxmax2, where xmaxxmax is the maximum displacement.

Now, in SHM, the maximum displacement (xmaxxmax) and maximum velocity (vmaxvmax) occur at the same points in the motion, namely at the extremities. This is due to the relationship between displacement, velocity, and acceleration in SHM.

Since kinetic and potential energies both reach their maxima at the extremities, the average kinetic energy over a period equals the average potential energy over the same period.

Thus, in the realm of linear SHM, the balance between kinetic and potential energies is not only a theoretical construct but a practical reality, demonstrating the elegant harmony in the dynamics of oscillatory motion. If you're keen on further exploring such concepts or need assistance with related problems, don't hesitate to reach out. Remember, UrbanPro is your gateway to the best online coaching and tuition experiences!