Introduction:
In simple harmonic motion (SHM), a particle oscillates back and forth about an equilibrium position under the influence of a restoring force. The restoring force is proportional to the displacement from the equilibrium position and acts in the opposite direction, trying to bring the particle back to equilibrium. The power supplied by the restoring force to the particle can be calculated using the equation for power.

Calculation of Power:
The power supplied by a force can be calculated using the formula: Power = Force × Velocity. In SHM, the force acting on the particle can be expressed as F = -kx, where k is the force constant and x is the displacement from the equilibrium position. The velocity of the particle can be expressed as v = ω√(A^2 - x^2), where ω is the angular frequency and A is the amplitude of the motion.

Derivation:
To find the maximum power supplied by the restoring force, we need to find the maximum value of the magnitude of the force and the velocity. The maximum value of the force occurs at the extreme points of the motion, where x = ±A. Thus, the maximum force is F_max = kA.

The maximum value of the velocity occurs when x = 0, which corresponds to the equilibrium position. At this point, the velocity is maximum and equal to the angular frequency multiplied by the amplitude. Thus, v_max = ωA.

Calculation of Power:
Using the formulas for maximum force (F_max) and maximum velocity (v_max), we can calculate the maximum power supplied by the restoring force: Power_max = F_max × v_max.

Substituting the values, we get: Power_max = (kA) × (ωA).

Since ω = √(k/m), where m is the mass of the particle, we can rewrite the equation as: Power_max = (kA) × (√(k/m)A).

Simplifying the equation further, we get: Power_max = kA^2 × √(k/m).

Conclusion:
The maximum power supplied by the restoring force to the particle during simple harmonic motion is given by the equation Power_max = kA^2 × √(k/m). This equation shows that the power supplied is directly proportional to the force constant (k), the square of the amplitude (A^2), and the square root of the inverse of the mass (√(k/m)).