Energy Exchange in SHM

In simple harmonic motion, the total mechanical energy $E = \frac{1}{2}kA^2$ remains constant throughout the motion.

Potential Energy

The elastic potential energy at displacement $x$ from equilibrium is

$$U(x) = \frac{1}{2}kx^2$$

This parabolic function equals zero at $x = 0$ and reaches maximum at $x = \pm A$.

Kinetic Energy

By conservation of energy, kinetic energy equals total energy minus potential energy:

$$K(x) = E - U(x)$$

Substituting the expressions for $E$ and $U(x)$ gives

$$K(x) = \frac{1}{2}kA^2 - \frac{1}{2}kx^2 = \frac{1}{2}k(A^2 - x^2)$$

Kinetic energy is maximum at $x = 0$ and zero at $x = \pm A$.

Verification

Adding both energy components at any position:

$$K(x) + U(x) = \frac{1}{2}k(A^2 - x^2) + \frac{1}{2}kx^2 = \frac{1}{2}kA^2 = E$$

This confirms energy conservation.