OSCILLATIONS AND WAVES OBJECTIVE 3

Solution to Problem 3

Solution: Pendulum Confined Between Walls

Effect of Inelastic Collisions

The natural amplitude of the pendulum is $\alpha$, and the walls are at angle $\beta$ where $\beta < \alpha$. Normally, the pendulum hits the walls and rebounds.

If a collision with even one wall is inelastic (coefficient of restitution $e < 1$), energy is dissipated during that impact. This causes the amplitude of oscillation to decrease with every cycle.

Long-Term Behavior

As the amplitude reduces, eventually the new amplitude $\alpha’$ will become less than the wall angle $\beta$ ($\alpha’ < \beta$).

Once $\alpha’ < \beta$, the pendulum no longer touches the walls. It oscillates freely between angles $-\alpha’$ and $+\alpha’$.

The period of a free simple pendulum is independent of amplitude (for small angles). Thus, the period settles to the natural period:

$$ T = T_0 = 2\pi\sqrt{\frac{l}{g}} $$

This explains why both (b) and (c) are correct: in both cases, energy loss leads to free oscillation.