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Two spheres of identical diameter but different masses hang from strings of equal length and are set oscillating as pendulums in a real (viscous) medium. After being displaced from equilibrium, which sphere has the greater period of oscillation, and which has the greater logarithmic decrement (faster relative damping)?

1. The heavier sphere has the larger period and the greater logarithmic decrement
2. The lighter sphere has the larger period and the greater logarithmic decrement
3. The lighter sphere has the larger period but the smaller logarithmic decrement
4. The heavier sphere has the larger period but the smaller logarithmic decrement

Correct answer: The lighter sphere has the larger period and the greater logarithmic decrement

Solution

Both spheres have the same diameter, so the viscous drag force (resistance constant) is the same, but the lighter sphere has a larger damping-per-unit-mass (b/m). The damped angular frequency omega_d = sqrt(omega0² - (b/2m)²) is smaller for larger b/m, so the lighter sphere has a longer period. The logarithmic decrement = (b/2m)*T also increases with b/m (and with the longer period), so the lighter sphere has the greater logarithmic decrement. Hence the lighter sphere has both the larger period and the greater logarithmic decrement.

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