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A ball oscillates in a viscous medium. If it is replaced by a ball of the same radius but greater density, how do the time period and the damping coefficient (b/2m) change?

1. Time period increases, but damping coefficient (b/2m) decreases.
2. Time period decreases, but damping coefficient (b/2m) increases.
3. Both time period and damping coefficient (b/2m) increase.
4. Both time period and damping coefficient (b/2m) decrease.

Correct answer: Time period increases, but damping coefficient (b/2m) decreases.

Solution

The drag depends on radius (via Stokes' law), so b is unchanged. Higher density at the same radius means larger mass m. The damping coefficient b/(2m) therefore decreases. The damped angular frequency omega = sqrt(omega0² - (b/2m)²); with smaller (b/2m) and (if restoring is via buoyancy/spring) the period generally increases because the heavier ball responds more slowly and the damping term shrinks. So the time period increases while the damping coefficient b/(2m) decreases.

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