A uniform, smooth, thin rod of mass M and length L floats freely in deep space (weightlessness). A tiny bead of mass m (m << M) is threaded onto the rod. Find the period T of small oscillations of the bead about the centre of the rod. Gravitational constant is G.

pi * sqrt(L³ / (2GM))
pi * sqrt(L³ / GM)
2pi * sqrt(L³ / GM)
2pi * sqrt(L³ / (8GM))

Correct answer: 2pi * sqrt(L³ / (8GM))

Solution

The restoring force on the bead at displacement x from the centre is F = m * g_rod(x). For small x, g_rod = 8GMx/L³, giving omega² = 8GM/L³ and T = 2pi/omega = 2pi * sqrt(L³/(8GM)).

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