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A block of mass M is attached to a thread wound around a pulley that is free to rotate about a fixed horizontal axis. The pulley is a uniform conducting disc of radius L placed in a horizontal magnetic field B, and is connected to a resistance R (forming a Faraday-disc generator). The block is released from rest. Find its terminal velocity.

1. 4*M*g*R/(B²*L²)
2. 3*M*g*R/(4*B²*L²)
3. 2*M*g*R/(B²*L²)
4. 3*M*g*R/(2*B²*L²)

Correct answer: 4*M*g*R/(B²*L²)

Solution

The Faraday disc generates emf e = (1/2)*B*omega*L². With v = omega*L, e = (1/2)*B*v*L. Current i = e/R = B*v*L/(2R). The magnetic force on the disc current acts at effective radius L/2 giving a retarding torque; at terminal velocity the gravitational torque M*g*L equals the magnetic braking torque. Working through the disc-generator torque balance yields v_terminal = 4*M*g*R/(B²*L²).

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