A solid cone A of mass m = 3.2 kg and half-angle alpha = 10 deg rolls without slipping over a fixed conical surface B, with its apex O held fixed. The cone's centre of mass lies at the same height as O, a distance l = 17 cm away. The cone axis sweeps round with angular velocity omega. (a)

Question

A solid cone A of mass m = 3.2 kg and half-angle alpha = 10 deg rolls without slipping over a fixed conical surface B, with its apex O held fixed. The cone's centre of mass lies at the same height as O, a distance l = 17 cm away. The cone axis sweeps round with angular velocity omega. (a) Find the friction force on cone A when omega = 1.0 rad/s. (b) For coefficient of friction k = 0.25 between the surfaces, find the omega values for which rolling occurs without sliding.

Accepted Answer

(a) F = 0.06 N; (b) omega <= 6 rad/s. The standard Irodov result gives F = (4/3) m omega² l tan(alpha) ~= 0.06 N, and the no-slip condition F <= k m g yields omega <= sqrt(3 k g / (4 l tan alpha)) ~= 6 rad/s.

Solution

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