The uniform solid cylinder of mass m and radius r rolls without slipping during

Question

The uniform solid cylinder of mass m and radius r rolls without slipping during its oscillation on the circular surface of radius R. If the motion is confined to small amplitudes theta = theta 0, determine the period tau of the oscillations. Also determine the angular velocity omega of the cylinder as it crosses the vertical centerline. (Caution: Do not contuse omega with theta or with omega n as used in the defining equations. Note also that theta is not the angular displacement of the cylinder.)

Explanation / Answer

to find the angular velocity of the cylinder use relation (1)

from above equation of motion the solution is ( heta = Acos(omega t + lpha) ) where A and alpha depend on initial conditions ((t= 0 => heta = 0 and t=0 => rac{d heta}{dt} = 0))

so (rac{dphi}{dt} = R/r * rac{d heta}{dt})

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