A solid uniform cylinder of mass m , radius R is placed on a plane inclined at angle θ relative to the horizontal as shown in fig. Let g denote the usual acceleration due to gravity, and let a be the acceleration along the incline of the axis of the cylinder the coefficient of friction between cylinder and plane is µ .
For θ less than some critical angle θ c , the cylinder will roll down the incline without slipping.
Text Solution
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Sol. Let f denote the frictional force and α the angular acceleration about the axis of the cylinder. The equations of motion are
mg sin θ – f = ma
fR = I α ,
with I = 
MR 2 .
If there is no slipping, we require a = R α , f < µN, where N, the normal reaction of the inclined plane, equals mg cos θ . The equations of motion give
F = 
mg sin θ .
Hence we require µmg cos θ > 
mg sin θ ,
or 3µ > tan θ .
Let tan θ c = 3µ. Then we require tan θ < tan θ c for no slipping. Therefore the crictical angle is θ c = arc tan 3µ.
a = g sin θ – 
= 
g sin θ .
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