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Question

A block is in contact with the inner wall of a hollow cylindrical drum as shown in figure. The minimum angular velocity (in rad/s) needed for the cylinder to keep the block stationary relative to itself, when the cylinder is vertical and rotating about its axis will be


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Solution

Let us suppose angular velocity needed for the cylinder to keep the block stationary is ω.
FBD and figure as per question -


For block, on applying F=ma along the vertical.
fmg=0 [ for vertical equilibrium ]
f=mg
μN=mg
N=mgμ .........(1)
[considering limiting friction for minimum angular velocity]

Similarly, for horizontal direction
N=Fc
N=mω2r [ for circular motion ]
mgμ=mω2r [ from (1) ]
ω=grμ
ω=102×0.2 [from figure]
=5 rad/s

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