Q24 of 63 Page 114

A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle θ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is μ, Find the range of the angular speed for which the block will not slip.


In this system there exists two physical forces, one is gravity and the other one is centrifugal force.


Normal reaction force is balanced as,



As the direction of friction is a variable, we will try both cases of friction, one is along mgsinθ and the other one is in the opposite direction.


Let us assume friction is along mgsinθ


So,



Here friction is at its maximum limit. Hence,



So,





Now, let us assume friction is in the opposite direction of mgsinθ


So,



Here friction is at its maximum limit. Hence,



So,





This implies that if the angular speed is in range from to then the block will not slip.


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