# Q. A cylinder of radius R is rotating with an angular acceleration such that speed of a point in its inner periphery varies with time as `v=at` .A particle of mass m is found to be at the inner...

Q. A cylinder of radius R is rotating with an angular acceleration such that speed of a point in its inner periphery varies with time as `v=at` .A particle of mass m is found to be at the inner periphery at time `t`. Find minimum coefficient of friction between cylinder and particle.

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A cylinder of radius R is rotating with an angular acceleration such that speed of a point in its inner periphery varies with time as v = a*t. A particle of mass m is found to be at the inner periphery at time t.

There are two forces acting on the particle. One of them is a downward force due to gravity that is equal to m*g. The other force is in the opposite direction and due to the friction between the cylinder and the mass. If the coefficient of friction is `mu` , the resistive force of friction is `N*mu` where N is the normal force. Here, the normal force is due to the centrifugal force acting on the particle and equal to `m*(v^2/R) = m*(a^2*t^2)/R` . The frictional force is `mu*m*(a^2*t^2)/R`

The particle does not move down when `m*g = mu*m*(a^2*t^2)/R`

=> `mu = (g*R)/(a^2*t^2)`

The minimum coefficient of friction required is `(g*R)/(a^2*t^2)`