A heavy homogeneous cyliner has mass `m` and radius `R`. It is accelerated by a force `F` which is applied through a rope wound around a light drum of radius r attached to the cylinder (figure) the coefficient of static friction is sufficient for the cylinder to roll without slipping.

Options

A. Find the friction force.
B. Find the acceleration a of the centre of the cylinder
C. it is possible to choose `r`, so that `a` is greater than `(F)/(m)` ? How ?
D. What is the direction of the friction in the circumstances of part(C)?

Explanation

(a). For pure rolling to take place,
`a=Ralpha` or `(F-f)/(m)+R[(Fr+fR)/((1)/(2)mR^(2))]`
Solving this equation we get
`f=(2)/(3)((1)/(2)-(r)/(R))`
(b). Acceleration `a=(F-f)/(m)`
substituting value of `f` from part (a), we get
`a=(2F)/(3mR)(R+r)`
(c). `agt(F)/(m)` if `(2)/(3R)(R+r)gt1` or `rgt(R)/(2)`
(d). In this case force of friction is in forward direction.