A uniform cylinder of mass `M` and radius `R` is placed on a rough horizontal board, which in turn is placed on a smooth surface. The coefficient of friction between the board and the cylinder is `mu`. If the board starts accelerating with constant acceleration a, as shown in the figure, then

Option1
for pure rolling motion of the cylinder, direction of frictional force is forward and its magnitude is M a / 3
Option2
the maximum value of a, so that cyliner performs pure rolling is 3 μ g
Option3
The acceleration of the centre of mass of te cylinder under pure rolling condition for the given a is a / 3
Option4
none of these

Options

A. for pure rolling motion of the cylinder, direction of frictional force is forward and its magnitude is `Ma//3`
B. the maximum value of a, so that cyliner performs pure rolling is `3mug`
C. The acceleration of the centre of mass of te cylinder under pure rolling condition for the given a is `a//3`
D. none of these

Answer

A, B, C, D

Explanation

Let the frictional force be in the forward direction then

`f=ma_(0)`
and `fR=(MR^(2))/2alpha_(0)`
`a_(0)=f/M` and `alpha_(0)=(2f)/(MR)`
For pure rolling `a_(0)+Ralpha_(0)=a`
`f/K+m+(2f)/M=aimplies(Ma)/3`
For pure rolling `f,f_(L)=mumg`
`(Ma)/3lt muMgimpliesalt3mug`