Question

A uniform solid cylindrical roller of mass 'm' is being pulled on a horizontal surface F parallel to the surface and applied at its centre. If the acceleration of the cylinder is 'a' and it is rolling without slipping then the value of 'F' is

• A. $ma$
• B. $\frac{5}{3}ma$
• C. $2ma$
• D. $\frac{3}{2}ma$

Answer 1

The correct option is D $\frac{3}{2}ma$

Writing equation of motion of center of mass of cylinder
$F-f=m{a}_{Centerofmass}=ma$...(i)
Writing torque equation for cylinder about center:
$fR=I_{Centerofmass}\alpha$ ...(ii)
where $\alpha$ is angular acceleration and $I_{Centerofmass}=\frac{M{R}^2}{2}$
substituting value of f in equation (i)
$F=ma+\frac{MR\alpha }{2}$...(iii)
As the cylinder is rolling without slipping, the acceleration of point of contact will be zero
$a_{Centerofmass}-R\alpha =0\Rightarrow \alpha =\frac{a_{Centerofmass}}{R}=\frac{a}{R}$
substituting value of $\alpha$ in equation (iii)
$F=ma+\frac{MR\frac{a}{R}}{2}$
$F=\frac{3}{2}ma$