Question

If hollow cylinder is performing pure rolling then frictional force $f$ will be

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

Answer 1

The correct option is B $\frac{F}{2}$

If slipping does not occur, point P is an instant center of rotation for the cylinder.
The equation of motion of the cylinder about point P is
${\tau }_P=I_P\alpha$ where
${\tau }_P$ is the torque acting about point P
$I_P$ is MI of the cylinder about point P.
$\alpha$ is the angular acceleration about point P.
$\therefore FR=(I_O+M{R}^2)\alpha$ .....(1) by parallel axis theorem, $I_P=I_O+M{R}^2$, $I_O$ is MI of the cylinder about cm O.
$\therefore \alpha =\frac{FR}{I_O+M{R}^2}$ ......(2)
Frictional force$F_{fr}$ is responsible for angular acceleration about cm O.
${\tau }_{fr}=F_{fr}R=I_O\alpha$ .....(3)
Substituting $\alpha$ from (2) in (3)
$F_{fr}R=I_O\frac{FR}{I_O+M{R}^2}$
$\Rightarrow F_{fr}=I_O\frac{F}{I_O+M{R}^2}$ .....(4)
For a hollow cylinder $I_O=M{R}^2$
Substituting $I_O=M{R}^2$ in (4) we get
$F_{fr}=M{R}^2\frac{F}{M{R}^2+M{R}^2}=\frac{F}{2}$