First slide

Rolling motion

Question

A solid cylinder of mass 3 kg rests on a rough horizontal plane. A horizontal force F is applied to the top most point of the cylinder as shown in the figure. Coefficient of friction between the cylinder and the surface is 0.6. Then the maximum value of F for which the cylinder will roll without slipping is

Moderate

A

54 N
B

18 N
C

28 N
D

68 N

Solution

Frictional force f is given by
$f = F (1 - \frac{h}{rp})$
Where $r_{P} = r_{C} + \frac{I_{C}}{{mr}_{C}}$
$r_{C} =$ Distance of centre of mass from instantaneous axis of rotation O.
For solid cylinder, $r_{P} = R + \frac{\frac{1}{2} {mR}^{2}}{m . R} = \frac{3 R}{2}$
$∴ f = F (1 - \frac{h}{r_{P}}) = F (1 - \frac{2 R}{3 / 2 R}) = - \frac{F}{3} .$
Since ‘f’ is negative, it must act in the forward direction. Now $f_{\max} = μmg$
No slipping takes place if $|f| < μmg$
i.e., if $\frac{F}{3} < μmg$
$∴ F_{\max} = 3 μmg = 3 \times 0 .6 \times 3 \times 10 N = 54 N$

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