A uniform disco of mass M and radius R, is resting on a table on its rim. The coeffecient of friction between disc and table is `mu`. Now the disc is pulled with a force F as shown in the figure. What is the maximum value of F for which the disc rolls without slipping?

Explanation

(i) We know that moment of inertia, `I=summr^(2)` . In case of a hollow cylinder of radius R, all its mass (m) lies at a distance R from the axis of symmetry. But in case of solid sphere of radius R and mass M, most of its mass lies at a distance smaller than R. Hence, `I_("cylinder") gt I_("solid sphere")` (about the axes of symmetry)
(ii) If f is the force of friction acting on the disc and a is the acceleration produced in it due to applied force F, then
Ma= F-t............(i)
When the disc does not slide, `a = R alpha` or `alpha = a/R`
As `I alpha = Rf`
`(1/2 MR^(2)) (a/R) = Rf` or Ma = 2f............(ii)
From eqns.(i) and (ii), 2f=F-f or F=3f since there is no sliding, `f le mu m g` of `F le 3 mu mg`