question_answer 59)

                  A uniform disc of radius R, is resting on a table on its rim. The coefficient 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?

Answer:

                  Let a be the acceleration of the C.M of the disc. The equation of motion of the disc is given by \[ma=F-f\]                                          ... (i)                 \[\tau \,=fR\,\,=I\,\alpha \,=\frac{Ia}{R}\]                     \[\,(\because \,\,\alpha =\frac{a}{R})\]                 For a disc, \[I=\,\frac{1}{2}\,m{{R}^{2}}\]                 \[\therefore \] \[fR=\frac{maR}{2}\] or \[ma\,=\,2f\]                 Put this value in eqn. (i), we get                 \[F=3f\]                 The disc will roll without slipping if                 \[f\le \,\mu \,mg\]                 \[\therefore \]\[F\le \,3\,\mu \,mg.\]