question_answer 49)

                  A uniform sphere of mass m and radius R is placed on a rough horizontal surface (fig.) The sphere is struck horizontal at a height h from the floor. Match the following:                  

(a) \[h=R/2\]	(i)            Sphere rolls without slipping with a constant velocity and no loss of energy.
(b) \[h=R\]	(ii)           Sphere spring clockwise, loses energy by friction.
(c) \[h=3R/2\]	(iii)          Sphere spins anticlockwise, lose energy by friction.
(d) \[h=7R/5\]	(iv)         Sphere has only a translation motion, looses energy by friction.

                 

Answer:

                  (a) ® (iii), (b) ® (iv), (c) ® (ii), (d) ® (i)                                 Angular velocity \['\omega '\] and linear velocity of the C.M. of the sphere are related as                 \[\omega =\,\frac{5\upsilon (h-R)}{2\,{{R}^{2}}}\]. Since\[\omega =\,\frac{\upsilon }{R}\]                 \[\therefore \] \[\,h=\frac{7}{5}\,R\]                 Thus, if force is applied at \[h=\frac{7}{5}\,R,\] sphere will roll without slipping.                 Torque due to applied force about C.M., \[\tau \,=\,F(h-R).\] Sphere will have only translation motion, if \[z=0\] or \[h=R\]. Sphere will spin clockwise if forque is positive and it will spin anticlockwise if torque is negative.