question_answer

A uniform sphere of mass m and radius R is placed on a rough horizontal surface. The sphere is struck horizontally at a height h from the floor. Show that the sphere rolls without slipping with a constant velocity, when h = 7 R/5.

Answer:

                The sphere will roll without slipping with constant velocity when \[\omega =\frac{\upsilon }{R}\] Now, angular momentum of sphere about c.m, Fig. 5(HT).12, is  \[m\upsilon \left( h-R \right)=I\omega =\left( \frac{2}{5}m{{R}^{2}} \right)\left( \frac{\upsilon }{R} \right)\] \[m\upsilon \left( h-R \right)=\frac{2}{5}m\upsilon R\] \[h-R=\frac{2}{5}R\] \[h=\frac{2}{5}R+R=\frac{7}{5}R\]