A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights h s p h and h c y l on the inline. The ratio h s p h h c y l is given by:

Step-by-step solution

From conservation of mechanical energy, For solid sphere: 1 2 m s p h V 2 + 1 2 I s p h ω 2 = m s p h g h s p h V 2 2 + V 2 5 = g h s p h 7 V 2 10 = g h s p h …(i) For cylinder: 1 2 m c y l V 2 + 1 2 I c y l ω 2 = m c y l g h c y l V 2 2 + V 2 4 = g h c y l 3 V 2 4 = g h c y l …(ii) From (i) and (ii) h s p h h c y l = 14 15