A solid sphere and a solid cylinder roll down without slipping along an inclined plane. If they start from rest from the top of the inclined plane, the ratio of the velocities of the solid sphere and solid cylinder when they reach the bottom of the inclined plane is

Step-by-step solution

Apply the conservation of energy aligned & mgh = 1 2 ~m v^2+ 1 2 I ^2 & mgh = 1 2 mv ^2+ 1 2 ~m k ^2 v ^2 R ^2 & mgh = 1 2 mv ^2 [1+ K ^2 R ^2 ] & 2 g h 1+ K^2 R^2 =v^2 & aligned For solid sphere v_ ss ^2= 2 g h 1+ 2 5 = 5 2 g h 7 for solid cylinder aligned & v_ sc ^2= 2 gh 1+ 1 2 = 2 2 gh 3 & Ratio = v_ ss ^2 v_ sc ^2 = 5 2 gh 7 2 2 gh 3 = 15 14 & Ratio = v_ ss v_ sc = 15 14 aligned