Amarble and a cube are placed at the top of a ramp. starting from rest at the same height, the marble rolls without slipping and the cube slides (no kinetic friction) down the ramp. determine the ratio of the center-of-mass speed of the cube to the center-of-mass speed of the marble at the bottom of the ramp.

Explanation:

For the marble :

= height from which the marble is released

= speed of center of mass of marble at the bottom

= angular speed of marble =

= mass of the marble

= radius of the marble

Moment of inertia of marble (solid sphere) is given as

Using conservation of energy between Top and bottom

Kinetic energy at bottom + rotational kinetic energy at bottom = potential energy at top

                                                 Eq-1

For Cube :

= height from which the cube is released

= speed of center of mass of cube at the bottom

= mass of the cube

Using conservation of energy between Top and bottom

Kinetic energy at bottom  = potential energy at top

                                                             Eq-2

Using Eq-1 and Eq-2

Taking square-root both side