A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is 
where 
______.
Text Solution
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To determine the ratio of velocities for a circular ring and a solid sphere rolling down an inclined plane without slipping, we apply the principle of mechanical energy conservation:
Conservation of Mechanical Energy:
Initially (at the top), we have:
Initial kinetic energy, 
(since they start from rest)
Initial potential energy, 
Finally (at the bottom), we have:
Final potential energy, 
Final kinetic energy, 
This gives:
Final potential energy, 
Final kinetic energy, 
This gives:
Solving for the velocity 
:
Ratio of Velocities:
For the circular ring (moment of inertia
For the solid sphere (moment of inertia
The ratio is:
Thus, 
. Rounding off, 
.
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