Question

A hollow sphere and a solid sphere having same mss and same radii are rolled down a rough inclined plane.

विकल्प

• The hollow sphere reaches the bottom first.

• The solid sphere reaches the bottom with greater speed

• The solid sphere reaches the bottom with greater kinetic energy.

• The two spheres will reach the bottom with same linear momentum.

Answer 1

The solid sphere reaches the bottom with greater speed.

 

Acceleration of a sphere on the incline plane is given by

\[a = \frac{g\sin\theta}{1 + \frac{I_{COM}}{m r^2}}\]
\[ I_{COM}\] for a solid sphere \[= \frac{2}{5}m r^2 \]
\[\text{So, }a = \frac{g\sin\theta}{1 + \frac{2m r^2}{5m r^2}} = \frac{5}{7}g\sin\theta\]

\[I_{COM}\] for a hollow sphere \[= \frac{2}{3}m r^2 \] 

\[\text{So, }a' = \frac{g\sin\theta}{1 + \frac{2m r^2}{3m r^2}} = \frac{3}{5}g\sin\theta\]

The acceleration of the solid sphere is greater; therefore, it will reach the bottom with greater speed.