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Rotational Mechanics — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsRotational Mechanics1 Mark
MCQ
A hollow sphere and a solid sphere having same mass and same radii are rolled down a rough inclined plane:
  • A
    The hollow sphere reaches the bottom first.
  • ✓
    The solid sphere reaches the bottom with greater speed.
  • C
    The solid sphere reaches the bottom with greater kinetic energy.
  • D
    The two spheres will reach the bottom with same linear momentum.
✓

Answer

Correct option: B.
The solid sphere reaches the bottom with greater speed.
Acceleration of a sphere on the incline plane is given by:
$\text{a}=\frac{\text{g}\sin\theta}{1+\frac{\text{I}_{\text{COM}}}{\text{mr}^2}}$
$I_\text{COM}$ for a solid sphere $=\frac{2}{5}\text{mr}^2$
So, $\text{a}=\frac{\text{g}\sin\theta}{1+\frac{2\text{mr}^2}{5\text{mr}^2}}=\frac{5}{7}\text{g}\sin\theta$
$I_\text{COM}$ for a hollow sphere $=\frac{2}{3}\text{mr}^2$
So, $\text{a}'=\frac{\text{g}\sin\theta}{1+\frac{2\text{mr}^2}{3\text{mr}^2}}=\frac{3}{5}\text{g}\sin\theta$
The acceleration of the solid sphere is greater;
therefore, it will reach the bottom with greater speed.

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