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Rotational Mechanics — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsRotational Mechanics1 Mark
Question
Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping:
  1. The heavier sphere reaches the bottom first.
  2. The bigger sphere reaches the bottom first.
  3. The two spheres reach the bottom together.
  4. The information given is not sufficient to tell which sphere will reach the bottom first.
✓

Answer

  1. The two spheres reach the bottom together.

Explanation:

Acceleration of a sphere on the incline plane is given by:

$\text{a}=\frac{\text{g}\sin\theta}{1+\frac{\text{I}_{\text{COM}}}{\text{mr}^2}}$

ICOM for a solid sphere $=\frac{2}{5}\text{mr}^2$

So, $\text{a}=\frac{\text{g}\sin\theta}{1+\frac{2\text{mr}^2}{5\text{mr}^2}}=\frac{5}{7}\text{g}\sin\theta$

a is independent of mass and radii; therefore, the two spheres reach the bottom together.

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