The Question is from neet-physics rotational-motion rolling-motion, and it was created by the author VENKATESWARARAO

Question

A solid sphere and a hollow sphere of the same mass and radius roll down an inclined plane without slipping.  Considering only rolling friction, which sphere will reach the bottom first?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
 **Primary Concept**: Rotational Kinetic Energy and Moment of Inertia
 **Related Topics**: Rolling motion, conservation of energy
 **Prerequisites**: Understanding of moment of inertia, rotational kinetic energy, and linear kinetic energy.
## SOLUTION PATHWAY
**Given Information**:
  * Two spheres (solid and hollow) with the same mass (m) and radius (r).
  * Rolling down an inclined plane without slipping.
  * Only rolling friction is considered.
**Method & Steps**:
 1. Analyze the energy conservation during rolling.
  Reasoning: The potential energy at the top of the incline is converted into translational and rotational kinetic energy at the bottom.  The sphere with the lower rotational inertia will have more translational kinetic energy, thus reaching the bottom faster.
  Formula/Rule: $PE = mgh$; $KE_{trans} = \frac{1}{2}mv^2$; $KE_{rot} = \frac{1}{2}I\omega^2$;  $v = r\omega$ (no slipping condition)
 2. Determine the moment of inertia for each sphere.
  Working: For a solid sphere, $I_{solid} = \frac{2}{5}mr^2$. For a hollow sphere, $I_{hollow} = \frac{2}{3}mr^2$. Since $I_{solid} < I_{hollow}$, the solid sphere has a smaller moment of inertia.
  Key Point: A smaller moment of inertia means less energy is stored in rotation, leaving more energy for translation (linear motion).
 3. Compare the final velocities.
  Result: The solid sphere will have a greater translational velocity at the bottom of the incline due to its lower rotational inertia, hence reaching the bottom first.
  Verification:  We can verify this by using energy conservation: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$. Solving for v, we find that the solid sphere has a higher final velocity.
 ## OPTION ANALYSIS
A ✓: The solid sphere reaches the bottom first because it has a lower moment of inertia, meaning less energy is used for rotation and more for translation.
B :x: The hollow sphere has a larger moment of inertia, resulting in a slower descent.
C :x:  They will not reach simultaneously due to the difference in their moments of inertia.
D :x: The angle of inclination affects the time taken, but the solid sphere will still reach the bottom first regardless of the angle (assuming the same mass and radius).
 ## LEARNING AIDS
 **Common Mistakes**:
  1.Ignoring rotational kinetic energy:  For rolling motion, both translational and rotational kinetic energies must be considered.
  2.Confusing moment of inertia:  Failing to distinguish between the moments of inertia of solid and hollow spheres.
 **Quick Tips**:
  1.Remember that a smaller moment of inertia leads to a faster descent in rolling motion.
  2.Visualize the mass distribution: The mass of a solid sphere is concentrated closer to the center, leading to a lower moment of inertia compared to a hollow sphere.

VENKATESWARARAO · Answer

The correct Option is "The solid sphere"

VENKATESWARARAO · Answer

The hollow sphere

VENKATESWARARAO · Answer

Both will reach simultaneously

VENKATESWARARAO · Answer

It depends on the angle of inclination

NEET Physics Systems Of Particles And Rotational Motion Rolling Motion MCQs

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