The Question is from neet-physics rotational-motion rotational-inertia-of-solid-bodies, and it was created by the author VENKATESWARARAO

Question

A solid sphere and a hollow sphere of the same mass and radius are rolling down an incline without slipping. Which one will reach the bottom first?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [A]
## CONCEPT FRAMEWORK
**Primary Concept**: Conservation of Energy and Rotational Inertia
**Related Topics**: Rolling Motion, Kinetic Energy (Translational and Rotational), Moment of Inertia
**Prerequisites**: Understanding of energy conservation, rotational motion, and moment of inertia.
## SOLUTION PATHWAY
**Given Information**:
  * Both spheres have the same mass (m) and radius (r).
  * Both start at the same height (h).
  * Both roll without slipping.
**Method & Steps**:
 1. **Energy Conservation**: The potential energy at the top of the incline is converted into kinetic energy (both translational and rotational) at the bottom.
  Reasoning: The total mechanical energy (Potential + Kinetic) is conserved in the absence of non-conservative forces like friction (which is assumed to be static here for rolling without slipping).
  Formula/Rule: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}Iω^2$
 2. **Relationship between Linear and Angular Velocity**: Since the spheres are rolling without slipping, the linear velocity (v) and angular velocity (ω) are related by $v = rω$ or $ω = \frac{v}{r}$.
  Working: Substitute $ω = \frac{v}{r}$ into the energy conservation equation: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}I(\frac{v}{r})^2$
 3. **Moment of Inertia**: The moment of inertia (I) for a solid sphere is $I_{solid} = \frac{2}{5}mr^2$ and for a hollow sphere is $I_{hollow} = \frac{2}{3}mr^2$.
  Working: Substitute the moment of inertia for each sphere into the energy equation: 
  * Solid sphere: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{5}mr^2)(\frac{v}{r})^2 = \frac{1}{2}mv^2 + \frac{1}{5}mv^2 = \frac{7}{10}mv^2$
  * Hollow sphere: $mgh = \frac{1}{2}mv^2 + \frac{1}{2}(\frac{2}{3}mr^2)(\frac{v}{r})^2 = \frac{1}{2}mv^2 + \frac{1}{3}mv^2 = \frac{5}{6}mv^2$
 4. **Solving for Linear Velocity**: Solve for v in both cases:
  * Solid sphere: $v_{solid} = \sqrt{\frac{10}{7}gh}$
  * Hollow sphere: $v_{hollow} = \sqrt{\frac{6}{5}gh}$
 5. **Comparison**: Since $\frac{10}{7} > \frac{6}{5}$, $v_{solid} > v_{hollow}$. The solid sphere will reach the bottom first because it will have a higher linear velocity.
  Result: The solid sphere reaches the bottom first.
  Verification: The solid sphere has a lower moment of inertia, so a larger fraction of its kinetic energy is translational, making it faster.
## OPTION ANALYSIS
[A] ✓: The solid sphere has a smaller moment of inertia than the hollow sphere, meaning it will convert more of its potential energy into translational kinetic energy resulting in higher linear speed and thus reaches the bottom first.
[B] :x: The hollow sphere has a larger moment of inertia, so it will have a lower linear speed and will reach the bottom last.
[C] :x: Since the moment of inertia is different for the spheres, they will not reach the bottom at the same time.
[D] :x: The material of the sphere does not influence the outcome, as it is not involved in the calculation of the final velocity in rolling without slipping. The moment of inertia depends on mass distribution not the material itself.
## LEARNING AIDS
**Common Mistakes**:
 1. [Error 1]: Forgetting the rotational kinetic energy term in the conservation of energy equation.
  How to avoid: Always remember that rolling objects have both translational and rotational kinetic energy.
 2. [Error 2]: Using the wrong moment of inertia for solid and hollow spheres.
  How to avoid: Refer to the standard formulas for the moment of inertia for different shapes.
**Quick Tips**:
 1. [Helpful shortcut]: Remember that objects with lower moments of inertia will roll faster down an incline.
 2. [Memory aid]: Think of the solid sphere as having its mass more concentrated towards the center, making it easier to accelerate linearly.

VENKATESWARARAO · Answer

The correct Option is "The solid sphere"

VENKATESWARARAO · Answer

The hollow sphere

VENKATESWARARAO · Answer

Both will reach at the same time

VENKATESWARARAO · Answer

It depends on the material of the spheres

NEET Physics Systems Of Particles And Rotational Motion Rotational Inertia Of Solid Bodies MCQs

Select Topic

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

NEET Subjects

Our Bots

Company