---
title: "A uniform solid sphere and a thin hollow spherical shell of identical mass \\(M\\) and radius \\(R\\) are released simultaneously from rest at the top of an incline of angle \\(\\theta\\). Both spheres roll down the incline to the bottom without slipping, but the hollow spherical shell consistently takes a longer time to reach the bottom than the solid sphere. Which of the following explanations correctly accounts for this difference in travel time?"
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url: "https://nerd-notes.com/ubq/124223/"
date_modified: "2026-09-28T14:04:15+00:00"
---

# A uniform solid sphere and a thin hollow spherical shell of identical mass \(M\) and radius \(R\) are released simultaneously from rest at the top of an incline of angle \(\theta\). Both spheres roll down the incline to the bottom without slipping, but the hollow spherical shell consistently takes a longer time to reach the bottom than the solid sphere. Which of the following explanations correctly accounts for this difference in travel time?

A uniform solid sphere and a thin hollow spherical shell of identical mass \(M\) and radius \(R\) are released simultaneously from rest at the top of an incline of angle \(\theta\). Both spheres roll down the incline to the bottom without slipping, but the hollow spherical shell consistently takes a longer time to reach the bottom than the solid sphere. Which of the following explanations correctly accounts for this difference in travel time?

![A right triangle representing a fixed ramp inclined at an angle \(\theta\) to the horizontal ground. Near the top of the incline, two circles of equal radius \(R\) rest side by side along the incline surface. The upper-left circle has a solid gray interior, representing the solid sphere, labeled \(M\). The upper-right circle is an unfilled ring with a thin dark perimeter line, representing the hollow spherical shell, also labeled \(M\). A dashed line extends horizontally from the base of the ramp to indicate the angle \(\theta\) between the ground and the ramp surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604255-brhb1d.jpg)

- **A.** Because its mass is distributed farther from the center of mass, the hollow sphere has a greater rotational inertia and requires a larger net torque about its center to maintain the non-slip condition, demanding a larger uphill static friction force that reduces its translational acceleration.
- **B.** Because static friction exerts a backward torque that does negative work on the rolling hollow sphere, energy is dissipated from the mechanical system at a higher rate due to the shell's greater rotational inertia, reducing its speed at every point.
- **C.** Because the parallel component of gravity exerts equal torque about the center of mass for both spheres, the hollow sphere's greater rotational inertia results in a smaller angular acceleration, which directly restricts its translational acceleration via the rolling constraint.
- **D.** Because the hollow sphere has a greater rotational inertia, it experiences a smaller static friction force along the incline, which prevents sufficient torque from being generated to accelerate its center of mass down the ramp at the same rate as the solid sphere.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124223/*