Which of the following objects has the largest moment of inertia, assuming they all have the same mass and the same radius?
a solid sphere
a solid disk
a circular hoop
a solid cylinder
The Correct Answer and Explanation is :
The circular hoop has the largest moment of inertia among the given objects, assuming all have the same mass and radius. Here’s why:
Formula for Moment of Inertia
The moment of inertia ((I)) depends on how mass is distributed relative to the axis of rotation. For objects rotating about their central axis:
- Solid sphere: (I = \frac{2}{5} M R^2)
- Solid disk: (I = \frac{1}{2} M R^2)
- Circular hoop: (I = M R^2)
- Solid cylinder: (I = \frac{1}{2} M R^2)
Here, (M) is the mass of the object and (R) is the radius.
Key Idea
Moment of inertia increases when more mass is distributed farther from the axis of rotation.
- For the circular hoop, all the mass is concentrated at the outer radius ((R)), which maximizes the moment of inertia.
- The solid sphere and solid cylinder have significant amounts of mass closer to the axis, reducing their moments of inertia.
- The solid disk has an intermediate distribution of mass between that of a sphere and a hoop, but its moment of inertia is still less than that of the hoop.
Comparison
When expressed relative to (M R^2):
- Solid sphere: (I = 0.4 M R^2)
- Solid disk: (I = 0.5 M R^2)
- Solid cylinder: (I = 0.5 M R^2)
- Circular hoop: (I = 1.0 M R^2)
Conclusion
The circular hoop has the largest moment of inertia because all its mass is concentrated at the maximum possible distance ((R)) from the axis. This property makes it the hardest to rotate, given the same mass and radius as the other objects.