Which of the following objects has the largest moment of inertia, assuming they all have the same mass and the same radius? solid sphere solid disk circular hoop a solid cylinder
The Correct Answer and Explanation is:
Correct Answer: Circular hoop
Explanation:
The moment of inertia (denoted I) measures an object’s resistance to rotational acceleration about an axis. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. When comparing objects with the same mass and same radius, the distribution of mass becomes the key factor.
Here are the standard formulas for the moment of inertia of each object rotating about its central axis:
- Solid Sphere:
I=25MR2I = \frac{2}{5}MR^2I=52MR2 - Solid Disk:
I=12MR2I = \frac{1}{2}MR^2I=21MR2 - Circular Hoop (thin ring):
I=MR2I = MR^2I=MR2 - Solid Cylinder:
I=12MR2I = \frac{1}{2}MR^2I=21MR2
Where:
- M is the mass of the object
- R is the radius of the object
As seen from the formulas, the circular hoop has the largest moment of inertia, because all of its mass is located at the maximum possible distance from the axis (at radius R). This means the entire mass contributes maximally to the rotational inertia.
By contrast, the solid sphere has the smallest moment of inertia among the listed objects. In the solid sphere, most of the mass is concentrated closer to the center, which lowers the moment of inertia.
Thus, even if all the objects have equal mass and radius, the circular hoop resists rotational acceleration the most due to its greater moment of inertia.
Summary of Values:
- Solid Sphere: 25MR2=0.4MR2\frac{2}{5}MR^2 = 0.4MR^252MR2=0.4MR2
- Solid Disk: 12MR2=0.5MR2\frac{1}{2}MR^2 = 0.5MR^221MR2=0.5MR2
- Solid Cylinder: 12MR2=0.5MR2\frac{1}{2}MR^2 = 0.5MR^221MR2=0.5MR2
- Circular Hoop: 1MR2=1MR21MR^2 = 1MR^21MR2=1MR2
Conclusion: The circular hoop has the largest moment of inertia.
