The moments of inertia of the two balls will not be equal because the distribution of mass significantly affects the moment of inertia. Let's clarify this with a step-by-step explanation:
1. Moment of Inertia Definition:
The moment of inertia is a measure of an object's resistance to changes in its rotation rate. It depends on the mass of the object as well as the distribution of that mass with respect to the axis of rotation.
2. Formulas for the Moments of Inertia:
For spheres, we have specific formulas depending on whether the sphere is solid or hollow:
- For a solid sphere, the moment of inertia (I) about an axis through its center is given by the formula:
[tex]\[ I_{\text{solid}} = \frac{2}{5}mr^2 \][/tex]
where [tex]\( m \)[/tex] is the mass and [tex]\( r \)[/tex] is the radius of the sphere.
- For a hollow sphere (i.e., a spherical shell), where all the mass is assumed to be concentrated at the surface, the moment of inertia (I) about an axis through its center is given by the formula:
[tex]\[ I_{\text{hollow}} = \frac{2}{3}mr^2 \][/tex]
3. Comparing the Moments of Inertia:
By simply looking at the coefficients in the respective formulas, we can compare the moments of inertia of a solid sphere and a hollow sphere:
- Solid sphere: Coefficient is [tex]\( \frac{2}{5} \)[/tex]
- Hollow sphere: Coefficient is [tex]\( \frac{2}{3} \)[/tex]
Since [tex]\( \frac{2}{3} > \frac{2}{5} \)[/tex], we can conclude that the moment of inertia of the hollow sphere is greater than that of the solid sphere, assuming that the mass and radius are the same.
4. Reason for the Difference:
The reason for the higher moment of inertia in the hollow sphere is that its entire mass is distributed far away from the axis of rotation, causing it to have a greater "rotational inertia" compared to the solid sphere where more mass is located closer to the axis.
In summary, the hollow sphere, with all its mass distributed at its surface, will have a greater moment of inertia than an identical solid sphere with the same mass and radius.
