Answer:
The ranking of the frequency of small oscillations for the given objects, from least to greatest, is as follows:
1. Hollow sphere
2. Solid sphere
3. Point mass
To understand this ranking, let's consider the factors that affect the frequency of small oscillations for pendulum-like systems.
The frequency of small oscillations (f) for a pendulum is given by:
f = (1 / 2π) * √(g / L)
Where:
- g is the acceleration due to gravity
- L is the effective length of the pendulum (distance from the pivot point to the center of mass)
Now, let's analyze each of the given objects:
1. Hollow sphere:
The hollow sphere has the largest moment of inertia due to its mass distribution away from the axis of rotation. This leads to a larger effective length (L) when it swings as a pendulum. The larger effective length results in a smaller frequency of oscillation compared to the other objects.
2. Solid sphere:
The solid sphere has a smaller moment of inertia compared to the hollow sphere due to its mass being more concentrated towards the center of rotation. The smaller moment of inertia gives a smaller effective length (L), resulting in a slightly higher frequency of oscillation compared to the hollow sphere.
3. Point mass:
The point mass is a theoretical idealization with all the mass concentrated at a single point. It has the smallest moment of inertia and, therefore, the smallest effective length (L). As a result, it has the highest frequency of oscillation among the three objects.
Therefore, the ranking of the frequency of small oscillations, from least to greatest, is:
1. Hollow sphere
2. Solid sphere
3. Point mass