We have a hollow sphere and a solid sphere equal radii and of the same material. They are heated to raise their temperature be equal amounts. How will the cange in their volumes, due to volume expansions, be related? Consider two cases (i) hollow sphere is filled with air, (ii) there is vaccum inside the hollow sphere.

Explanation

To solve the problem, we will analyze the volume expansion of both the hollow sphere and the solid sphere under two different conditions: when the hollow sphere is filled with air and when it is a vacuum.

### Step-by-Step Solution:

1. **Understanding Volume Expansion**:
- Volume expansion of a material is given by the formula:
\[ \Delta V = \beta V_0 \Delta T \]
where \( \Delta V \) is the change in volume, \( \beta \) is the coefficient of volume expansion, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature.

2. **Initial Volumes of the Spheres**:
- For both spheres (solid and hollow), since they have the same radius \( R \), the initial volume \( V_0 \) can be calculated using the formula for the volume of a sphere:
\[ V_0 = \frac{4}{3} \pi R^3 \]
- Therefore, both the solid sphere and the hollow sphere have the same initial volume.

3. **Case 1: Hollow Sphere Filled with Air**:
- When the hollow sphere is filled with air and heated, both the air inside and the material of the hollow sphere will expand.
- The volume expansion of the solid sphere is:
\[ \Delta V_{\text{solid}} = \beta V_0 \Delta T \]
- The volume expansion of the hollow sphere will also be:
\[ \Delta V_{\text{hollow}} = \beta V_0 \Delta T \]
- However, the air inside the hollow sphere will also expand, applying pressure on the walls of the hollow sphere, leading to additional expansion.
- Therefore, the total expansion of the hollow sphere will be greater than that of the solid sphere:
\[ \Delta V_{\text{hollow}} > \Delta V_{\text{solid}} \]

4. **Case 2: Hollow Sphere with Vacuum**:
- In this case, the hollow sphere is empty (vacuum), meaning there is no air inside to exert pressure.
- The volume expansion of the solid sphere remains the same:
\[ \Delta V_{\text{solid}} = \beta V_0 \Delta T \]
- The hollow sphere, despite having a cavity, will also expand according to the same volume expansion formula:
\[ \Delta V_{\text{hollow}} = \beta V_0 \Delta T \]
- Since both spheres are made of the same material and have the same initial volume, their volume expansions will be equal:
\[ \Delta V_{\text{hollow}} = \Delta V_{\text{solid}} \]

### Conclusion:
- **Case 1**: When the hollow sphere is filled with air, the change in volume of the hollow sphere is greater than that of the solid sphere.
- **Case 2**: When the hollow sphere is a vacuum, the change in volume of the hollow sphere is equal to that of the solid sphere.