A disc of metal is melted to recast in the form of a solid sphere. The moment of inertia about a vertical axis passing through the centre would

Options

A. decrease
B. increase
C. remains same
D. nothing can be said

Answer

To solve the problem of how the moment of inertia changes when a disc is melted and recast into a solid sphere, we can follow these steps:

### Step 1: Understand the moment of inertia of the disc
The moment of inertia (I) of a solid disc about an axis passing through its center and perpendicular to its plane is given by the formula:
\[ I_{\text{disc}} = \frac{1}{2} m r^2 \]
where \(m\) is the mass of the disc and \(r\) is the radius of the disc.

### Step 2: Understand the moment of inertia of the sphere
The moment of inertia of a solid sphere about an axis through its center is given by the formula:
\[ I_{\text{sphere}} = \frac{2}{5} m R^2 \]
where \(R\) is the radius of the sphere.

### Step 3: Consider the mass and volume conservation
When the disc is melted and recast into a sphere, the mass remains constant. Therefore, the mass of the disc is equal to the mass of the sphere:
\[ m_{\text{disc}} = m_{\text{sphere}} = m \]

### Step 4: Relate the dimensions of the disc and sphere
Assuming the volume of the disc is equal to the volume of the sphere, we can express this as:
\[ \text{Volume of disc} = \text{Volume of sphere} \]
The volume of the disc can be expressed as:
\[ V_{\text{disc}} = \pi r^2 h \]
where \(h\) is the thickness of the disc. The volume of the sphere is:
\[ V_{\text{sphere}} = \frac{4}{3} \pi R^3 \]
Setting these equal gives us:
\[ \pi r^2 h = \frac{4}{3} \pi R^3 \]
From this, we can find a relationship between \(r\), \(h\), and \(R\).

### Step 5: Analyze how the moment of inertia changes
Since the radius of the disc \(r\) is typically much larger than the radius of the sphere \(R\) when the same mass is used, we can conclude:
- The moment of inertia of the disc is larger than that of the sphere because it depends on the square of the radius.

### Step 6: Conclusion
Since the moment of inertia of the disc is greater than that of the sphere, we can conclude that the moment of inertia decreases when the disc is recast into a sphere.

Thus, the final answer is:
**The moment of inertia decreases.**