When the temperature of a body increases

Options

A. density and moment of inertia increases
B. density and moment of inertia decreases
C. density decreases and moment of inertia increases
D. density increases and moment of inertia decreases

Answer

To solve the question regarding the effects of temperature increase on density and moment of inertia, we can follow these steps:

### Step 1: Understand the Relationship Between Temperature and Density
- **Density** is defined as the mass of an object divided by its volume:
\[ \text{Density} (\rho) = \frac{\text{Mass} (m)}{\text{Volume} (V)} \]
- When the temperature of a body increases, it undergoes **thermal expansion**.

### Step 2: Analyze the Effect of Thermal Expansion on Volume
- During thermal expansion, the **volume** of the body increases while the **mass** remains constant.
- Therefore, if the volume increases and mass remains the same, the density will decrease:
\[ \text{If } V \text{ increases, then } \rho = \frac{m}{V} \text{ decreases.} \]

### Step 3: Understand the Moment of Inertia
- The **moment of inertia** (I) of a body is given by the formula:
\[ I = m r^2 \]
where \( r \) is the distance from the axis of rotation.
- For a rod rotating about its center, the moment of inertia is given by:
\[ I = \frac{1}{12} m l^2 \]
where \( l \) is the length of the rod.

### Step 4: Analyze the Effect of Thermal Expansion on Moment of Inertia
- When the temperature increases, the rod expands linearly, resulting in a new length \( l' \) which is greater than the original length \( l \).
- The new moment of inertia will be:
\[ I' = \frac{1}{12} m (l')^2 \]
- Since \( l' > l \), it follows that \( (l')^2 > l^2 \), thus:
\[ I' > I \]
indicating that the moment of inertia increases.

### Conclusion
- From the analysis, we can conclude that:
- The **density** of the body **decreases** when the temperature increases.
- The **moment of inertia** of the body **increases** when the temperature increases.

### Final Answer
- The correct statement is: **Density decreases and moment of inertia increases.**

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