Moment of inertia of a rigid body is expressed in units of kg `m^(2)`. There are two rods A and B made of same metal. Both of them have equal cross-sectional area bur rod A is double in length as compared to rod B. What is the ratio of moment of ineria of rod A to that of rod B?

Options

A. 1
B. 2
C. 4
D. 8

Answer

To find the ratio of the moment of inertia of rod A to that of rod B, we can follow these steps:

### Step 1: Understand the Moment of Inertia Formula
The moment of inertia \( I \) for a rod about its end is given by the formula:
\[ I = \frac{1}{3} m L^2 \]
where \( m \) is the mass of the rod and \( L \) is its length.

### Step 2: Define the Lengths and Masses of the Rods
Let:
- Length of rod B = \( L \)
- Length of rod A = \( 2L \) (since rod A is double the length of rod B)

### Step 3: Calculate the Mass of Each Rod
Since both rods are made of the same material and have the same cross-sectional area \( A \), their masses can be expressed in terms of their volumes and densities.
The mass \( m \) of a rod can be calculated as:
\[ m = \text{Density} \times \text{Volume} \]
The volume \( V \) of a rod is given by:
\[ V = A \times L \]
Thus, we can express the masses as:
- Mass of rod B:
\[ m_B = \rho \cdot A \cdot L \]
- Mass of rod A:
\[ m_A = \rho \cdot A \cdot (2L) = 2\rho \cdot A \cdot L \]

### Step 4: Substitute Masses into the Moment of Inertia Formula
Now we can substitute the masses into the moment of inertia formula for both rods:
- Moment of inertia of rod A:
\[ I_A = \frac{1}{3} m_A (2L)^2 = \frac{1}{3} (2\rho A L)(4L^2) = \frac{8}{3} \rho A L^3 \]
- Moment of inertia of rod B:
\[ I_B = \frac{1}{3} m_B L^2 = \frac{1}{3} (\rho A L)(L^2) = \frac{1}{3} \rho A L^3 \]

### Step 5: Find the Ratio of the Moments of Inertia
Now, we can find the ratio of the moment of inertia of rod A to that of rod B:
\[ \frac{I_A}{I_B} = \frac{\frac{8}{3} \rho A L^3}{\frac{1}{3} \rho A L^3} \]
The terms \( \frac{1}{3} \), \( \rho \), and \( A L^3 \) will cancel out, leading to:
\[ \frac{I_A}{I_B} = \frac{8}{1} = 8 \]

### Conclusion
The ratio of the moment of inertia of rod A to that of rod B is:
\[ \frac{I_A}{I_B} = 8 \]