A hammer strikes a long metallic rod at one end. A listener, whose ear is close to the other end of the rod, hears the sound of blow twice within a time interval 120 ms. Speed of sound in air is 350 m/s and speed of sound in metal is 6000 m/s. Find the approximate value of the length of the rod.

Options

A. 45m
B. 90m
C. 30m
D. 130m

Answer

To solve the problem, we need to find the length of the metallic rod based on the time interval in which the listener hears the sound twice. The key points to consider are the speeds of sound in air and metal, and the time difference in the sounds reaching the listener.

### Step-by-step Solution:

1. **Understanding the Problem**:
- A hammer strikes one end of a metallic rod.
- The listener at the other end hears the sound twice within a time interval of 120 ms (0.120 seconds).
- The speed of sound in air is 350 m/s, and in metal, it is 6000 m/s.

2. **Defining Variables**:
- Let \( L \) be the length of the rod.
- \( T_1 \) is the time taken for sound to travel through the metal.
- \( T_2 \) is the time taken for sound to travel through the air.
- The difference in time between the two sounds reaching the listener is given as \( \Delta T = T_2 - T_1 = 0.120 \) seconds.

3. **Calculating Time in Metal**:
- The time taken for sound to travel through the metal rod can be expressed as:
\[ T_1 = \frac{L}{6000} \]

4. **Calculating Time in Air**:
- The time taken for sound to travel through the air can be expressed as:
\[ T_2 = \frac{L}{350} \]

5. **Setting Up the Equation**:
- From the difference in time, we have:
\[ T_2 - T_1 = 0.120 \]
- Substituting the expressions for \( T_1 \) and \( T_2 \):
\[ \frac{L}{350} - \frac{L}{6000} = 0.120 \]

6. **Finding a Common Denominator**:
- The common denominator for 350 and 6000 is 21000. We can rewrite the equation as:
\[ \frac{6000L - 350L}{21000} = 0.120 \]
- Simplifying the numerator:
\[ \frac{5650L}{21000} = 0.120 \]

7. **Solving for \( L \)**:
- Cross-multiplying to solve for \( L \):
\[ 5650L = 0.120 \times 21000 \]
- Calculating the right side:
\[ 5650L = 2520 \]
- Now, divide both sides by 5650:
\[ L = \frac{2520}{5650} \]

8. **Calculating the Length**:
- Performing the division:
\[ L \approx 0.445 \text{ meters} \approx 45 \text{ meters} \]

### Final Answer:
The approximate value of the length of the rod is **45 meters**.