A copper rod 1 m long and clamped at a point distant 25 cm, from its end is set in longitudinal vibrations and is used to produce stationary waves is a Kundt's tube containing air at `30^@C.` Heaps of lycopodium dust are found to be 4.95 cm apart. What is the velocity of longitudinal waves in copper ? Velocity of sound in air at `0^@C"= 332 m/s."`

Answer

To find the velocity of longitudinal waves in copper, we can use the relationship between the velocities of sound in different media and the physical parameters of the system. Here’s a step-by-step solution:

### Step 1: Identify Given Values
- Length of the copper rod (L) = 25 cm = 0.25 m
- Distance between heaps of lycopodium dust (ΔL) = 4.95 cm = 0.0495 m
- Velocity of sound in air at 0°C (V_air) = 332 m/s

### Step 2: Use the Relationship Between Velocities
The relationship between the velocities of sound in copper (V_copper) and air is given by the formula:
\[ \frac{V_{copper}}{V_{air}} = \frac{2L}{\Delta L} \]

### Step 3: Rearrange the Formula to Solve for V_copper
We can rearrange the formula to find the velocity of sound in copper:
\[ V_{copper} = V_{air} \times \frac{2L}{\Delta L} \]

### Step 4: Substitute the Known Values
Now, substitute the known values into the equation:
\[ V_{copper} = 332 \, \text{m/s} \times \frac{2 \times 0.25 \, \text{m}}{0.0495 \, \text{m}} \]

### Step 5: Calculate the Value
First, calculate the fraction:
\[ \frac{2 \times 0.25}{0.0495} = \frac{0.5}{0.0495} \approx 10.101 \]
Now, multiply by the velocity of sound in air:
\[ V_{copper} \approx 332 \, \text{m/s} \times 10.101 \approx 3353.53 \, \text{m/s} \]

### Final Answer
The velocity of longitudinal waves in copper is approximately **3353.53 m/s**.

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