A wave moves with speed `300 m//s` on a wire which is under a tension of `500 N`. Find how much tension must be changed to increase the speed to `312 m//s`?

Explanation

To solve the problem of how much tension must be changed to increase the speed of a wave on a wire from 300 m/s to 312 m/s, we can follow these steps:

### Step-by-step Solution:

1. **Understand the relationship between wave speed and tension**:
The speed of a transverse wave on a wire is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( v \) is the wave speed, \( T \) is the tension in the wire, and \( \mu \) is the mass per unit length of the wire.

2. **Identify the initial conditions**:
We know the initial speed \( v_1 = 300 \, \text{m/s} \) and the initial tension \( T_1 = 500 \, \text{N} \).

3. **Calculate the mass per unit length (\( \mu \))**:
Rearranging the wave speed formula, we can express \( T \) in terms of \( v \) and \( \mu \):
\[ T = \mu v^2 \]
For the initial conditions:
\[ T_1 = \mu v_1^2 \implies 500 = \mu (300)^2 \]
\[ \mu = \frac{500}{90000} = \frac{1}{180} \, \text{kg/m} \]

4. **Calculate the new tension for the new speed**:
Now, we want to find the new tension \( T_2 \) when the speed is \( v_2 = 312 \, \text{m/s} \):
\[ T_2 = \mu v_2^2 \]
Substituting \( \mu \) and \( v_2 \):
\[ T_2 = \frac{1}{180} (312)^2 \]
\[ T_2 = \frac{1}{180} \times 97344 = 540 \, \text{N} \]

5. **Calculate the change in tension**:
The change in tension \( \Delta T \) is given by:
\[ \Delta T = T_2 - T_1 \]
\[ \Delta T = 540 \, \text{N} - 500 \, \text{N} = 40 \, \text{N} \]

### Final Answer:
The tension must be increased by **40 N** to achieve the new wave speed of 312 m/s.