if tension of a wire is increased to 3 times, the wave speed becomes____the initial speed.

Options

A. 3 times
B. 9 times
C. `(1)/(3)` times
D. `sqrt(3)` times

Answer

To solve the problem of how the wave speed changes when the tension in a wire is increased to three times its initial value, we can follow these steps:

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

### Step 2: Establish the initial conditions
Let the initial tension be \( T_1 \) and the initial wave speed be \( v_1 \). According to the formula:
\[ v_1 = \sqrt{\frac{T_1}{\mu}} \]

### Step 3: Determine the new tension
If the tension is increased to three times its initial value, we have:
\[ T_2 = 3T_1 \]

### Step 4: Calculate the new wave speed
Using the new tension \( T_2 \), the new wave speed \( v_2 \) can be expressed as:
\[ v_2 = \sqrt{\frac{T_2}{\mu}} = \sqrt{\frac{3T_1}{\mu}} \]

### Step 5: Relate the new wave speed to the initial wave speed
Now, we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = \sqrt{3} \cdot \sqrt{\frac{T_1}{\mu}} = \sqrt{3} \cdot v_1 \]

### Step 6: Conclusion
Thus, the new wave speed \( v_2 \) is:
\[ v_2 = \sqrt{3} \cdot v_1 \]

So, if the tension of a wire is increased to three times, the wave speed becomes \( \sqrt{3} \) times the initial speed.

### Final Answer
The wave speed becomes \( \sqrt{3} \) times the initial speed.

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