Speed of transverse wave on string is v . If tension is increased by factor of 4 and radius of the string is increased by factor of 2, then the new wave speed will be :

Options

A. (a)`sqrt(2) v`
B. (b)v
C. (c)4v
D. (d)`2sqrt(2)v`

Explanation

To solve the problem, we need to find the new speed of the transverse wave on a string when the tension is increased by a factor of 4 and the radius of the string is increased by a factor of 2.

### Step-by-Step Solution:

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

2. **Express mass per unit length \( \mu \):**
The mass per unit length \( \mu \) can be expressed in terms of the density \( \rho \) and the radius \( r \) of the string:
\[ \mu = \frac{m}{l} = \frac{\rho \cdot V}{l} = \frac{\rho \cdot (\pi r^2 l)}{l} = \rho \cdot \pi r^2 \]
where \( V \) is the volume of the string.

3. **Substitute \( \mu \) into the wave speed formula:**
Now substituting \( \mu \) back into the wave speed formula:
\[ v = \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]

4. **Determine the new tension and radius:**
- The tension is increased by a factor of 4, so the new tension \( T' = 4T \).
- The radius is increased by a factor of 2, so the new radius \( r' = 2r \).

5. **Calculate the new mass per unit length \( \mu' \):**
With the new radius, the new mass per unit length \( \mu' \) becomes:
\[ \mu' = \rho \cdot \pi (r')^2 = \rho \cdot \pi (2r)^2 = \rho \cdot \pi \cdot 4r^2 = 4\rho \cdot \pi r^2 \]

6. **Substitute the new values into the wave speed formula:**
The new wave speed \( v' \) can be calculated as:
\[ v' = \sqrt{\frac{T'}{\mu'}} = \sqrt{\frac{4T}{4\rho \cdot \pi r^2}} = \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]

7. **Conclusion:**
Notice that the expression for \( v' \) simplifies to:
\[ v' = \sqrt{\frac{T}{\rho \cdot \pi r^2}} = v \]
Therefore, the new wave speed \( v' \) is equal to the original wave speed \( v \).

### Final Answer:
The new wave speed will be \( v \).