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)`sqrt2v`
B. (b)v
C. (c)4v
D. (d)`2 sqrt2v`

Explanation

To find the new speed of the transverse wave on the string after changing the tension and radius, we will follow these steps:

### Step 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.

### Step 2: Express mass per unit length in terms of density and radius
The mass per unit length \( \mu \) can be expressed as:
\[ \mu = \frac{m}{L} = \frac{\rho V}{L} = \frac{\rho (A \cdot L)}{L} = \rho A \]
where \( \rho \) is the density of the material, \( V \) is the volume, and \( A \) is the cross-sectional area. For a circular cross-section, the area \( A \) is given by:
\[ A = \pi r^2 \]
Thus, we can write:
\[ \mu = \rho \pi r^2 \]

### Step 3: Substitute \( \mu \) into the wave speed formula
Substituting \( \mu \) into the wave speed formula gives:
\[ v = \sqrt{\frac{T}{\rho \pi r^2}} \]

### Step 4: Analyze the changes in tension and radius
We are given that:
- The tension is increased by a factor of 4, so \( T_2 = 4T_1 \).
- The radius is increased by a factor of 2, so \( r_2 = 2r_1 \).

### Step 5: Write the new wave speed in terms of the new tension and radius
The new speed \( v_2 \) can be expressed as:
\[ v_2 = \sqrt{\frac{T_2}{\mu_2}} = \sqrt{\frac{4T_1}{\rho \pi (2r_1)^2}} \]
Substituting \( r_2 \):
\[ v_2 = \sqrt{\frac{4T_1}{\rho \pi (4r_1^2)}} = \sqrt{\frac{4T_1}{4\rho \pi r_1^2}} = \sqrt{\frac{T_1}{\rho \pi r_1^2}} \]

### Step 6: Relate the new speed to the original speed
Notice that:
\[ v_2 = \sqrt{\frac{T_1}{\rho \pi r_1^2}} = v_1 \]
Thus, the new wave speed \( v_2 \) is equal to the original wave speed \( v_1 \).

### Conclusion
The new speed of the wave \( v_2 \) is:
\[ v_2 = v \]

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

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