The density of the stretched string is changed by 2% without change in tension and radius. The change in transverse wave velocity.

Options

A. 2% increase
B. 1% increase
C. 1% increase or decrease
D. 4% change

Answer

To find the change in transverse wave velocity when the density of a stretched string is changed by 2%, we can follow these steps:

### Step 1: Understand the formula for wave velocity
The velocity \( V \) of a transverse wave on a string is given by the formula:

\[ V = \sqrt{\frac{T}{\rho}} \]

where \( T \) is the tension in the string and \( \rho \) is the density of the string.

### Step 2: Identify the changes
In this problem, we are told that the density \( \rho \) changes by 2% while the tension \( T \) remains constant.

### Step 3: Calculate the percentage change in velocity
The percentage change in velocity can be approximated using the following relationship:

\[ \frac{\Delta V}{V} \approx \frac{1}{2} \frac{\Delta T}{T} - \frac{1}{2} \frac{\Delta \rho}{\rho} \]

Since there is no change in tension (\( \Delta T = 0 \)) and the area of cross-section does not change, the first term becomes zero. The only term that contributes is the change in density.

Given that the density changes by 2%, we have:

\[ \frac{\Delta \rho}{\rho} = 0.02 \]

Substituting this into our equation gives:

\[ \frac{\Delta V}{V} \approx -\frac{1}{2} \cdot 0.02 \]

### Step 4: Simplify the expression
Calculating the above expression:

\[ \frac{\Delta V}{V} \approx -0.01 \]

This means that the percentage change in velocity is approximately -1%.

### Step 5: Interpret the result
The negative sign indicates that the velocity decreases. Therefore, the change in transverse wave velocity is a decrease of 1%.

### Final Answer
The change in transverse wave velocity is a decrease of 1%.

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