The diameter of a stretched string is increased 3%, keeping the other parameters constant then the velocity is x% decreases what is the value of x ?

Answer

To solve the problem, we need to understand how the velocity of a stretched string is affected by changes in its diameter. The formula for the velocity \( v \) of a stretched string is given by:

\[ v = \sqrt{\frac{T}{\rho A}} \]

where:
- \( T \) is the tension in the string,
- \( \rho \) is the density of the string,
- \( A \) is the cross-sectional area of the string.

The cross-sectional area \( A \) of a circular string can be expressed in terms of its diameter \( d \):

\[ A = \frac{\pi d^2}{4} \]

Substituting this into the velocity equation gives:

\[ v = \sqrt{\frac{T}{\rho \left(\frac{\pi d^2}{4}\right)}} = \sqrt{\frac{4T}{\pi \rho d^2}} \]

From this, we can see that the velocity \( v \) is inversely proportional to the square of the diameter \( d \):

\[ v \propto \frac{1}{d} \]

Now, if the diameter is increased by 3%, we can express this mathematically. If the original diameter is \( d \), the new diameter \( d' \) after a 3% increase is:

\[ d' = d + 0.03d = 1.03d \]

Since the velocity is inversely proportional to the diameter, we can express the new velocity \( v' \) in terms of the old velocity \( v \):

\[ v' = k \cdot \frac{1}{d'} = k \cdot \frac{1}{1.03d} = \frac{v}{1.03} \]

Now, to find the percentage decrease in velocity, we can calculate:

\[ \text{Percentage decrease} = \frac{v - v'}{v} \times 100\% \]

Substituting \( v' \):

\[ \text{Percentage decrease} = \frac{v - \frac{v}{1.03}}{v} \times 100\% = \left(1 - \frac{1}{1.03}\right) \times 100\% \]

Calculating \( \frac{1}{1.03} \):

\[ \frac{1}{1.03} \approx 0.9709 \]

Thus,

\[ 1 - 0.9709 \approx 0.0291 \]

Now, converting this to a percentage:

\[ \text{Percentage decrease} \approx 0.0291 \times 100\% \approx 2.91\% \]

So, the value of \( x \) (the percentage decrease in velocity) is approximately 2.91%.

Rounding this, we can say:

\[ x \approx 3\% \]

### Final Answer:
The value of \( x \) is approximately 3%.