The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by 4%, will be ______%. 

CONCEPT:

The transverse speed of the stretched string is written as;

v = \(\sqrt {\frac{T}{μ}}\)

Here we have T as the tension of the string, μ is the mass per unit length and v is the velocity of the string.

CALCULATIONS:

Given: \(\frac{\Delta T}{T}\times 100\) = 4 %

As we know that the velocity of the string with tension T is written as;

v = \(\sqrt {\frac{T}{μ}}\) 

v = \(\frac{T^{1/2}}{μ^{1/2}}\)    ----(1)

Let us calculate the relative change of equation (1) we have;

\( {\frac {\Delta v}{v}} = \frac {1}{2} \frac{\Delta T}{T}\)    -----(2)

Here we have \(\frac{\Delta μ}{μ}\) is constant.

Now multiplying equation (2) by 100 on both sides we have;

\( {\frac {\Delta v}{v}}\times 100 = \frac {1}{2} \frac{\Delta T}{T} \times 100\)

⇒ \( {\frac {\Delta v}{v}}\times 100 = \frac {1}{2}\times 4 \) %

⇒ \( {\frac {\Delta v}{v}}\) % = 2 %

Therefore percentage increase in the speed of transverse waves produced in a stretched string is 2%

Hence 2% increase in the speed of transverse waves.