Question 127: Waves

Question

Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section R is $v_1$, and that in the other string having radius of cross section R/2 is $v_2$. Then $\frac{v_2}{v_1}$ =

Accepted Answer

To find the ratio of the velocities of transverse waves in two strings with different radii but identical materials and tension, consider the following:
The wave velocity $ v $ in a string is given by the formula:
$ v = \sqrt{\frac{T}{\mu}} $
where $ T $ is the tension and $ \mu $ is the linear mass density of the string, defined as:
$ \mu = \rho \pi R^2 $
Here, $ \rho $ is the density of the material, and $ R $ is the radius of the string. Given that both strings have the same tension $ T $ and material, we compare their wave velocities $ v_1 $ and $ v_2 $ for radii $ R_1 = R $ and $ R_2 = \frac{R}{2} $.
The velocity ratio is expressed as:
$ \frac{v_2}{v_1} = \frac{\sqrt{\frac{T}{\rho \pi R_2^2}}}{\sqrt{\frac{T}{\rho \pi R_1^2}}} $
Simplifying this expression:
$ \frac{v_2}{v_1} = \frac{\sqrt{R_1^2}}{\sqrt{R_2^2}} = \frac{R_1}{R_2} = \frac{R}{\frac{R}{2}} = 2 $
Thus, the ratio $ \frac{v_2}{v_1} $ is 2.

Answer

8

Answer

4

Answer

2

Answer

$\sqrt{2}$

Waves

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