Question

Wire A and wire B are made from the same material and are under the same tension, but waves travel along string B at one-third the wave speed of string A. The radius of wire A is 0.100 mm. What is the radius of wire B

Answer 1

Answer:

0.3 mm

Explanation:

$\rho$ = Density of material

$\mu$ = Linear density of the material = $\rho A$

Area = Area = $\pi r^2$

r = Radius

The velocity of a wave is given by

$v=\sqrt{\dfrac{T}{\mu}}\\\Rightarrow v=\sqrt{\dfrac{T}{\rho A}}\\\Rightarrow v=\sqrt{\dfrac{T}{\rho \pi r^2}}$

It can be seen that the velocity is inversly proportional to the radius

$v\propto \sqrt{\dfrac{1}{r^2}}\\\Rightarrow v\propto \dfrac{1}{r}$

So,

$\dfrac{v_a}{v_b}=\dfrac{r_b}{r_a}$

From the question

$v_b=\dfrac{1}{3}v_a$

$\\\Rightarrow \dfrac{v_a}{\dfrac{1}{3}v_a}=\dfrac{r_b}{0.1}\\\Rightarrow 3=\dfrac{r_b}{0.1}\\\Rightarrow r_b=3\times 0.1\\\Rightarrow r_b=0.3\ mm$

The radius of wire B is 0.3 mm