Question

Two wires of different densities but same area of cross section are sodered together at one end and are stretched to a tension T. The velocity of a transverse wave in the first wire is double of that in the second wire. Find the ratio of the density of the first wire to that of the second wire.

Answer 1

Let, $v_1$ = velocity in the $1^{st}$ string

$\Rightarrow v_1=\sqrt{\left(\frac{T}{m}\right)}$

Because, $m_1=$ mass per unit length

$=\left(\frac{p_1{a}_1}{I_1}\right)$

$=p_1{a}_1$

where $a_1=$ Area of cross section

$\Rightarrow v_1=\sqrt{\left(\frac{T}{p_1{a}_1}\right)}$ ...(1)

Let $v_2=$ velocity in the second string

$\Rightarrow v_2=\sqrt{\left(\frac{T}{m_2}\right)}$

$\Rightarrow v_2=\sqrt{\left(\frac{T}{p_2{a}_2}\right)}$ ...(2)

Given that,

$\Rightarrow v_1=2v_2$

$\Rightarrow \sqrt{\left(\frac{T}{a_1{p}_1}\right)}=2\sqrt{\left(\frac{T}{a_2{p}_2}\right)}$

$\Rightarrow \frac{p_1}{p_2}=\frac{1}{4}$

$\Rightarrow p_1:p_2=1:4$

$a_1=a_2$