Two wires of different densities but same area of cross-section are soldered 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.
Let the velocity of the 1st string will be $$V_1$$
$$\Rightarrow v_1= \sqrt{\dfrac{T}{m_1}}$$
because $$m_1$$= mass per unit lenght = $$\dfrac{\rho_1a_1l_1}{l_1}=\rho_1a_1$$
where $$a_1$$ = area pf the cross section
$$\Rightarrow v_1 =\sqrt{\dfrac{T}{\rho_1a_1}}$$
Let the velocity of the 2nd string will be $$V_2$$
$$\Rightarrow v_2= \sqrt{\dfrac{T}{m_2}}$$
$$\Rightarrow v_2 =\sqrt{\dfrac{T}{\rho_2a_2}}$$
given that $$V_1=V_2$$
$$\Rightarrow \sqrt {\dfrac{T}{\rho_1a_1}}$$=$$2 \sqrt{\dfrac{T}{\rho_2a_2}}$$
$$\Rightarrow\dfrac{\rho_1}{\rho_{2}}=\dfrac{1}{4}$$
$$\Rightarrow \rho_1:\rho_2= 1:4$$ ( because $$a_1=a_2$$)
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