Two wires are kept tight between the same pair of supports. The tensions in the wires are in the ratio 2 : 1, the radii are in the ratio 3 : 1 and the densities are in the ratio 1 : 2. Find the ratio of their fundamental frequencies.

Answer

2:3

Explanation

Frequency `f=1/(Id)sqrt(T/(piP))`
`rarr f_1=1/(I_1D_1)sqrt(T_1/(piP_1))`
`rarr f_2=1/(I_2D_2)sqrt(T/(piP_2))`
Given `T_1/T_2=2`
`r_1/r_2=3=D_1/D_2`
`P_1/P_2=1/2`
`So, f_1/f_2=(I_2D_2)/(I_1D_1)sqrt(T_1/T_2) sqrt((piP_2)/(piP_1))`
`I_1=I_2`=length of string)
`rarr f_1/f_2=1/3 sqrt(2/1xx2/1)`
`rarr f_1:f_2=2:3`