A musical instrument is made using four different metal strings, 1, 2, 3 and 4 with mass per unit length
`mu,2mu,3mu and 4mu` respectively. The instrument is played by vibrating the strings by varying the free length in between the range
`L_0 and 2L_0`. It is found that in string-1
`(mu)` at free length `L_0` and tension `T_0` the fundamental mode frequency is `f_0` .
List-I gives the above four strings while list-II lists the magnitude of some quantity.

If the tension in each string is `T_0`, the correct match for the highest fundamental frequency in `f_0` units will be -

Options

A. `ItoP,IItoQ,IIItoT,IVtoU`
B. `ItoP,IItoR,IIItoS,IVtoQ`
C. `ItoP,IItoQ,IIItoR,IVtoT`
D. `ItoT,IItoQ,IIItoR,IVtoU`

Answer

A

Explanation

`f_1=f_0=(c_1)/(2L_0)=1/(2L_0)sqrt((T_0)/(mu))`
`f_2=3 fo_2=3(C_2)/(2L_2)=(3C_2)/(2xx(3L_0)/2)=(c_2)/L_0=f_0`
`implies (C_2)/L_0=(C_1)/(2L_0)=implies C_2=(C_1)/2implies (T_2)/(2mu)=1/4xx(T_0)/(mu)`
`f_3=5 fo_3=(5C_3)/(2xx(5L_0)/4)=(2C_3)/(L_0)implies fo=(2C_3)/(L_0)`
`implies C_3=(C_1)/4implies (T_3)/(3mu)=1/16xx(T_0)/(mu)impliesT_3-(3)/16T_0`
`f_4=14 fo_4=14(C_4)/(2L_4)=(14xxC_4)/(2xx(7L_0)/4)=(4C_4)/(L_0)implies f_0=(C_1)/(2L_0)`
`implies C_4=(C_1)/8implies(T_4)/(4mu)=1/64xx(T_0)/(mu)impliesT_4=(T_4)/16`
Correct option is (A)