Question 518: Waves

Question

A musical instrument is made using four different metal strings, 1, 2, 3 and 4 with mass per unit length $\mu $, 2$\mu $, 3$\mu $ and 4$\mu $ 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
![](https://imagex.cdn.examgoal.net/1kt3lkino/c2d1d6dd-7da2-412d-8c49-e9283e26ccf2/ce8fb340-0c4a-11ec-a7b0-c3f7d2fd429a/file-1kt3lkinp.jpeg)

Accepted Answer

Fundamental frequency is maximum when length is minimum i.e. L
0
,
Case 1. $L = {L_0},\,T = {T_0},\,f = {f_0}$;
${f_1} = {1 \over {2{L_0}}}\sqrt {{{{T_0}} \over \mu }} $
Case 2. ${f_2} = {1 \over {{L_0}}}\sqrt {{{{T_2}} \over {2\mu }}}  = {{{f_0}} \over {\sqrt 2 }}$
Case 3. ${f_3} = {1 \over {{L_0}}}\sqrt {{{{T_2}} \over {3\mu }}}  = {{{f_0}} \over {\sqrt 3 }}$
Case 4. ${f_4} = {1 \over {{L_0}}}\sqrt {{{{T_2}} \over {4\mu }}}  = {{{f_0}} \over 2}$

Answer

I $ 	o $ P, II $ 	o $ Q, III $ 	o $ T, IV $ 	o $ S

Answer

I $ 	o $ P, II $ 	o $ R, III $ 	o $ S, IV $ 	o $ Q

Answer

I $ 	o $ Q, II $ 	o $ S, III $ 	o $ R, IV $ 	o $ P

Answer

I $ 	o $ Q, II $ 	o $ P, III $ 	o $ R, IV $ 	o $ T

Waves

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