Question 57: 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 L0 and 2L0. It is found that in string-1$$\mu $$ at free length L0 and tension T0 the fundamental mode frequency is f0.List-I gives the above four strings while list-II lists the magnitude of some quantity.  If the tension in each string is T0, the correct match for the highest fundamental frequency in f0 units will be

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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