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

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

Choose the correct answer: