JEE MAIN - Physics (2013 (Offline) - No. 19)
A sonometer wire of length $$1.5$$ $$m$$ is made of steel. The tension in it produces an elastic strain of $$1\% $$. What is the fundamental frequency of steel if density and elasticity of steel are $$7.7 \times {10^3}\,kg/{m^3}$$ and $$2.2 \times {10^{11}}\,N/{m^2}$$ respectively ?
$$188.5$$ $$Hz$$
$$178.2$$ $$Hz$$
$$200.5$$ $$Hz$$
$$770$$ $$Hz$$
Explanation
Fundamental frequency,
$$f = {v \over {2\ell }} = {1 \over {2\ell }}\sqrt {{T \over \mu }} = {1 \over {2\ell }}\sqrt {{T \over {A\rho }}} $$
$$\left[ {\,\,} \right.$$ as $$v = \sqrt {{T \over \mu }} $$ $$\,\,\,\,\,\,$$ and $$\,\,\,\,\,\,$$ $$\left. {\mu = {m \over \ell }\,\,} \right]$$
Also, $$Y = {{T\ell } \over {A\Delta \ell }} \Rightarrow {T \over A} = {{Y\Delta \ell } \over \ell }$$
$$ \Rightarrow f = {1 \over {2\ell }}\sqrt {{{\gamma \Delta \ell } \over {\ell \rho }}} ....\left( i \right)$$
Putting the value of $$\ell ,{{\Delta \ell } \over \ell },\rho $$ $$\,\,\,\,\,\,$$ and
$$\,\,\,\,\,\,$$ $$\gamma $$ in $$e{q^n}.\left( i \right)$$ we get,
$$f = \sqrt {{2 \over 7}} \times {{{{10}^3}} \over 3}$$ $$\,\,\,\,\,\,$$ or, $$\,\,\,\,\,\,$$ $$f \approx 178.2\,Hz$$
$$f = {v \over {2\ell }} = {1 \over {2\ell }}\sqrt {{T \over \mu }} = {1 \over {2\ell }}\sqrt {{T \over {A\rho }}} $$
$$\left[ {\,\,} \right.$$ as $$v = \sqrt {{T \over \mu }} $$ $$\,\,\,\,\,\,$$ and $$\,\,\,\,\,\,$$ $$\left. {\mu = {m \over \ell }\,\,} \right]$$
Also, $$Y = {{T\ell } \over {A\Delta \ell }} \Rightarrow {T \over A} = {{Y\Delta \ell } \over \ell }$$
$$ \Rightarrow f = {1 \over {2\ell }}\sqrt {{{\gamma \Delta \ell } \over {\ell \rho }}} ....\left( i \right)$$
Putting the value of $$\ell ,{{\Delta \ell } \over \ell },\rho $$ $$\,\,\,\,\,\,$$ and
$$\,\,\,\,\,\,$$ $$\gamma $$ in $$e{q^n}.\left( i \right)$$ we get,
$$f = \sqrt {{2 \over 7}} \times {{{{10}^3}} \over 3}$$ $$\,\,\,\,\,\,$$ or, $$\,\,\,\,\,\,$$ $$f \approx 178.2\,Hz$$
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