Question 206: Waves

Question

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 $$7.7 	imes {10^3}\,kg/{m^3}$$ and $$2.2 	imes {10^{11}}\,N/{m^2}$$ respectively ?

Accepted Answer

Fundamental frequency,
$$f = {v \over {2\ell }} = {1 \over {2\ell }}\sqrt {{T \over \mu }}  = {1 \over {2\ell }}\sqrt {{T \over {Aho }}} $$
$$\left[ {\,\,} ight.$$ as $$v = \sqrt {{T \over \mu }} $$ $$\,\,\,\,\,\,$$ and $$\,\,\,\,\,\,$$ $$\left. {\mu  = {m \over \ell }\,\,} ight]$$
Also,  $$Y = {{T\ell } \over {A\Delta \ell }} \Rightarrow {T \over A} = {{Y\Delta \ell } \over \ell }$$
$$ \Rightarrow f = {1 \over {2\ell }}\sqrt {{{\gamma \Delta \ell } \over {\ell ho }}} ....\left( i ight)$$
Putting the value of $$\ell ,{{\Delta \ell } \over \ell },ho $$ $$\,\,\,\,\,\,$$ and
$$\,\,\,\,\,\,$$ $$\gamma $$ in $$e{q^n}.\left( i ight)$$ we get,
$$f = \sqrt {{2 \over 7}}  	imes {{{{10}^3}} \over 3}$$ $$\,\,\,\,\,\,$$ or, $$\,\,\,\,\,\,$$ $$f \approx 178.2\,Hz$$

Answer

$$188.5$$ $$Hz$$

Answer

$$178.2$$ $$Hz$$

Answer

$$200.5$$ $$Hz$$

Answer

$$770$$ $$Hz$$

Waves

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 ?

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