Waves

JEE Main level, numerical answer, medium

66 of 142

Question 66 of 142

A guitar string of length 90cm90cm vibrates with a fundamental frequency of 120Hz120Hz. The length of the string producing a fundamental of 180Hz180Hz will be _____ cmcm

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Answer: 60

The formula to calculate the fundamental frequency produced in a vibrating string is given by
f0=v2L...(1){f}_{0}=\dfrac{v}{2L}...\left(1\right)
For the first case, it can be written that
120=v2(90)...(2)120=\dfrac{v}{2(90)}...\left(2\right)
Similarly, for the second case, it can be written that
180=v2(L)...(3)180=\dfrac{v}{2(L)}...\left(3\right)
Divide equation (3) by equation (2) and solve to obtain the required length of the string.
180120=v2(L)v2(90)=90L⇒L=90×120180=60\begin{matrix}\dfrac{180}{120} & = & \dfrac{\dfrac{v}{2\left(L\right)}}{\dfrac{v}{2\left(90\right)}} \\ & = & \dfrac{90}{L} \\ & \Rightarrow & L=90\times \dfrac{120}{180} \\ & = & 60\end{matrix}

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