Question

A string clamped at both ends, vibrates in a pattern of six loops at a frequency of 240 Hz. What frequency will cause it to vibrate in four loops?

Answer 1

Explanations & Calculations

• Hello Ahmad, explanation to this question also refers to that made for the question Q279421.
• For a string clamped at both ends, the formula $\small f = \large\frac{nv}{2L}$ gives the frequency of each mode of vibration.
• Hoping you are conversant with the derivation of that formula, it could be said that, that $\small n$ represents the number of loops present while in a vibration.
• Therefore, for the 6 loops,

$\qquad\qquad \begin{aligned} \small 240&=\small \frac{6v}{2L}\implies 80=\frac{v}{L}\cdots(1) \end{aligned}$

• For the frequency at 4 loops,

$\qquad\qquad \begin{aligned} \small f&=\small \frac{4v}{2L}\implies f=2\Big(\frac{v}{L}\Big) \end{aligned}$

• Now you know the value of the $\small v/L$ ratio, you can give it a try to get the answer.