Question

The frequency of sound can remain unchanged even if we double the length of the wire, provided:

• A. Tension of the wire is doubled
• B. Tension of the wire is decreased four-fold
• C. Tension of the wire is halved
• D. Tension of the wire is increased four-fold

Answer 1

The correct option is C Tension of the wire is halved

Answer is C.
the dependence of the frequency of the fundamental mode of vibration of the string on its wavelength: $f=\frac{v}{\lambda }$.
The length of the string to the wavelength of the fundamental mode: $\lambda =2L$.
Substituting for $\lambda$ in the first equation, we get, $f=\frac{v}{2L}$.
Tthe dependence of the speed of waves on the string on the tension in the string: $v=\sqrt{\frac{F}{\mu }}$.
Substituting for v in the above expression for f, we get:$f=\frac{1}{2L}\sqrt{\frac{F}{\mu }}$.
From the above equation, we can infer that, the frequency of sound can remain unchanged even if we double the length of the wire, provided the tension of the wire is halved.