(A) The cord vibrates in its simplest standing-wave mode (fundamental mode),where the length of the cord $d$ corresponds to half a wavelength,i.e.,$d

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(A) The cord vibrates in its simplest standing-wave mode (fundamental mode),where the length of the cord $d$ corresponds to half a wavelength,i.e.,$d = \frac{\lambda}{2}$,or $\lambda = 2d$.The frequency of the standing wave $f$ is equal to the frequency of the $AC$ current.The wave speed $v$ on a stretched string is given by $v = \sqrt{\frac{T}{\mu}}$.Using the wave equation $v = f\lambda$,we substitute $\lambda = 2d$ to get $v = f(2d) = 2fd$.Equating the two expressions for $v$: $\sqrt{\frac{T}{\mu}} = 2fd$.Squaring both sides: $\frac{T}{\mu} = 4f^2 d^2$.Therefore,the relationship is $T = 4\mu f^2 d^2$.

Class 11 Physics Waves and Sound Transverse Stationary Waves and Sonometer

Difficult

One insulated conductor from a household extension cord has a mass per unit length of $\mu$. $A$ section of this conductor is held under tension between two clamps. $A$ subsection is located in a magnetic field of magnitude $B$ directed perpendicular to the length of the cord. When the cord carries an $AC$ current of $i$ at a frequency of $f$,it vibrates in resonance in its simplest standing-wave vibration state. Determine the relationship that must be satisfied between the separation $d$ of the clamps and the tension $T$ in the cord.

A
$T = 4\mu f^2 d^2$
B
$T = 2\mu f^2 d^2$
C
$T = \frac{\mu f^2 d^2}{2}$
D
$T = \frac{\mu f^2 d^2}{4}$

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Waves and Sound

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Transverse Stationary Waves and Sonometer

The string of a violin has a frequency of $440 \,cps$. If the violin string is shortened by one fifth,its frequency will be changed to ........... $cps$.

Medium

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In an experiment to study standing waves,you use a string whose mass per unit length is $\mu = (1.0 \pm 0.1) \times 10^{-4} \ kg/m$. You look at the fundamental mode,whose frequency $f$ is related to the length $L$ and tension $T$ of the string by the equation $L = \frac{1}{2f} \sqrt{\frac{T}{\mu}}$. You make a plot with $L$ on the $y$-axis and $\sqrt{T}$ on the $x$-axis,and find that the best-fitting line is $y = (8.0 \pm 0.3) \times 10^{-3}x + (0.2 \pm 0.04)$ in $SI$ units. What is the value of the frequency of the wave (including the error)? Express your result in $SI$ units $(Hz)$.

Difficult

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$A$ uniform wire of length $L$,diameter $D$,and density $\rho$ is stretched under a tension $T$. The correct relation between its fundamental frequency $f$,the length $L$,and the diameter $D$ is:

EasyKCET 2010

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$A$ wire under tension vibrates with a fundamental frequency of $600 \,Hz$. If the length of the wire is doubled, the radius is halved and the wire is made to vibrate under one-ninth the tension. Then the fundamental frequency will become (in $\,Hz$)

MediumKCET 2011

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Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$, the ratio of the lengths of the strings is :

DifficultAP EAMCET 2003

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The string of a violin has a frequency of $440 \,cps$. If the violin string is shortened by one fifth,its frequency will be changed to ........... $cps$.

$A$ uniform wire of length $L$,diameter $D$,and density $\rho$ is stretched under a tension $T$. The correct relation between its fundamental frequency $f$,the length $L$,and the diameter $D$ is:

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