A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $45 \;Hz$. The mass of the wire is $3.5 \times 10^{-2} \;kg$ and its linear mass density is $4.0 \times 10^{-2} \;kg m ^{-1} .$ What is
$(a) $ the speed of a transverse wave on the string, and
$(b)$ the tension in the string?
A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $45 \;Hz$. The mass of the wire is $3.5 \times 10^{-2} \;kg$ and its linear mass density is $4.0 \times 10^{-2} \;kg m ^{-1} .$ What is
$(a) $ the speed of a transverse wave on the string, and
$(b)$ the tension in the string?
Mass of the wire, $m=3.5 \times 10^{-2} \,kg$
Linear mass density, $\mu=\frac{m}{l}=4.0 \times 10^{-2} \,kg\, m ^{-1}$
Frequency of vibration, $v=45 \,Hz$
$l=\frac{m}{\mu}=\frac{3.5 \times 10^{-2}}{4.0 \times 10^{-2}}=0.875\, m$
$l-$ Iength of the wire,
The wavelength of the stationary wave ( $\lambda$ ) is related to the length of the wire by the relation:
$\lambda=\frac{2 l}{n}$
Where, $n=$ Number of nodes in the wire
For fundamental node, $n=1:$
$\lambda=2 l \Rightarrow\lambda=2 \times 0.875=1.75\, m$
The speed of the transverse wave in the string is given as:
$v=v \lambda=45 \times 1.75=78.75\, m / s$
The tension produced in the string is given by the relation:
$T=v^{2} \mu$
$=(78.75)^{2} \times 4.0 \times 10^{-2}=248.06 \,N$
Similar Questions
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- [JEE MAIN 2019]
The transverse displacement of a string (clamped at its both ends) is given by
$y(x, t)=0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$
where $x$ and $y$ are in $m$ and $t$ in $s$. The length of the string is $1.5\; m$ and its mass is $3.0 \times 10^{-2}\; kg$
Answer the following:
$(a)$ Does the function represent a travelling wave or a stationary wave?
$(b)$ Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?
$(c)$ Determine the tension in the string.
The transverse displacement of a string (clamped at its both ends) is given by
$y(x, t)=0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$
where $x$ and $y$ are in $m$ and $t$ in $s$. The length of the string is $1.5\; m$ and its mass is $3.0 \times 10^{-2}\; kg$
Answer the following:
$(a)$ Does the function represent a travelling wave or a stationary wave?
$(b)$ Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave?
$(c)$ Determine the tension in the string.
If tension in a wire is made four times, then what will be the change in speed of wave propagating in it ?
If tension in a wire is made four times, then what will be the change in speed of wave propagating in it ?
A uniform rope of mass $6\,kg$ hangs vertically from a rigid support. A block of mass $2\,kg$ is attached to the free end of the rope. A transverse pulse of wavelength $0.06\,m$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top is (in $m$ )
A uniform rope of mass $6\,kg$ hangs vertically from a rigid support. A block of mass $2\,kg$ is attached to the free end of the rope. A transverse pulse of wavelength $0.06\,m$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top is (in $m$ )
Write definition and dimensional formula of linear mass density of string.
Write definition and dimensional formula of linear mass density of string.