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A taut string for which $\mu =5.00\times {{10}^{-2}}kg/m$ is under a tension of 80.0 N. How much power must be supplied to the string to generate sinusoidal waves at a frequency of 60.0 Hz and an amplitude of 6.00 cm?

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Last updated date: 13th Jun 2024
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Answer
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Hint:To find the power needed to generate sinusoidal wave in a taut string, we first find total energy for sinusoidal wave associated with the string using the formula $E=\dfrac{1}{2}\mu x{{\omega }^{2}}{{A}^{2}}$ and then take a differentiation of it with respect to time to find power.

Formula used:
$E=\dfrac{1}{2}\mu x{{\omega }^{2}}{{A}^{2}}$

Complete step-by-step answer:
For a sinusoidal wave –
Total energy is given by $E=\dfrac{1}{2}\mu x{{\omega }^{2}}{{A}^{2}}$
Where,
$\mu $ is the linear mass density of string
x is the length element of string
$\omega $ is the angular frequency of wave, and A is the amplitude
So, power will be given by
$P=\dfrac{dE}{dt}$
After putting value of Energy (E) into power formula we get
$P=\dfrac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}v$
Where, v will be the speed of the sinusoidal wave, i.e., $\dfrac{dx}{dt}$ along the string.
Velocity can be found by $v=\sqrt{\dfrac{T}{\mu }}$, where T is the tension applied on string.
We want to find the power needed to generate sinusoidal waves of frequency 60 Hz and 6 cm amplitude.
At first, we find the velocity of wave along string, which is $v=\sqrt{\dfrac{T}{\mu }}$
Substituting the values, we get
$v=\sqrt{\dfrac{80}{5\times {{10}^{-2}}}}=40m/\sec $
Now we need to convert frequency into angular frequency
$\begin{align}
  & \omega =2\pi \theta \\
 & \Rightarrow \omega =2\pi \times 60=377{{\sec }^{-1}} \\
\end{align}$
Now it is given the amplitude is 6cm, converting it to metre, we get
A = 0.06m
Substituting these values into the formula of power, we get
$\begin{align}
  & P=\dfrac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}v \\
 & \Rightarrow P=\dfrac{1}{2}\times \left( 5\times {{10}^{-2}} \right)\times {{\left( 377 \right)}^{2}}\times {{\left( 0.06 \right)}^{2}}\times 40 \\
 & \Rightarrow P=512W \\
\end{align}$
Hence, 512W power must be supplied to the string to generate sinusoidal waves at a frequency of 60.0 Hz and an amplitude of 6.00 cm

Note: To generate sinusoidal waves in a string, we generate an impulse at one end of it, which travels along the string.
These kinds of waves generally need a medium to travel, in this question wave is travelling through a string.
Power is generally defined as the rate at which work is done. So mathematically it is a time differential of Energy or work done.
In the above question, our aim is to find the power required to generate a sinusoidal wave along a string which is under tension. We first find the total energy contained in this wave and take a time differentiation of it to find the power. After putting all the values in the power formula, we find the numerical value of it.