Question

A string of mass $2.50kg$ is under a tension of $200N$. The length of the stretched string is $20.0m$. If the transverse jerk is struck at one end of the string, how long does his disturbance take to reach the other end?

Answer 1

The velocity of longitudinal waves on the string depends on mass per unit length and the tension applied to the string. The time taken to cover the length can be calculated using the basic formula of motion,
$\text{time}=\dfrac{\text{Distance}}{\text{speed}}$

Complete step by step solution:
It is given that the mass of the string is $m=2.50kg$
The magnitude of the tension applied is $T=200N$
Length of the string is $L=20.0m$
The mass per unit length of the string, $\mu =\dfrac{\text{Mass}}{\text{Length}}$
Putting the values $m=2.50kg$ and $L=20.0m$, we get
$\begin{aligned} & \mu =\dfrac{2.5}{20}kg/m \\\ & =0.125kg/m \end{aligned}$
Hence, the mass per unit length of string is$\mu =0.125kg/m$.
Speed of the longitudinal wave on the string is given by the formula,
$v=\sqrt{\dfrac{T}{\mu }}$
Putting the values$T=200N$ and $\mu =0.125kg/m$, we get
$\begin{aligned} & v=\sqrt{\dfrac{200}{0.125}}m/s \\\ & =\sqrt{1600}m/s \\\ & =40m/s \end{aligned}$

Hence, the speed of the longitudinal wave on the string is $40m/s$
Now, the distance to cover on the string for the longitudinal wave is $L=20m$
The time taken by the longitudinal wave can be given as,
$t=\dfrac{\text{distance}}{\text{speed}}$
Putting the values$\text{distance=20m}$ and $\text{speed}=40m/s$, we get
$\begin{aligned} & t=\dfrac{20}{40}s \\\ & =0.5s \end{aligned}$

Therefore, the time taken by the longitudinal wave to cover the string length is 0.5s.

Note: It is assumed that the mass is uniformly distributed all over the length of the string. It is assumed that the tension applied to the string is uniform all over the string along the length of the string.