Question

The time taken by pulse to reach the other end of rope is:
A) $\sqrt {\dfrac{{2L}}{g}} $.
B) $\sqrt {\dfrac{L}{g}} $.
C) $2\sqrt {\dfrac{L}{g}} $.
D) $\sqrt {\dfrac{L}{{2g}}} $.

Answer 1

Hint: The fundamental vibration of a string has $\lambda = 2L$ also $v = \lambda f$ where, the velocity is $v$, $\lambda $ is the wavelength and $f$ is frequency of the pulse also the rope will have some tension which will be equal to the weight of rope as there is no other mass on the rope. Also the relation between the velocity of pulse and tension of the rope which can be used in solving this problem.

Formula used: The velocity of a pulse on the rope of mass $M$, length $L$ and tension $T$ is given by $v = \sqrt {\dfrac{{T \cdot L}}{M}} $. Also the velocity of a pulse is given by $v = \lambda \cdot f$ where $v$ is the velocity of pulse, $\lambda $ is the wavelength and $f$ is the frequency of the pulse.

Step by step solution:
Step 1.
The velocity of a pulse under tension $T$, mass of rope $M$ and length of rope as $L$ is given by $v = \sqrt {\dfrac{{T \cdot L}}{M}} $.
Step 2.
Since $v = \lambda f$ and for fundamental vibration of a string $\lambda = 2L$ and velocity is $v = \sqrt {\dfrac{{T \cdot L}}{M}} $.
Therefore,
$
  v = \sqrt {\dfrac{{T \cdot L}}{M}} \\
  \lambda \cdot f = \sqrt {\dfrac{{T \cdot L}}{M}} \\
  2L \cdot f = \sqrt {\dfrac{{T \cdot L}}{M}} \\
  f = \dfrac{1}{{2L}} \cdot \sqrt {\dfrac{{T \cdot L}}{M}} \\
  f = \dfrac{1}{2} \cdot \sqrt {\dfrac{T}{{L \cdot M}}} \\
 $………eq.(1)
Step 3.
Since the tension in the string has to be equal to the weight of the body $T = M \cdot g$.
Replace the value of $T = M \cdot g$ in the equation (1).
$
  f = \dfrac{1}{2} \cdot \sqrt {\dfrac{T}{{L \cdot M}}} \\
  f = \dfrac{1}{2} \cdot \sqrt {\dfrac{{M \cdot g}}{{L \cdot M}}} \\
  f = \dfrac{1}{2} \cdot \sqrt {\dfrac{g}{L}} \\
 $………eq.(2)
Step 4.
As we know that the frequency is the inverse of the time therefore taking the inverse of frequency from equation (2).
$
  f = \dfrac{1}{2} \cdot \sqrt {\dfrac{g}{L}} \\
  T = 2 \cdot \sqrt {\dfrac{L}{g}} \\
 $
So, the time taken by the pulse to reach to the other end of the rope is given by $T = 2 \cdot \sqrt {\dfrac{L}{g}} $.

So the correct answer for this problem is option C.

Additional information: A wave of wavelength $\lambda $ and velocity $v$ and time taken as $T$ is given as $
  {\text{speed = }}\dfrac{{{\text{distance}}}}{{{\text{time}}}} \\
  v = \dfrac{\lambda }{t} \\
  v = \lambda \cdot f \\
 $
As we know that the inverse of the time is nothing but frequency $f$.

Note: Students should remember the formula of the velocity for fundamental vibration of a string also the formula which can be used for similar problems is the relation between the velocity of the pulse and the tension of the string. The tension in the string will be equal to the weight of the rope as there is no mass except the mass of the rope and there are only two forces which are in opposite directions.