Question

A heavy uniform rope hangs vertically from the ceiling with its lower end free .A disturbance on the rope travelling upward from the lower end has a velocity v a distance $x$ from the lower end, then?
A. $V \propto 1/x$
B. $V \propto x$
C. $V \propto \sqrt x$
D. $V \propto 1/\sqrt x$

Answer 1

The question talks about the vibratory or oscillatory motion (vibration) set up in a rope with certain parameters such the velocity after being disturbed from a specific distance. It also speaks of a body exhibiting simple harmonic motion.

Complete step by step answer: We are given a heavy uniform rope which is hanging vertically from the ceiling with its lower end free.
We have to find how velocity of a disturbance travelling on the rope upwards is related to the distance from the lower end of the rope.
We represent $\mu$ as mass per unit length of the rope, T is the tension set up in the rope and at a fixed distance x from the lower end, basically in the vertical direction.
We know that Tension is a force which is the product of mass and acceleration due to gravity (because the rope is fixed at a point and left to move freely at the end only in the vertical direction)
And velocity can be written as
$V = \sqrt {\dfrac{T}{\mu }}$
The length is x and $\mu$ is $\dfrac{m}{x}$. T is the tension which is equal to $m \times g$
On substituting the values of $\mu$ and T in the velocity equation, we get
$V = \sqrt {\dfrac{{m \times g}}{{\left( {\dfrac{m}{x}} \right)}}} \\\ \to V = \sqrt {g \times x} \\\ \therefore V = \sqrt {gx} \\\ \to V \propto \sqrt x \\\$
Therefore the velocity is directly proportional to square root of distance (x).
$V \propto \sqrt x$
Hence, the correct option is Option C.

Note: The same kind of motion occurs in springs and strings where the area of focus is the tension set up in the spring or string and most especially the kind of energy set up in the process, which could be set up throughout the material. Mechanical energy set up in the spring or string can be kinetic energy or potential energy as the case.