Question

A uniform wire of length 20 m and weighing 5 kg hangs vertically. If $g = 10m{s^{ - 2}}$ ,then the speed of transverse waves in the middle of the wire is

$$A.{\text{ }}10m{s^{ - 1}} \\\ B.{\text{ }}10\sqrt 2 m{s^{ - 1}} \\\ C.{\text{ }}15m{s^{ - 1}} \\\ D.{\text{ }}2m{s^{ - 1}} \\\$$

Answer 1

In order to deal with this question first we will evaluate tension in the middle of the wire and also the density of the given wire then we will proceed further by applying formula of velocity of wave in terms of tension and density of the wire to get the answer.
Formula used:
${\text{Density}} = \dfrac{{{\text{Mass}}}}{{{\text{Length}}}},v = \sqrt {\dfrac{T}{\mu }}$

Complete answer:
Given that:
Mass of wire $m = 5kg$
Length of wire $l = 20m$
Density of the wire is given as:
$\because {\text{Density}} = \dfrac{{{\text{Mass}}}}{{{\text{Length}}}} \\\ \Rightarrow \mu = \dfrac{m}{l} \\\ \Rightarrow \mu = \dfrac{{5kg}}{{20m}} \\\ \Rightarrow \mu = 0.25kg/m \\\$
As we know that tension in the middle of the wire is given as
$T = \dfrac{m}{2}g$
Where m is the mass of the wire and g is the acceleration due to gravity.
Substitute the values in above formula we have
$T = \dfrac{5}{2} \times 10 \\\ T = 25N \\\$
Formula of velocity of transverse wave is given as
$v = \sqrt {\dfrac{T}{\mu }}$
Substitute the values in above formula we have velocity of transverse wave is:
$\Rightarrow v = \sqrt {\dfrac{{25}}{{0.25}}} \\\ \Rightarrow v = \sqrt {100} m/s \\\ \Rightarrow v = 10m/s \\\$
Hence, speed of transverse waves in the middle of the wire is $10m/s$

So, the correct answer is option A.

Additional information:
A transverse wave is a traveling wave whose oscillations are perpendicular to the wave direction of the propagation line. A basic example is provided by the waves which can be produced by anchoring one end and pushing the other end up and down on a horizontal string length. A further example is the waves produced on a drum 's membrane. The waves travel in directions similar to the membrane plane but the membrane itself, perpendicular to the plane, is moved up and down.

Note:
Tension is a force along a medium's length, particularly a force carried by a flexible media such as a rope or cable. Stress is a force but it produces no displacement and stress function is zero. A transverse wave is a traveling wave whose oscillations are perpendicular to the wave direction of propagation path.