Question

A string has a mass per unit length of ${{10}^{-6}}kgc{{m}^{-1}}$. The equation of the simple harmonic wave produced in it is 4=0.2$\sin \left( 2x+80t \right)$. The tension in the string is
A. 0.0016 N
B. 0.16 N
C. 16 N
D. 1.6 N

Answer 1

Hint: We are provided with an equation of simple harmonic wave, which we can compare with the general equation of SHM of simple harmonic wave. We can compare with standard form$y=a\sin \left( kx+\omega t \right)$. Tension acts on anybody when it is subjected to a pulling force. Additionally, because the string is under SHM has a velocity associated with it. Use the formula of tension in the string which has relation between velocity and mass, to find the required answer. When it is subjected to pull and while performing motion it has velocity so use a formula of tension in the string which will give a relation between velocity and mass.

Formula used:
Formula of tension in the string is given by,
$T={{v}^{2}}\mu$
Where,
$T$ = tension in the string
V = velocity of the waves
$\mu$ = linear density/mass per unit length

Complete step by step answer:
The definition of simple harmonic motion is given as a body of linear periodic motion in which the restoring forces/acceleration is always directly proportional to the displacement from the mean position (magnitude of restoring force) and directed towards the mean position.
The standard equation of S.H.M is given as,
$y=a\sin \left( kx+\omega t \right)$ --$(1)$
Where, y= displacement
$k=$ Spring constant/force constant
$\omega =$ Angular velocity
$t=$ Time
$a=$ Amplitude
The equation of simple Harmonic motion or wave produced in it is given as,
$y=0.2\sin \left( 2x+80t \right)$ --- (2)
Compare equation (2) with (1), we get
$k=2n/m$ ; $w=80rad/s$ ; $a=0.2$
Now we need to calculate the velocity of the wave. So from above data we can calculate velocity of wave which is given as
$v=\dfrac{w}{k}$ ---- (3)
Put the value in above equation (3), we get
$v=\dfrac{80}{2}=40m/s$
Hence velocity of the wave is $40m/s$
In question we have given mass of string per unit length which is donated as $\mu$
Therefore $\mu =m/l={{10}^{-6}}$ $kg/cm$
Convert $kg/cm$ into $kg/m$
We get $\mu ={{10}^{-4}}$ $kg/m$
Let $T$be the tension then tension in the string
$T={{v}^{2}}\mu$
$T={{\left( 40 \right)}^{2}}\times {{10}^{-4}}=0.16N$
Therefore tension in the string is given as $T=0.16N$

Therefore option (B) is the correct option.

Note:
Tension does not possess any kind of component perpendicular to it. Tension (T) is always directed towards the line of string. Tension acts on the body on a string when it is subjected to pull. Tension is one of the types of force which acts on a string or body when weight or mass is applied to it and it gets pulled by that mass.