Question 297: Waves

Question

A string has mass per unit length of $10^{-6} \mathrm{~kg} / \mathrm{cm}$ The equation of simple harmonic wave produced in it is $\mathrm{Y}=0.2 \sin (2 \mathrm{x}+80 \mathrm{t}) \mathrm{m}$. The tension in the string is

Accepted Answer

The tension in the string can be determined using the formula for wave velocity, which is:
$ v = \sqrt{\frac{T}{\mu}} $
From this, we can express the tension $ T $ in terms of wave velocity $ v $ and mass per unit length $ \mu $:
$ T = v^2 \mu $
Given that:
$ \mu = 10^{-6} \, 	ext{kg/cm} = 10^{-4} \, 	ext{kg/m} $
Substituting into the tension formula, we have:
$ T = v^2 	imes 10^{-4} 	ag{Equation (i)} $
The wave equation given is:
$ Y = 0.2 \sin(2x + 80t) $
To compare this with the standard form of a wave equation:
$ y = A \sin(Kx + \omega t) $
we identify that:
$ \omega = 80 $
Using the relationship between $ \omega $, wave number $ K $, and velocity $ v $:
$ \omega = vK $
$ v = \frac{\omega}{K} $
Given $ K = 2 $, the velocity $ v $ is:
$ v = \frac{80}{2} = 40 \, 	ext{m/s} $
Now, substituting $ v = 40 \, 	ext{m/s} $ back into Equation (i):
$ T = (40)^2 	imes 10^{-4} $
Calculating the tension:
$ T = 1600 	imes 10^{-4} $
$ T = 0.16 \, 	ext{N} $

Answer

0.16 N

Answer

0.0016 N

Answer

$\quad 1.6 \mathrm{~N}$

Answer

16 N

Waves

A string has mass per unit length of 10^{-6} \mathrm{~kg} / \mathrm{cm} The equation of simple harmonic wave produced in it is \mathrm{Y}=0.2 \sin (2 \mathrm{x}+80 \mathrm{t}) \mathrm{m}. The tension in the string is

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