Question

the equation of a transverse wave travelling along a string is given by Y=1.8sin(23.8X+ 317t) where X is in meter and Y is in millimeter and is in seconds .The string under tension of 16.3N. Find linear mass density?

Answer 1

We can find the linear mass density from the formula:

$$v=\sqrt{\dfrac{T}{\mu}},$$ $$\mu=\dfrac{T}{v^2}.$$

We can find the speed of the wave from the wave speed equation:

$$v=f\lambda.$$

We can find the frequency from the formula:

$$f=\dfrac{\omega}{2\pi}=\dfrac{317\ \dfrac{rad}{s}}{2\pi}=50.45\ Hz.$$

We can find the wavelength of the wave from the formula:

$$\lambda=\dfrac{2\pi}{k}=\dfrac{2\pi}{23.8\ m^{-1}}=0.26\ m.$$

Finally, we can find the linear mass density:

$$\mu=\dfrac{T}{(f\lambda)^2}=\dfrac{16.3\ N}{(50.45\ Hz\cdot0.26\ m)^2}=9.47\cdot10^{-2}\ \dfrac{kg}{m}.$$