One end of a horizontal rope is attached to a prong of another end passes over a pulley and supports a 1.50-kg mass. The linear mass density of the rope is 0.0480 kg/m. (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would you answer to parts (a) and (b) change if the mass were increased to 3.00 kg?

A transverse wave is a wave which has oscillations that are perpendicular to the direction in which the wave is advancing.

Use the formula for velocity, that is v =$\sqrt{\frac{T}{\mu }}$,where T is the tension of the rope and$\mu$ is the linear mass density.

Once the velocity is found, apply the formula for wavelength to find the wavelength of the transverse wave,$\lambda =\frac{v}{f}$ , where v is the velocity of the wave and f is the frequency.

Finally, apply both the velocity and wavelength formula after changing the value of tension of the rope. Due to increasing the mass of the object by 3.00 kg, the tension of the rope will increase to 30N.

Here,weight of the mass is 1.5 kgand frequency is 120Hz.

Tension of the rope is$1.5\times 10=15\mathrm{N}$

Velocity$=\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}=\sqrt{\frac{15}{0.0480}}=17.67\mathrm{m}/\mathrm{s}$

Wavelength$=\lambda =\frac{v}{f}$

$\lambda =\frac{17.67}{120}=0.147m$

If mass was 3 kg new tension will be 30N

Velocity $=\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}=\sqrt{\frac{30}{0.0480}}=25\mathrm{m}/\mathrm{s}$

Wavelength$=\lambda =\frac{v}{f}=\frac{25}{120}=0.208m$

Hence, the speed of the transverse wave on the rope is 17.67m/s. The wavelength is 0.147m. If the mass were increased to 3.00 kg, then the speed of the transverse wave will be 25m/s and its wavelength will be 0.208m/s.