Question

A string that is under 50.0 N of tension has linear density

5.0 g/m. A sinusoidal wave with amplitude 3.0 cm and wavelength 2.0 m travels along the string. What is the maximum speed of a particle on the string?

Short answer

Maximum speed of a particle on the string is,$9.4m/s.$

Step-by-step solution

Given data

The string has a linear density of $\mu =5gm/m=0.005kg/m$

The tension force $T=50N$

Wavelength of wave, $\lambda =2m$

Wave amplitude,$A=3cm$

Determination of wave frequency

Speed of the wave,

$$\begin{gathered} v=\sqrt{\frac{T}{\mu }} \\ v=\sqrt{\frac{50.0N}{0.005kg/m}} \\ v=100m/s \end{gathered}$$

Therefore, the wave frequency is,

$$\begin{gathered} f=\frac{v}{\lambda } \\ f=\frac{100m/s}{2m} \\ f=50Hz \end{gathered}$$

Determination of maximum speed

The angular frequency can be determined by,

$$\begin{gathered} \omega =2\pi f \\ \omega =\left(2\pi \right)(50Hz) \\ \omega =314rad/s \end{gathered}$$

The maximum velocity of the particle is,

$$\begin{gathered} v_{max}=A\omega \\ {v}_{max}=(3.0cm)(314rad/s) \\ {v}_{max}=940cm/s \\ {v}_{max}=9.4m/s \end{gathered}$$