Question

A string that is under 50N of tension has a linear density of 5.0 g/m. A sinusoidal wave with amplitude 3.0cm and wavelength 2.0 m travels along the wave. What is the maximum velocity of a particle on the string?

Answer 1

Using the wave equation for transverse waves on a string, the maximum velocity of a particle is calculated from the given tension, linear density, amplitude, and wavelength.

To find the maximum velocity of a particle on the string, we can use the wave equation for transverse waves on a string. The wave equation relates the wave speed (v), frequency (f), wavelength
`(\(\lambda\))`, and maximum displacement (A):

`\[ v = f \cdot \lambda \]`

The frequency can be determined from the tension (T) and linear density (mu) of the string using the equation:

`\[ v = \sqrt{(T)/(\mu)} \]`

First, convert the linear density from grams per meter
`(\(g/m\))` to kilograms per meter
`(\(kg/m\))` by dividing by 1000:

`\[ \mu = 5.0 \, \text{g/m} * \frac{1 \, \text{kg}}{1000 \, \text{g}} \]`

Now, plug in the values of
`\(T\) (tension), \(\mu\),` and
`\(\lambda\)` into the wave equation:

`\[ v = \sqrt{(T)/(\mu)} \]`

Next, calculate the frequency (f) using the wave equation:

`\[ f = (v)/(\lambda) \]`

Once the frequency is known, we can use it to find the maximum velocity
`(\(v_{\text{max}}\))` of a particle on the string:

`\[ v_{\text{max}} = A \cdot 2\pi f \]`

Now, substitute the values into the equation to find
`\(v_{\text{max}}\)`.