# PHY 141 Lab 8 - Standing Waves

The purpose of this lab is to study standing waves on a vibrating string.

## Equipment

• DC Motor with vibrating reed on moveable stand
• Power supply for motor
• photogate
• pulley
• ruler
• box with weights
• an elastic string
• a roll of thick white non-stretching string
• bulldog clip
• scale to weigh the string

Make sure that you do not attach the end of the string directly to the motor, have the string pass through the motor reed eye and then attach the end of the string to the post using the bulldog clip.

## Introduction

In this lab you will study standing waves on a string and a wire. The excitation frequency is determined by the DC voltage applied to the motor. Position the photogate so that is blocked by the flag on top of the motor once per cycle. You can measure the time for one period of rotation of the motor using the pendulum timer program on the computer and from this determine the driving frequency.

The velocity of a traveling wave is determined by the tension in the wire and the linear mass density.

The equation for the velocity of a travelling wave on a string is

$\Large v=\sqrt{\frac{T}{\mu}}$
(8.1)

The linear mass density $\mu$ for the system above can be found using

$\Large \mu=\frac{m}{L'}$
(8.2)

and the the tension in the string will be given by the gravitational force

$\Large T=Mg$
(8.3)

There are two parts of this experiment that can be completed in either order if you prefer to swap setups instead of dismantling and reassembling the setup for the two parts.

## Changing the frequency

Using the elastic string you can very easily excite standing waves. We can use the setup with the elastic string to see how adjusting the frequency changes the number of nodes and antinodes you can generate.

The equation for the frequencies $f_{n}$ of the standing waves on a string is:

$\Large f_{n}=n\frac{v}{2L}$
(8.4)

With a fixed mass on the end of the string and fixed distance from puller to motor adjust the frquency using the DC power supply voltage. Identify several of the standing wave frequencies and make a plot of $f_{n}$ vs $n$ to determine the speed of the traveling wave in the elastic string. Use this to determine the mass density of the elastic string under the experimental condition.

## Changing the tension

Set up the experiment with the non-stretching string.

1. Adjust the power supply voltage until the frequency is about 20Hz, record the exact frequency you end up using for the experiment as this is an important parameter!.
2. Place a 100 g mass on the end of the string. Adjust the position of the vibrating reed until a stable, standing wave pattern is observed. Measure the wavelength $\lambda$ of the wave, and estimate the uncertainty.
3. Try to make standing wave patterns for several different positions of the reed (each different by 1/2 wavelength).
4. Add some more mass to the end of the string and repeat the above procedures for at least 3 different values of $M$, for each value of $M$ calculate the tension $T$.
5. Plot $\lambda^{2}$ as a function of $T$, the tension of the string, and from the slope of this curve (and your knowledge of the frequency!), determine $\mu$, the mass per unit length of the string. Estimate the uncertainty in this result. Compare with the mass per unit length written on the door at the back of the lab.

Why could we not use the elastic string for both parts of this experiment?