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phy133:labstandingwaves

# PHY 133 Lab 9 - Standing Waves

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

## Equipment

• electric motor with flag and metal reed
• variable power supply for motor
• photogate
• pulley
• meter stick
• various masses (at least 50g and 100g)
• golden “stretchy” string
• white “non-stretchy” string
• clip
• digital scale

## Introduction

In this lab, you will study the transverse standing waves formed along a vibrating taut string attached to a rotating electric motor. For your measurements, you will use two different kinds of string: a golden “stretchy” (or elastic) string, or a white “non-stretchy” (or inelastic) string. At certain speeds, the motor will create transverse standing wave patterns along the string length between the motor and the pulley. The string is held taut by the motor on one end and the mass dangling below the pulley at the other end.

As you may know, “transverse” indicates that the displacement of the wave medium (the string) from equilibrium is perpendicular to the direction of propagation of the wave (which is parallel to the string). “Standing” indicates that the waves do not appear to be moving in either direction; rather, a seemingly stationary pattern appears due to constructive and destructive interference of two counter-propagating waves. Hence, the anti-nodes (the “maxima” in the amplitude of transverse oscillations) and the nodes (the “minima” in the amplitude of transverse oscillations) do not move. These transverse standing waves will only appear under certain conditions, which is what you will investigate in this lab.

In this lab, a standing wave pattern is produced by an electric motor that vibrates one end of the string up and down. As this happens, the string displacement is sent from one end of the string to the other. At this other end, which is fixed, the incoming wave reflects and bounces back in the other direction. When it reaches the end with the motor, it is reflected back, and this repeats again and again. However, if the reflected wave traveling to the left is in phase with the original wave traveling to the left, then their amplitudes constructively interfere. Similarly, the leftward moving wave and a rightward moving wave may interfere destructively. The combination of all of these interactions yields a standing wave pattern, as shown in the diagram below, which looks like a stationary wave pattern, rather than many separate waves traveling to the left and to the right. Depending on the speed of the electric motor, different frequencies can be generated along the length of the string, so long as there are nodes at the two fixed endpoints. This constraint gives us a relationship between the wavelength $\lambda_{n}$ of the $n$th mode and the length $L$ of the string:

$$\lambda_{n} = \frac{2L}{n}$$

As shown in the diagram below, the higher the mode number $n$, the shorter the wavelength $\lambda_{n}$ of the standing wave pattern. In this lab, you will generate many standing wave modes like these, while exploring the relationship between the string's properties and those of the standing wave pattern produced.

(As you might have guessed, the physics to be discussed in this lab applies very well to stringed musical instruments like the guitar, violin, and cello. When playing one of these instruments, a musician is controlling the frequency of sound produced by the standing waves formed along the strings, generated by their fingers, a pick, or a bow. By striking a string freely, it is able to vibrate along the entire length, in the “lowest order mode,” which has the lowest frequency (or “note”) given the length, tension, and mass of that string. However, when the string is lightly pressed down along the neck of the instrument, the effective string length shortens, and the string no longer vibrates in its lowest order mode. Instead, it vibrates primarily at a higher frequency (called a “harmonic”) and produces a higher pitch of sound. If you're familiar with stringed musical instruments, you should be able to anticipate the outcome of the investigations you'll conduct in the lab today!)

## Procedure

In this lab, you will first predict what the traveling wave velocity $v$ should be for the golden stretchy string, using the stretched linear mass density $\mu$ of the string and the tension $T$ along the string due to a hanging mass $M$. Next, you will excite multiple orders (labeled as “order $n$”) of standing waves along the golden stretchy string, and calculate the traveling wave velocity using experimental data. (You can then compare your experimental value to your predicted value of wave velocity.) Lastly, you will investigate the dependence of the wave velocity on the tension in the white non-stretchy string, and use your data to estimate the gravitational acceleration $g$.

### Part I: Calculation of wave velocity $v$ from linear mass density $\mu$ and tension $T$

In this part of the lab, you will calculate the traveling wave velocity $v$ from the tension $T$ in the stretched golden string when a mass $M=150$ g is attached to its end. You will also need to compute its stretched linear mass density, $\mu_{s}$ for this calculation. When a string is fixed at both ends (with node points), at certain vibration frequencies $f_{n}$, traveling waves going left and right along the string add constructively and destructively to form standing wave patterns. The mode number $n$ associated with each frequency corresponds to the number of maximum amplitudes (or anti-node points) between the two fixed ends. An example with 11 nodes (not counting the end points) and 12 anti-nodes is shown in the picture below.

The equation for the velocity of these counter-propagating waves along the string is:

$$v = \sqrt{\frac{T}{\mu}} \tag{1}$$

where the linear mass density $\mu$ for a string is the ratio of its mass $m$ and its length $L'$:

$$\mu = \frac{m}{L'} \tag{2}$$

The tension $T$ in the string is created by the weight of the hanging mass $M$, so that:

$$T = Mg \tag{3}$$

If the string is stretchy (or elastic), then its stretched length $L'$ depends on $T$. Since the stretching of the string does not change its total mass $m$, but it does change its length $L'$, its stretched linear mass density $\mu_{s}$ is not equal to its un-stretched linear mass density $\mu$. As the string stretches, then, you should expect that $\mu_{s} < \mu$.

To predict a value for the velocity $v$ along the stretched golden string, do the following:

• Measure the mass $m$ of the golden string using the digital scale in the lab room. Assume an uncertainty of $\sigma_{m}=0.1$ g.
• Loop one end of the golden string, and hang a mass $M = 150$ g from it. (Assume an uncertainty for this mass of $\sigma_{M}=0.1$ g. Pass the free end up and over the pulley, towards the electric motor. It is essential that you then pass the string through the hole in the motor reed, then through the hole in the motor post, and clamp it to the side of the post using the clip. Do NOT tie the string end directly to the motor reed! The mass should now hang slightly below the pulley.
• Measure the length $L$ from the end of the string at the hole in the motor reed to the end of the string where it first touches the pulley. This portion of the string is where your standing waves will appear. Also, obtain a value for the full length of the stretched string $L'$ by measuring the distance from the top of the pulley down to the loop where the mass is hanging. Adding this distance to $L$ should give $L'$. Assume uncertainties of $\sigma_{L}=\sigma_{L'}=1$ cm.
• Using equations (2) and (3) above, calculate the values of tension $T$ and stretched linear mass density $\mu_{s}$, as well as their uncertainties, $\sigma_{T}$ and $\sigma_{\mu_{s}}$. You will need to use the uncertainty propagation formulas from the uncertainty guide. How does the golden string's stretched linear mass density $\mu_{s}$ compare to the “accepted” value of its unstretched linear mass density of $\mu = 2.7\times 10^{-3}$ kg/m? As mentioned before, why should it be that $\mu_{s} < \mu$?
• Lastly, using equation (1) above, calculate the predicted value of wave velocity $v$ (and its uncertainty $\sigma_{v}$) along the stretched golden string.

### Part II: Measurement of wave velocity $v$ using the frequencies $f_{n}$ of standing wave patterns

In this part of the experiment, you will now use the electric motor to excite standing wave patterns with frequencies $f_{n}$ along the horizontal length $L$ of the stretched golden string. The integer $n$ is the number of anti-nodes (or half-wavelengths, as shown in the figure above) between the two fixed endpoints. Then, by plotting the frequency $f_{n}$ vs. mode number $n$, you will determine the traveling wave velocity and compare it to the value predicted in Part I.

The excitation frequency $f_{n}$ is determined by the DC voltage applied to the motor. Higher voltages produce higher frequencies, since more energy is sent into the string. First, position the photogate so that its beam is blocked by the small metal flag on the motor when it is vertical, which will happen once per cycle. Note: Because this occurs once per cycle, you will NOT need to find and enter into LoggerPro a characteristic distance $d$ for this experiment! By combining the equation for the wavelength $\lambda_{n}$ of the $n$th mode (given in the Introduction) with the wave equation $v=f\lambda$, we can write an expression for the frequency $f_{n}$:

$$f_{n} = \left(\frac{v}{2L}\right)n \tag{4}$$

Now, you're ready to collect some data:

• With the photogate aligned properly, connect the photogate output wire to the input of the interface box labeled “DIG/SONIC 1.” Turn on the computer and double-click the Desktop icon labeled “Exp8_Period.” A “Sensor Confirmation” window should appear, so make sure that “Photogate” is selected, and click “Connect.” A LoggerPro window with a spreadsheet (having columns labeled “State” and “Pulse Time) on the left and an empty graph of Pulse Time (or Period) vs. Time on the right should appear. The “Pulse Time” is the period of a single rotation of the electric motor. Test the photogate by passing your finger through its beam. The red LED on the photogate should flash on/off when the beam is blocked/unblocked.
• Turn on the power supply, and turn both the coarse and fine tuning voltage knobs fully counter-clockwise (to zero). You must make sure that the current knob is turned up (clockwise) enough to run the motor. If the current knob is set too low, then increasing the voltage will not increase the speed of the motor!
• Now, with the motor barely turned on, you should have no wave pattern along the string. Turn up the coarse voltage knob until a single anti-node appears to vibrate on the string. Then, adjust the fine voltage knob until this amplitude is maximized. This is the $n=1$ mode, vibrating at a frequency $f_{1}$.
• Once you see a stable standing wave pattern, click the green “Collect” button in the LoggerPro window. The time values appearing in the spreadsheet are the period values of the motor, which we assume is also the period of the waves along the string. The frequency $f_{n}$ at which the stretched golden string is vibrated to produce the $n$th standing wave mode is the inverse of the period $T_{n}$: $f_{n} = \frac{1}{T_{n}}$. Do not confuse the period values $T_{n}$ with the string tension $T$ from before!
• The graph of Pulse Time (or Period) vs. Time on the right should fill with data points along a horizontal line. If no data points are appearing, then right-click near the y-axis of the graph, and click “Autoscale” (not from 0). This should change the scale of your y-axis to better show the period values measured by the photogate. Note: The plot may now be “zoomed in,” so even a large variation in these data points does not imply the values are spread far apart. Look at the y-axis to verify this.
• Under the “Analyze” tab in LoggerPro, click “Linear Fit.” A small box should appear on the plot, containing the linear fit information. You can click and drag the end points of your selection for the fit if necessary. Record the mean of these “Pulse Time” (or period) values, which will be the period value for $T_{1}$.
• Now, increase the coarse and fine voltage knobs until a stable standing wave pattern appears with two anti-nodes, indicating mode $n=2$, and repeat the steps above to find the period $T_2$.
• Repeat these steps until you have five period values for modes $n=1$ through $n=5$.

To find an estimate of the wave velocity using these data, you will need to plot $f_{n}$ vs. $n$ and determine its slope, then apply equation (4) above. Use the uncertainty propagation methods to find the uncertainty $\sigma_{v}$ in your estimate. For plotting, you should use the web-based Plotting Tool. (Note: Assume no uncertainty in your values of $f_{n}$ or $n$.)

Is your experimentally-determined estimate for the wave velocity $v$ consistent with the predicted value from Part I? Use the “overlap method” with each of the uncertainty ranges to do this. Regardless of your findings, what potential sources of error may have influenced your results?

### Part III: Verifying the dependence of wave velocity $v$ on tension $T$, and obtaining an estimate of $g$

In this last part of the lab, you will measure the dependence of one of the frequencies $f_{n}$ on the string tension $T$ in a non-stretchy string. You will vary the tension by suspending various amounts of mass $M$ from the string. During this part of the experiment, you must keep the standing wave pattern with exactly 1 anti-node, so that you examine the $n=1$ mode with frequency $f_{1}$ only.

Because the elastic golden string would change its length for different amounts of mass attached, and thus its linear mass density would vary, you must first replace the golden string with the white non-stretchy string. For various attached masses $M$, and thus various tensions $T$, the linear mass density of this white string will always remain constant.

The equation for the standing wave frequency $f_{1}$ (for mode $n=1$) may be expressed in terms of the hanging mass $M$ by combining equations (1), (3), and (4) above:

$${f_{n}}^{2} = \frac{g}{4 \mu L^{2}} M \tag{5}$$

For data collection for this part, do the following:

• Measure the mass $m$ of the white string using the digital scale in the lab room. Assume an uncertainty of $\sigma_{m} = 0.1$ g.
• Loop one end of the white string, and hang a mass $M=50$ g from it. (Assume an uncertainty for this mass of $\sigma_{M}=0.1$ g. Pass the free end up and over the pulley, towards the electric motor. Again, it is essential that you pass the string through the holes in the motor reed and the motor post, then clip it to the motor post. Do NOT tie it to the hole in the motor reed! The mass should now hang slightly below the pulley.
• Measure the length $L$ from the end of the string at the hole in the motor reed to the end of the string where it first touches the pulley. (It should be the same value you measured in Part I.) Also, obtain a value for the full length $L'$ of the string by measuring the distance from the top of the pulley down to the loop where the mass is hanging. Adding this distance to $L$ should give $L'$. Assume uncertainties of $\sigma_{L} = \sigma_{L'} = 1$ cm.
• Using equation (2) above, calculate the linear mass density of this white non-stretchy string, using its mass $m$ and its full length $L'$. Compare this value to its “accepted” value of $\mu = 0.47\times 10^{-3}$ kg/m. They should be close; if they are not, try measuring the length $L'$ again, and recalculate your value of $\mu$.
• Align the photogate with the flag on the motor as in Part II, and repeat the same process as in Part II to create a stable $n=1$ standing wave pattern. Using the same procedure as before, obtain the period $T_{1}$, and take the inverse of this value to get the frequency $f_{1}$.
• Increase the hanging mass to $M = 100$ grams, make slight adjustments to the voltage knobs on the power supply to create a stable $n=1$ standing wave pattern with maximum amplitude, and repeat the measurement process to find the period $T_{1}$ and frequency $f_{1}$.
• Repeat this process for hanging masses $M$ of 150, 200, and 250 grams. Record all mass $M$ values and their corresponding standing wave frequencies $f_{1}$.

To find an estimate of the gravitational acceleration $g$, you need to plot ${f_{1}}^{2}$ vs. $M$ and determine its slope, then apply equation (5) above. Use the uncertainty propagation methods to find the uncertainty $\sigma_{g}$. For plotting, you should use the web-based Plotting Tool. (Note: Assume no uncertainty in your values of $f_{1}$ or $M$ here.)

Is your experimentally-determined estimate for the gravitational acceleration $g$ consistent with its accepted value of 9.81 m/s2? Does your estimate equal this value within experimental uncertainty? Regardless of your findings, what potential sources of error may have influenced your results? 