Video Experiments 10-11

Note that video clips are all on the Physics II CD labeled EPS01.

Begin by downloading the program String Simulation under Simulations (button at top of page) and watch it for a few minutes (choose Default or just hit Enter at the beginning). The simulation is actually pretty soothing and looks fairly good. It shows you what happens in a string when it is plucked randomly. Certain oscillations disappear quickly, leaving the first few harmonics (modes of oscillation). Different plucks have different end results. Note the differences for yourself. When you finish these experiments you should watch the simulation again.

These experiments will show you how these individual modes arise. The string oscillations you see near the end of the simulation are superpositions of these modes of vibration.

Clip 1: You see 100-grams masses at 25 cm intervals with 12.5 cm at each end.

What is the string's length?
If each centimeter of rubber-band stretch (stretch is measured as the excess over the unstretched length) corresponds to approximately .9 Newtons of force, for each rubber band, then what is the tension in the string at each of the rubber-band lengths: 8.7 cm (unstretched), 9 cm and 9.7 cm?
Note that pulses are somewhat difficult to see. What is the approximate velocity of the pulse at each tension?

Clip 2:

Make the same calculations for the new rubber band lengths. Be sure to note the lengths.
Make a table of pulse velocity vs. tension.
Calculate the pulse velocity that would result from each tension, using the equation v = `sqrt( T / (m/L) ), where T is tension, m is the total mass and L is the length of the string.

Are the predictions of this equation close to the observed pulse velocities?

To what extent do the results of this experiment confirm the equation v = `sqrt(T / (m/L) )?

Phy 232 and 242 (optional extra credit for Phy 202):

Construct a graph of log(velocity) vs. log(tension) for your velocity vs. tension observations, approximate the best-fit straight line for this graph, and find the line's vertical-axis intercept, which we call v0, and the slope of the graph, which we will call p.
The straight line model for your graph is thus of the form log(v) = p log(T) + v0. This tells you that 10^(log(v) ) = 10^( p log(T) + v0) so that v = 10^ (p log(T)) * 10^v0 = 10^v0 * 10^(log(T)) ^ p = 10^v0 * T^p.
Thus v = A * T^p, where A = 10^v0 and p is the slope of your graph.

How well does your model correlate with the theoretical relationship v = `sqrt(T / (M/L) ), which is derived using the Impulse-Momentum Theorem as in your text?

Note that the ideal model can be written v = `sqrt ( L / M) `sqrt ( T) = `sqrt (L / M) * T^.5.
How close is your exponent p to the theoretical exponent .5?
How close is your A = 10^v0 to `sqrt(L / M)?

To what extent do your results support the theory?

Taking experimental errors into account are your results consistent with theory?

Video Experiment 11

Observe but do not take data on Clips 3 and 4.

On clip 5, use the TIMER program, hitting the t key in faster and faster rhythm every time the string is nudged, determine

the approximate time between nudges in the fundamental harmonic, which emerges when the string first becomes coherent with the string moving smoothly back and forth in one motion with a single antinode (the greatest oscillation of any weight) occurring at the middle of the string
the approximate time between nudges in the second harmonic, which emerges when the string first becomes coherent with two antinodes, one near each end of the string, moving smoothly back and forth in opposite directions, and a node (a point of little or no oscillation) in the middle.
the approximate time between nudges in the third harmonic, which emerges when the string first becomes coherent with three antinodes, one near each end of the string and one in the middle, with the ends moving smoothly back and forth together and the middle moving smoothly in the opposite direction, noting that nodes occur between antinodes.

Note that the string tension results from rubber bands stretched to lengths of 11.5 cm. What is the approximate tension in this string?

What therefore should be the velocity of a pulse in this string?

In the fundamental mode of oscillation you see half of a complete wave, from a node at one end to an antinode at the middle and a node at the other end, comprising the entire length of the string. The wavelength is therefore twice the length of the string and the period of the wave is the period of oscillation of the string from an extreme on one side to the extreme on the other and back.

From the wavelength and period of the wave what velocity do we infer?
How well does this agree with the velocity predicted from the mass, length and tension in the string?

In the second harmonic you see a complete wave, from a node at one end to a 'peak' on one side to a node in the middle and a 'opposite peak' (or if you prefer to think of it that way a 'valley') on the other side to a node at the other end. Note that the wavelength therefore consists of four quarter-wavelengths, from end node to 'peak', then from 'peak' to middle node, from middle node to 'valley' and finally from 'valley' to the far-end node.

From the wavelength and period of the second harmonic what wave velocity is inferred?
How well does this agree with the velocity inferred from the first harmonic?

Recall noting that the distance between any node and an adjacent antinode is a quarter-wavelength.

For the third harmonic, explain how we can see that there are six node-antinode distances and therefore 3/2 of a complete wavelength in the length of the string.
From the wavelength and period of the third harmonic what wave velocity is inferred?
How well does this agree with the velocity inferred from the first and second harmonics?

What do you think should be the relationship between the periods of the first, second and third harmonics?

What do you think should be the relationship between the frequencies of the first, second and third harmonics?

How well are these relationships confirmed by your results?

Are these relationships consistent with your results?

Now measure the period of the fundamental mode only on Clip #8.

Note that the tension is the same as in Clip #5 but that the mass per 25 cm is about 1/6 as great.

How should this difference in mass affect the velocity of the wave?

How should it affect the period of the fundamental oscillation?

Was the period affected as you predicted?

Observe Clips 12 and 14-17.

Describe in your own words the difference between a transverse and a longitudinal pulse.
Briefly describe what you saw when longitudinal pulses were sent down a slinky mounted on a PVC pipe and on a thin metal pipe.

Which setup did the better job of showing you how the longitudinal pulse moved?

Which did the better job of showing you how longitudinal standing waves are formed?

What was the greatest number of 'bunched-up' antinodes (i.e., antinodes formed when the coils of the slinky were much closer together than the average separation) you saw for a standing longitudinal wave? Were there nodes between these 'bunched-up' antinodes? Were there antinodes between them?