Lab 2 - Transverse Mechanical Waves and Resonance PDF

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Name: Danny Stein Course: PHYS 2450L-801 Instructor: Wangyao Li Lab Title: Transverse Mechanical Waves and Resonance Date: 4 October 2016 Partners: Keegan Donahue & James Rothenberg

Objective The goal of the experiment is to examine a string’s wave velocity and tension, causing two traveling waves to combine and create resonance.

Introduction A mechanical wave is a wave that requires a medium in order for its energy to get transferred from one point to another. Even though a wave can reach long distances, the medium is not capable of traveling far. `Although there are a number of common types of mechanical waves, this experiment will be focusing on transverse waves. A transverse wave is a wave that makes the makes the motion of the medium perpendicular to the direction of the wave. In other words, if energy causes a wave to move from left to right, then the medium would be moving up and down. The velocity (v) of this pulse along the wave can be represented using the equation 𝐹

𝑣 = √𝜇

(1)

where F is the string tension, and μ is equal to the mass per unit length of the wave. In this experiment, a heavy string and a light string will be representing the transverse waves. If one end up the string begins to move up and down while the other is held down with some weight, then a “loop” will be created throughout the string. By increasing the speed of the end moving up and down, more “loops” will begin to form, but they will become shorter. The frequency (f), or the rate at which something is repeated over a given amount of time, of the traveling wave is equal to the frequency of the source that makes the end of the string go up and down. Likewise, the wavelength (λ) is the horizontal distance of each “loop” during one full cycle. The wav velocity can then be determined with the equation v=f ●λ which can be rewritten as

(2)

𝜆=

1

𝑓

𝐹

● √𝜇

(3)

Finally, the length of the string (L) can also be determined if the wavelength and number of loops (n) are known by using the equation 𝐿=𝑛●

𝜆

(4)

2

Procedure A) Resonance at constant tension Locate the heavy string (Figure 1, Part 1) and measure the mass and length. Position the mechanical vibrator (Figure 1, Part 2) 110 cm away from the pully (Figure 1, Part 3) and loop the end around the pin (Figure 1, Part 4). Hang a 250 g weight at the opposite end (Figure 1, Part 5) and measure the length of the string between part 2 and part 3. Set the vibrating source to the maximum amplitude, adjust the frequency on the same source, and record the frequency once one full loop (n=1) in the string is formed. Record two more trials and repeat up to n=8.

Figure 1: A diagram explaining the setup of the experiment. The figure represents the string's third harmonic motion (n = 3 loops)

B) Determining uncertainty in M at resonance Set the vibrating source to 60 Hz and adjust the hanging mass to make five loops (n = 5). Record the mass uncertainty. C) Effect of string linear density Replace the heavy string with the lighter string. Measure the mass and length of the new string and record the data. Repeat the procedure from part A using the same 250 g mass and record the frequencies from n = 1 to n = 9. Only one trial is necessary. Results and Analysis Once the heavy string was attached for the first part of the experiment, it became very obvious when a new loop was formed. This made it very simple to reach the desired amount of loops and determine the frequency to create each of the loops. The data for each loop is represented as: Table 1: Data collected through three trials using the heavy string with a constant hanging mass (0.250 kg) and length of resonating string (1.16 m)

n 0 1 2 3 4 5 6 7 8

Trial 1 0.00 11.53 22.83 34.09 45.26 56.92 68.16 80.07 91.23

Frequency (Hz) Trial 2 0.00 11.51 22.60 33.78 45.30 56.91 68.14 79.77 91.44

Trial 3 0.00 11.33 22.58 33.80 45.33 56.83 68.15 79.61 91.46

Average Frequency, f (Hz) 0.00 11.46 22.67 33.89 45.30 56.89 68.15 79.82 91.38

Table 1 demonstrates a very similar frequency during all three trials for each value of n. The following is a linear plot represents the data found in Table 1.

Loops Vs. Frequency Average Frequency, f (Hz)

100.00 80.00

60.00 40.00 20.00 0.00 0

2

4

6

8

10

Number of Loops, n

Figure 2: A linear scatter plot of the number of loops versus the average frequency for the heavy string

The slope of Figure 2 was then determined by using two points (n=7 and n=2). The resulting slope equaled 11.43 Hz; however, according to equations 2 and 4, the wave velocity is the product of the slope and 2L. This gives an experimental velocity of 26.518 m/s, which is not far from the theoretical value (calculated from equation 1) of 23.611 m/s. If the frequency on the heavy string remains constant at 60 Hz, and five loops need to be created, then an increase in the hanging mass is required (part 2 of the experiment). By increasing the hanging mass, the string tension increases as well. Instead of the original string tension equaling 2.453 N, the new value would be 2.65 N. This slight alteration results in the wave velocity increasing from 23.611 m/s to 24.541 m/s. If the original hanging mass was kept the same, then a frequency of 60 Hz would produce loops between n=5 and n=6, as seen in Table 1. After replacing the heavy string with the light string for part 3 of the experiment, the results became much more difficult. The light string required a smaller amplitude, so the number of loops had to be observed very closely, as it is nearly impossible for the human eye to make any

observations from a distance. It was also necessary to increase the frequency of the light string very slowly in order to make sure the number of loops does not get skipped. The data collected for the light string can be seen in Table 2, followed by its linear plot. Table 2: Data collected through one trial using the light string with a constant hanging mass (0.250 kg) and length of resonating string (1.16 m)

n 0 1 2 3 4 5 6 7 8 9

Frequency, f (Hz) 0.00 50.45 95.27 160.46 210.71 264.26 316.41 370.71 425.08 474.60

Loops Vs. Frequency 600.00

Frequency, f (Hz)

500.00 400.00 300.00 200.00 100.00 0.00 0

2

4

6

8

10

Number of Loops, n

Figure 3: A linear scatter plot of the number of loops versus frequency for the light string

Using two data points from Table 2 (n=7 and n=5), the slope of the graph is calculated out to be 53.225 Hz. The experimental wave velocity can once again be determined from equations 2 and

4, equaling 123.482 m/s. This velocity is very close to the theoretical velocity (calculated from equation 1) of 116.335 m/s. Discussion All three parts throughout the experiment were very successful. The heavy string’s experimental velocity of 26.518 m/s compared to the actual value of 23.611 m/s resulted in a 12.3% error. The percent error was expected to be slightly smaller, as a good percent error generally consists of anything under 10%. The most obvious reasoning behind this was the waves did not have a total equilibrium point. In other words, the center of the wave could still have been slightly moving back and forth, which would ultimately cause insufficient results. The mechanical vibrator also has the possibility of moving around, so it is unlikely that it constantly remained 110 cm from the pulley. Finally, the end of the string attached to the pin needed to be angled down to prevent any slack within the string. This could also have altered the results. Similarly, the light string resulted in an excellent 6.1% error. This inaccuracy could be due to the same factors as the heavy string, but the light string made it difficult to notice when the wave added another loop. For the last few values of n, the frequencies became more of a guess than actual values, due to the minimum movement throughout the string. The data could have been altered if the frequency was not stopped directly on the point of movement. Conclusion Transverse waves are one of the many types of mechanical waves and are typically the most common type. This lab proved to be a total success, as transverse waves were created with two different sized strings. With both strings having a low percent error, the results were very accurate and helps give a better understanding to waves altogether.

Questions 1) F (in N) = μ= F μ

=

kg ● m s2

kg m

(

2

kg ● m ) s2 kg m

√m2 = s

=

m2 s2

m s

2) v = f ● λ v

3)

v

L = n ● 2f

f=n●

y = mx + c

y = f, x = n, and c = 0, so slope (m) =

2L v

2L...