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Lab 2 Post-Lab Questions Question Post-1: You are using the experimental setup shown in Figure 2

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Lab 2 Post-Lab Questions

Question Post-1: You are using the experimental setup shown in Figure 2. 1. You attach a hanging mass of mass M to the end of the string, adjust the frequency of the wavedriver, and observe that the third harmonic is at 22.0 Hz.

a) Sketch what the fifth harmonic would look like.

b) At what frequency would you find the fifth harmonic for this string?

c) Suppose you changed the amount of mass hanging on the string to M/2, changing the tension in the string. What would the resonant frequency of the fifth harmonic be for this new tension?

Lab 2: Vibrating String

PHYS 2306

Objective

To study the behavior of standing waves on a string.

To develop a qualitative understanding of the relationship between the properties of a vibrating string and the harmonics that the string supports.

Recommended background reading

Knight, Chapter 16 and Chapter 17

Procedure 1.2.2

Now you will make some measurements to determine the wave speed v.

Hang a total of 150 g of mass on the end of the string.

Adjust the frequency of the wavedriver until you produce the second, third, and fourth harmonic. For each harmonic, record the frequency and measure the shortest distance between the nodes, I_n using the moveable position markers on the meter stick.

Record your data in Question 1.4.

Question 1.4: Record your data from Procedure 1.2.2 and calculate the wave speed for each harmonic using your answer to Question Pre-2b.

Answer for Question Pre-2b:

f₂= __________________ l₂= _________________   v= ______________

f₃= __________________ l₃= _________________   v=______________

f₄= __________________ l₄= _________________   v=______________

 

Question 1.5: Write down any patterns you noticed about the wave speed. Provide a Scientific explanation for the patterns you observed. Your explanation should coordinate appropriate physics concepts and principles.

Lab 2: Vibrating String

PHYS 2306

Pre-lab Assignment

Question Pre-1: A traveling sinusoidal wave is propagating in the +x direction on a string, you make the following observations about the wave:

• The maximum transverse displacement of the string from its equilibrium position is 5 cm.
• At a fixed instant time, the distance between two points where the string has zero displacement in the y-direction from its equilibrium position is 40 cm.
• At a fixed position along the string, you observe that the string has zero displacement in the y-direction from its equilibrium position, and then it starts to be displaced. After 0.1 sec. elapses, you again observe that the string has zero displacement in the y-direction from its equilibrium position (at that position along the string).
• At t=0 sec, the string is maximally displaced from its equilibrium position in the +y-direction.

Write down the mathematical equation and a diagram that describes this traveling wave.

Explain how these two representations are consistent with each other.

Question Pre-2:

1. For a given string under constant tension, find an expression for the frequency of the n^th harmonic, f_n. given the wave speed v and the length of the string L.
2. Derive the expression, v = 2l_nf_f. where L_n is the shortest distance between nodes for the n^th harmonic. Write your answer here and in Question 1.4 for reference during the tab.
3. Refer to Section 1.1. In Figure 2. 1, suppose a total mass M = 150 grams is hanging on the end of the string. Assume the linear mass density of the string is µ = 4.1 g/m and the length of the string is L = 1.8m.

Determine the expected frequency f₂ of the second harmonic (n=2). Write your answer here and in Question 1.1 for reference during the lab.

Lab 2: Vibrating String

This lab contains post-lab questions, When 10 minutes are remaining in the lab session, stop what you are doing, and answer the post-lab question.

1. Harmonics of a Vibrating Strings

1.1 Preparing your lab station

From the desktop of the computer, open the Class Notes folder and open the lab 2, cap file.

If needed, clean up your workstation and untangle any strings. Return any errant masses to the holder.

The vibrating string apparatus is shown in Figure 2. 1. You will use a wavedriver to vibrate the string. In other words, the wavedriver is the source of vibrations in the string.

Figure 2. 1

One end of the string is attached to the wavedriver. The string then passes over a pulley, and the other end is attached to a hanging mass, which creates a tension in the string. When the wavedriver moves up and down with a frequency f a periodic disturbance or wave travels down the string. The frequency of the wave is the frequency of the source (i.e. f).

Replicate the experimental setup shown in Figure 2. 1. The length of the string between the wavedriver piston and the pulley should be about 1.8 m.

1.2 Activity: Measuring Harmonics of a Vibrating String (150 grams)

Procedure 1.2.1

Hang a total of 150 g of mass on the end of the string. The mass of the hanger is 50 g.

Lab 2: Vibrating String

In Capstone, navigate to the Vibrating String tab. The left-most menu contains controls for the Signal Generator, shown in Figure 2. 2, which you will use to control the frequency and amplitude of the wavedriver. In Figure 2. 2, the frequency controls are labeled with a red and yellow arrow.

Set the amplitude to 5.00 V.

Slowly adjust the frequency setting until you observe the second harmonic.

• When the string approaches a harmonic, the string’s amplitude increases, and a distinctive wave envelope forms. Increase the frequency from 0 Hz until you find the first harmonic, which is indicated by the first appearance of a distinctive wave envelope. Continue increasing the frequency until another distinctive wave envelope forms. This is the second harmonic.

Lab 2: Vibrating String

Question 1.1: Write down the frequency you found for the second harmonic.

Use your measured value of the 2^nd harmonic frequency to predict the values of the 3^rd and 4^th harmonic frequencies. Show your work.

2^nd harmonic frequency (predicted) = __________________ (Question Pre-2c)

2^nd harmonic frequency (measured) = __________________

3^rd harmonic frequency (predicted) = __________________

4^th harmonic frequency (predicted) = __________________

Question 1.2: Adjust the frequency of the wavedriver until you observe the 3^rd and 4^th harmonics. Measure the frequencies of these harmonics.

3^rd harmonic frequency (measured) = ________________

4^th harmonic frequency (measured) = ________________

Question 1.3: Do your measurements in Question 1.2 agree with your predictions in Question 1.1? If there are discrepancies, discuss those discrepancies. Was there anything wrong with your assumptions or reasoning of your predicitons?

1.3 Activity: Measuring Harmonics of a Vibrating String (200, 250, 300, 350 grams)

Procedure

Repeat procedure 1.2.2 for total hanging masses of 200, 250, 300, 350 g. For each hanging mass, choose one harmonic to measure (n = 2.3 or 4).

Question 1.6: Enter your data into Table 2.1. For 150 g, copy your data from Question 1.4 Calculate the value of the wavespeed v for each measurement.

Table 2.1

Hanging mass	n	fn	ln	v
150 g	2
150 g	3
150 g	4
200 g
250 g
300 g
350 g

 

Question 1.7: For each hanging mass you used, calculate the tension in the string, and copy the wavespeed values from Table 2.1 into Table 2.2. For the 150 g, enter the average wavespeed.

In the space provided, show your calculations for how you computed the tension for the 150 g hanging mass.

Table 2. 2

Tension: T	Wavespeed: v

 

Question 1.8: The dependency of the wavespeed v on the string tension T can be modeled by Equation 2.1:

                                                V=t/µ

                                                                                Equation 2. 1

 

Use this equation and your data in Table 2.2 to calculate the linear mass density of the string,µ.

Enter your data and results in Table 2. 3, and compute the average µ.

Table 2. 3

Tension: T	Wavespeed: v	Linear mass density: µ
Average value of µ

 

Question 1.9: In Question 1.8, you calculated the linear mass density of the string using Equation 2.1. Now, you will measure the linear mass density directly. What two quantities do you need to measure to determine the linear mass density of the string?

Question 1.10: Go to the instructor's table. There is a sample piece of string there. Measure the quantities you listed in Question 1.9.

Record your measurements here and compute the linear mass density. Use SI units.

Linear mass density = µ =_________

Question 1.11: Based on your answers to Questions 1.8 and 1.10, is Equation 2.1 a good model for how the wavespeed v depends on the string tension T? Explain.

If there are discrepancies between the model and your data, discuss those discrepancies and their source.

2. Reflection

Question 2.1: Which of the following quantities depend on just the wave source (i.e. the wave driver). On just the properties of the wave medium (i.e. the string). Or on both the wade source and the wave medium? Explain.

Support your answers by using your data and observations from this lab

Frequency of Standing Wave

Wavelength of Standing Wave

Wavespeed of Standing Wave

The Second Harmonic of the Standing Wave

3. Clean up

Out of respect for the students and TAs that will use this lab station after you, please clean up your lab station.

• Detach the mass hanger from the string and place it on the table.
• Un-tangle and organize wires and strings.
• Collect small items and put them back in their containers or bags.