Labs

General Lab Guidelines

1. Use a composition-style notebook (preferably, but not necessarily, one containing graph paper).

2. Write in ink whose color contrasts that of paper (e.g. No green ink on green paper and no pencil)

3. Use single-line strikeouts on your mistakes: my mistake. Never use whiteout.

4. In entering answers into your notebook either write down the question first, or write down the answer in an informative sentence.

 Example: What is the error in your result?

 Acceptable: "What is the error in your result? -2.3%"

 Acceptable: "The error in my result is -2.3%"

 Unacceptable: "-2.3%"

5. Reserve 3 pages at the start of your notebook for its table of contents.

6. You may paste screen captures of simulations or printouts of instructions into your notebook.

7. You may paste printouts of Excel graphs into your notebook.

8. Please use a ruler or other straight edge in drawing tables and the axes of graphs.

9. You must complete each lab within 1 week after the Thursday on which it was posted online.

10. In graphing data always record a table of the data before graphing the table's contents.

11. Always summarize the results of your lab in a paragraph labeled "Conclusions".

12. Number every page.

13. Begin each lab write-up with the title of the lab and its assignment date.

14. Enter your data on the right-side page. Use the left-side page for scratch.

15. Draw an X through any blank pages or large blank areas in your writeup.

16. Neatness and legibility counts!

Format of Lab Report

1. Title

2. Objective

3. Screen capture or drawing of experiment (simulation)

4. Procedure (ok to copy and paste into notebook)

5. Data (tables)

6. Calculations and Analysis (graphs)

7. Summary of Error and Sources of Error

8. Conclusion (1 or 2 sentences)

Lab for 08-31-23

Understanding Vectors

Objective: Obtain an intuitive understanding of vectors and their addition.

Procedure

1. Click on "Explore 2D" and play with the simulator until you understand how to
   A) Drag vectors into the central grid from their box on the right.
   B) Change the orientation of a vector by dragging its tip in a direction perpendicular to its length.
   C) Change the length of a vector by dragging its tip in a direction parallel or antiparallel to its length.
   D) Sum the vectors on the central grid by clicking the "Sum" box.
   E) Display a vector's length, angle relative to the horizontal, and components by clicking on it.
   F) Display angles of all vectors relative to the horizontal by clicking the θ box.
   G) Display the lengths of all vectors by clicking the "Values" box.
   
2. Position 2 random vectors on the simulator. Visualize their sum.
3. Use simulator to add them. Repeat until your visualization matches the simulator's result.
4. Use the simulator to verify the Law of Cosines by calculating the length of the sum using the Law of Cosines and comparing it to the simulator's result for 5 distinct triangles. Follow the example shown to fill out the table below.



Example






b = (16.8² + 24.7² - 2(16.8)(24.7)cos(86°))^1/2 = 28.89

Error = (28.9 - 28.89)/28.9 x 100 = 0.035%

Trial	Vector a Length	Vector s Length	Vector a Angle	Vector s Angle	(Vector a Angle) — (Vector s Angle)	Length of b (Simulator)	Length of b (Law of Cosines)	Error
Example	16.8	24.7	107.4°	21.4°	86°	28.9	28.89	0.035%
1
2
3
4
5



5. Are the values in the two length-of-b columns identical to 3 significant digits?
6. If they are not identical, what might be the source of the error?
7. Can you conclude that the it's possible to train yourself to accurately visualize the sum of two vectors? Can you conclude the the simulator is perfectly precise?

Lab for 09-07-23

Projectile Motion

Objective: Confirm Kinematics Equations and Study Effect of Air Resistance

Procedure

1. Click on the "Labs" (cannon) icon above.
2. Turn air resistance OFF. Select "user choice", diameter = 0.1. For a 3 kg mass shot with an initial speed of 15 m/s, at what angle does it travel the greatest horizontal distance? [Use trial and error to find this angle by fill out the table below and graphing the results (use angle as the horizontal axis).]

   Angle (degrees)	Range (m)
   25
   30
   40
   45
   50
   55
   60

3. Turn air resistance ON. Answer question # 1 again.
4. With air resistance OFF and mass = 1 kg, find the speed and angle required to hit a target at 18 m (± 0.3 m) in 2.8 seconds (± 0.1 second).
5. With air resistance ON and mass = 1 kg, find the speed and angle required to hit a target at 18 m in 2.8 seconds.
6. With air resistance ON, plot max_distance vs mass. Use the angle you obtained in step 2 and a speed of 15 m/s.  Use mass values: 1, 5, 9, 13, 17, 21, 25, 29 kg.
7. With air resistance OFF, repeat step 5 (using the same angle used in step 5).

Analysis

1. How well does your range vs angle graph compare to the theory? [Range = V_o²sin(2θ)/(2g)].
2. Does the distance traveled depend on mass, when air resistance is on?
3. Does the distance traveled depend on mass, when air resistance is off?

Lab for 09-14-23

Friction

Click to Run

Objective: Confirm the Equations for Static and Kinetic Friction

Procedure

Part 1: Static Friction

1. Open the simulation by clicking on the image above and clicking on the .jnlp file that is downloaded.
2. Select the "Friction" tab. Use the following settings:
   Static Friction (μ_s) = 0.1,
   KineticFfriction (μ_k) = 0.1,
   Object Mass = 100 kg,
   Gravity = "Earth",
   Ramp Angle = 0.0 degrees,
   Object Position = 6 m
   Friction: Wood
   Walls: Brick
3. Create the following table.
   
   

   Static Friction (μs)	Ramp Angle (Observed) (deg)	Ramp Angle (Theory) (deg)
   0.1
   0.3
   0.5
   0.7
   1.0
   1.5
   2.0

4. For each choice of the coefficient of static friction, slowly increase the ramp angle until the object begins to slide down. Record this "Ramp Angle (Observed)".
5. Calculate arctan(μ_s), and enter it as the "Ramp Angle (Theory)".
6. Repeat steps 4 and 5 for an object mass of 200 Kg on the Moon.
7. Do your results seem to depend on object mass?
8. Do your results seem to depend on the acceleration of gravity?
9. For each row calculate the error percentage: ((Observed - Theory)/Theory) x 100
10. What is the average of the error percentages of the rows?

Part 2: Kinetic Friction

1. Create the following table.
   
   

   Kinetic Friction (μk)	Static Friction (μs)	Ramp Angle (deg)	XStart(m)	XEnd(m)(Observed)	XEnd(m)(Theory)
   0.1	0.1	10	8
   0.2	0.2	15	8
   0.2	0.2	20	8
   0.3	0.3	30	8
   0.3	0.3	40	6
   0.4	0.4	50	6
   0.4	0.4	60	4
   0.5	0.5	70	3

   
   where
   X_Start is the distance to the right of the center (0 m) point from which object begins its motion.
   X_End is the distance to the left of the center (0 m) point at which object ends its motion.
   
2. Starting witht the first row of the table, use the textbox controls on the right to set the indicated values for μ_k, μ_s, and ramp angle. [Leave gravity, object mass the same as in Part 1.]
3. Using the textbox control on the right, set the object position to the value shown in the X_Start column.
4. Press the "Enter" key to cause the object to be positioned on the ramp and begin to move.
5. Record, in the X_End(Observed) column, the position to the left of the center that at which the object comes to rest. [Ignore the negative sign.]
6. Calculate X_End(Theory) using the following formula:
   
       If μ_s < tan(θ)
   
          X_End = -((sin(θ) - μ_kcos(θ))/μ_k)X_Start
   
       Else
   
          X_End = X_Start
   
   where θ is the the ramp angle, and μ_k and μ_s are respectively the coefficients of kinetic and static friction.
7. Do the contents of this table seem to depend on the mass of the object?
8. Do the contents of this table seem to depend on the the acceralation of gravity?
9. For each row calculate the error percentage: ((Observed - Theory)/Theory) x 100
10. What is the average of the error percentages of the rows?

Part 3: Prediction

1. Select the "Robot Moving Company" tab.
2. Assume that the ramp angle is 30 degrees.
3. Estimate the starting position of the object at the top of the ramp (e.g. 9 m)
4. Use the theory equation of Part 2 to predict where each object, once nudged down the ramp by the robot, will come to rest.
5. For each object predict whether it will fall off of the cliff. Justisfy your prediction with the equation.


Example: The first object, the crate, has
μ_k = 0.3.
X_Start = 9 m (approximately)
So
     X_End = ((sin(30) - 0.3*cos(30))/0.3)*9
            = 7.2 m

Because 7.2 m < 10 m (the location of the cliff), the object will not go over the cliff.

As usual, please discuss any possible sources of error, and end your write up with a brief conclusion.

Lab for 09-21-23

Kepler's 3rd Law

Objective: Verify Kepler's Third Law

Procedure

1. Create a table like that below.

   Trial	2A (Mkm)	A3 (Mkm3)	T (days)	T2 (days2)	T2/A3 (day2)/(Mkm3)
   1
   2
   3
   4

   A = semi major axis T = period Mkm = Mega kilometer (1 million kilometers) day = Earth day
2. Double click on the "To Scale" icon above.
3. In the simulation that appears, select "Grid".
4. Move the star 1 grid spacing to the left.
5. Move the planet so that it is 1 grid spacing to the right of the star.
   [Star and planet should be on same horizontal grid line.]
6. Select "Velocity".
7. Make the velocity vector of the planet 3 grid spacings long and perfectly vertical.
   
   
8. Select "Path".
9. To run the simulation, click the blue start (arrow) button. Pause the sim, when the planet has completed about 95% of its orbit.
10. Use the step button (to immediate right of the start/pause button) to complete the planet's orbit. [Velocity vector should be vertical again.]
11. If you need to redo the orbit, use the restart button (immediately to the left of the start/pause button).[Use scale control in upper left, if path does not fit on screen.]
12. Select "Measuring Tape". Use the tape measure to measure the length of the major (horizontal axis) of the orbit.
    [Might help to use tape measure upside down.]
13. Enter your major axis measurement in the "2A" column of your table.
14. Enter the Earth Days shown in the "T" column of your table.
15. Deselect "Path".
16. Position the planet 2 grid spacing to the right of the star.
17. Make the velocity vector perfectly vertical and 2 grid spacings long.
18. Select "Path".
19. Have the planet complete another orbit.
20. Measure 2A. Enter its value and that of T in the table as before.
21. Deselect "Path".
22. Position the planet 3 grid spacing to the right of the star.
23. Make the velocity vector perfectly vertical and 1 grid spacing long .
24. Select "Path".
25. Have the planet complete another orbit.
26. Measure 2A. Enter its value and that of T in the table as before.
27. Deselect "Path".
28. Position the planet 4 grid spacing to the right of the star.
29. Make the velocity vector perfectly vertical and 0.5 grid spacing long .
30. Select "Path".
31. Have the planet complete another orbit.
32. Measure 2A. Enter its value and that of T in the table as before.
33. Deselect "Path".
34. Graph T²/A³ vs. Trial Number. [Make Trial Number the horizontal axis.]
35. Is your graph a horizontal line?
36. What is the difference between the largest value in the T²/A³ and the smallest?
37. Calculate the % error as the difference above divided by the average value of T²/A³.
38. Graph T² vs. A³. [Make the latter the horizontal axis.]
39. Is your graph a straight line? What is its slope?
40. Speculate: Does T²/A³ depend on the mass of the star?
41. Add a conclusion of the form: My findings support (or do not support) Kepler's Third with an error of ___%.

Lab for 09-28-23

Kinetic and Potential Energy

Objective: Confirm conservation of energy and formulas for its kinetic and potential forms.

Procedure

1. Double click on the "Intro" icon above. [You may set the simulation to full-screen by clicking on the "hamburger" menu in the lower right.]
2. Select "Bar Graph" and "Grid". Select "Slow Motion". Leave Mass at the position midway between small and large.
3. Enter a sketch or screen capture of the parabolic track into your lab notebook.
4. Position your skater near the highest position of your track. [If necessary, start the simulation my clicking the "Play/Pause" button near the bottom of the simulation.]
5. Is the sum of potential energy and kinetic energy the same at all points on the track? [Pause the skater at 3 different positions. Check to see if the sum of E_p and E_k in the bar graph is the same at each position.]
6. Where on the track where the potential energy is lowest?
7. Where on the track where the potential energy is highest?
8. Where on the track where the kinetic energy is lowest?
9. Where on the track where the kinetic energy is highest?
10. How is the skater's motion affected by changing his mass?
11. Click on the "Friction" icon at the bottom of the simulation. Select parabolic track.
12. How is the skater's motion affected by changing maximizing friction"?
13. Click on the "Playground" icon at the bottom of the simulation. Set the skater's mass to "small".
14. Use the red-gray track segments to create a track that consists of two hills, as shown below.
    
    We will refer to the hill on the left as the "first hill", that on the right as the "second hill".
15. Click on to set "Roller Coaster Mode" ON.
16. Select the "Grid" to measure the heights your hills. Set Friction to "none".
17. If the 1st hill has a height h₁, what is the greatest height h₂ of the 2nd hill that will allow the skater to traverse the 2nd hill?
18. How does the total energy of the skater at the top of the 1st hill compare with its total energy at the top of the 2nd hill?
19. How does the total energy of the skater at the top of the 1st hill compare with its total energy at the bottom of the 1st hill?
20. Potential energy of skater: E_p = mgh. Kinetic energy of skater: E_k = mv²/2. Total energy: E = E_p + E_k. At what locations is the skater's kinetic energy greatest?
21. Create a track with a loop that resembles the image below. [The loop's diameter should be about half the available space, as shown.]
    
22. Click on to set "Roller Coaster Mode" OFF.
23. At what locations is the skater's potential energy greatest?
24. Does the acceleration of the the skater around the loop increase or decrease if the radius of the loop is decreased?
25. If the loop has a radius r and the skater has a velocity v, what is the formula that you already know that gives the acceleration of the skater around the loop?
26. Let v_t and v_b be the speeds of the skater at the top of and bottom of the loop respectively. Set the hill to its maximum height (~8 m). What is the formula for the total energy of the skater at the bottom of the hill? [Hint: assume h = 0 at the bottom of the hill, and v = 0 at the top of the hill.]
27. How does v_b depend on h (the height of the hill)?
28. How does the total energy at the top of the loop compare with that at the bottom of the loop? Write an equation equating the energies at the top and bottom.
29. Solve the equation above for v_t
30. For the skater to barely remain on the track at the top of the loop, the acceleration of gravity g must equal the circular acceleration v_t²/r. Solve for the loop radius r_max at which this occurs.  Help
31. According to your calculation, how does the diameter of the largest loop on which the skater remains on the track compare with h₂?
32. Use trial and error to determine how closely the simulation matches your derived expression for the maximum diameter d_max = 2r_max in terms of h?

Lab for 10-05-23

Elastic and Inelastic Collisions

Objectives:
  Determine whether kinetic energy and momentum are conserved in collisions.
  Test formula for final velocities of colliding objects.

Procedure:

1. Click on the "Intro" icon above. Familiarize yourself with the basic operation of the simulation.
   [Note reset buttons and that the menu in the lower right includes "screen shot" and "full screen".]
2. Create a table that looks like this:
   
   

   Elastic Collisions
   Trial	m1 (kg)	m2 (kg)	v1initial (m/s)	v1final (m/s)	v2initial (m/s)	v2final (m/s)	KEinitial (J)	KEfinal (J)	Pinitial (kg-m/s)	Pfinal (kg-m/s)
   1	1	1	1		0		0.5		1
   2	3.0	0.1	1		0		1.5		3
   3	0.1	3.0	1		0		0.05		0.10

3. Click "Explore 1D" icon.
4. Select "More Data" checkbox to display the data panel. Place Ball 2 about 1 meter to the right of Ball 1.
5. For Trial 1, enter the values shown in the table for the masses and initial velocities for Balls 1 and 2.
6. Select "Kinetic Energy", "Velocity", and "Values" checkboxes.
7. Set elasticity to 100%.
8. Click the "Play" button. Click the "Pause" button immediately after the collision.
9. Record the final values for the velocities, total kinetic energy (KE), and total momentum (P).
   (Note that P is the sum of the momenta shown in the data panel).
10. Repeat Steps 4 - 8 for Trials 2 and 3.
11. Does total kinetic energy appear to be conserved?
12. Does total momentum appear to be conserved?
13. To what degree are your results consistent with the following formulas?
    
      v1_final = (m₁ - m₂)v1_initial/(m₁ + m₂)
      v2_final = (2m₁)v1_initial/(m₁ + m₂)
    
    
14. What approximate form do the above formulas take when m₁ = m₂ (Trial 1)?
15. What approximate form do the above formulas take when m₁ is much greater than m₂ (Trial 2)?
16. What approximate form do the above formulas take when m₁ is much less than m₂ (Trial 3)?
17. Create a table that looks like this:
    

    Inelastic Collisions
    Trial	m1 (kg)	m2 (kg)	v1initial (m/s)	v1final (m/s)	v2initial (m/s)	v2final (m/s)	KEinitial (J)	KEfinal (J)	Pinitial (kg-m/s)	Pfinal (kg-m/s)
    1	1	1	1		0		0.5		1
    2	3.0	0.1	1		0		1.5		3
    3	0.1	3.0	1		0		0.05		0.10

18. Set "Elasticity" to 50%.
19. Conduct three trials as above and fill out your table.
20. Does total kinetic energy appear to be conserved?
21. Plot the change in Kinetic energy (y-axis) vs Trial number (x-axis).
22. Do the equations in Step 12 seem to hold for inelastic collisions.
23. Does total momentum appear to be conserved?
24. Set "Elasticity" to 0% and cause the Balls to collide.
25. What happens in such a "completely inelastic" collision?

Lab for 10-12-23

Rotational Motion

Objectives:  Test the formulas for torque, moment of inertia, and angular momentum.

Click on the image above.

1. Select the "Intro" tab.
2. Figure out how to make the disk rotate. Record your findings.
3. Place the insects near the center of the disk. Figure out how to speed up the disk so that they fly off. Record your finding
4. Select the "Torque" tab.
5. Place the lady bug near the center of the disk.
6. Set the Applied Force (F) to about 1 N by moving the vertical slider (blue arrow).
7. Set the Radius of Force (r) to about 3 m by moving the vertical slider (green arrow).
8. Is the value of the Applied Torque (T) given by T = rF?
9. Sketch the position of the Applied Force vector on the disk.
10. Click on any of the "Go!" buttons.
11. Does the disk seem to spin faster and faster? [Does the lady bug fly off of it?]
12. Select the "Moment of Inertia" tab.
13. Set the Applied Torque to about 1 N-m by moving the vertical slider (blue arrow).
14. Set the inner radius to 0 m, outer radius to 4 m, mass to 0.15 kg, Force of brake to 0.
15. Click on any of the "Go!" buttons.
16. How does the moment of inertia (I) compare to the formula I = (MR²)/2 (where M =platform mass; R = outer radius) ?
17. Convert the angular acceleration of the platform to rad/s/s by dividing it by 57.3 deg/rad
18. How does the torque compare to the formula T = Iα (where α = angular acceleration in rad/s/s)?
19. Select the "Angular Momentum" tab.
20. Set the Angular Velocity to about 100.
21. Convert the angular velocity to rad/s by dividing by 57.3 deg/rad
22. Click on any of the "Go!" buttons.
23. How does the angular momentum (L) compare to the formula L = Iω (where I = moment of inertia, ω = angular velocity in rad/s)?

Lab for 10-19-23

Moments of Inertia and Rolling

Objective: Observe the effect of momentum of inertia on the speed of rolling objects

notes[]

Helpful facts in solving for acceleration a: Loss of potential energy = Gain in kinetic energy Kinetic energy = (translational kinetic energy) + (rotational kinetic energy) Rotational kinetic energy = Iω2/2 v = ωr, I = βmr2, H - h = ssin(θ), v2 = 2as

Please go to this link and do the following.

The simulation shows a race of 3 objects rolling down an inclined plane.
One of the objects is a solid sphere, on of them is a solid disk, and one of them is a hoop.

Height of ramp = 640 m. Base of ramp = 1500 m.

1. Use your browser's scale control (CNTL-) to size the simulation so that it fits on your screen.
2. Run the simulation and write down your guess for the identity of each rolling object.
3. Write down an equation stating that the loss in potential energy is the gain in total kinetic energy.
4. Use the equations listed above to solve the equation of the previous step for v².
5. Replace v² with 2as and solve for the acceleration a.
6. Use H - h = ssin(θ) to eliminate s from your equation for a.
7. Look up the moments of inertia for a hoop, disk, and sphere (i.e. find β for them). Use these to calculate the corresponding accelerations using the formula obtained in step #5.
8. Does your formula agree with the simulation?
9. Was your guess correct?
10. What qualitative conclusion can you make about the dependence of acceleration on moment of inertia?
11. Watch this brief video.

Lab for 10-26-23

Pulleys

Objective: Determine how the force required to lift an object depends on the number of pulleys used.

Procedure

1. Create a table that looks like this:
   
   

   Object's Mass = 700 grams
   Number of Pulleys Used	Force Required to Lift Object
   1
   2
   3
   4
   5
   6

   
   
   
2. On the simulation, click "Begin". Click "The hanging mass is" until it shows 700. Click "Number of Pulleys" until it shows 1. Ensure that "Experiment Conducted on:" shows "Earth".
3. Click on the black force sensor. Record (on the 1st row of your table) the force value in the red box that appears. Click outside the box to make the force sensor readout disappear.
4. Increase the number of pulleys by 1. Repeat steps 3 and 4 until your table is completely filled out.
5. Is there difference in the lift force required when 1 pulley is used with that required when 2 pulleys are used? What about between 3 and 4 pulleys? 5 and 6 pulleys?
6. Try to explain the reason for your answers in the previous step.
7. Plot your data. Put number of pulleys on the horizontal axis. Be sure to use only the even number of pulleys (2, 4, 6).
8. Calculate the weight in Newtons of the 700 gram object.
9. Use your results to write down a formula that obtains the force required to lift an object as a function of its weight and the (even) number of pulleys used.
10. Would your formula change if you had used a different mass?
11. Would your formula change if you had had located your experiment on a different planet?
12. Summarize your conclusions.

Lab for 11-02-23

Resonance

Objective: Confirm the condition under which resonance occurs.

Procedure

1. Use the following formula to calculate the natural frequency f of oscillation of each of the 3 spring/mass systems shown.
   
   
   f = ((k/m)^1/2)/(2π)
   
   Record these values as the "blue frequency", "pink frequency", and "yellow frequency".
2. Set the "Frequency of Forced Vibration" as closely as possible to the blue frequency. Click "Reset" then "Run".
3. Watch closely to see which mass decended the farthest from its equilibrium position.
   Observe continuosly until t = 20. Estimate in "grids" this mass' greatest deviation from its equilibrium position during this entire period (from t=0 to t=20), e.g. 1.2 grids.
   Record your observation.
4. Repeat step 3 with the pink and yellow frequencies.
5. Summarize your results in table.
6. Does resonance seem to result when the frequency of the forced vibration matches the natural frequency of the oscillating spring/mass system?

Lab for 11-09-23

Superposition and Standing Waves

Objective: Observe the consequences of the principle of superpostion. Test the formula for the frequency of standing waves on a string.

Part A: Principle of Superposition

Please go to Superposition of Waves. Use the animations there to answer the following questions.

1. Animation 1: What is the peak amplitude of the rightward-moving Gaussian wave?[Use a ruler to measure peak amplitude in mm.]
2. Animation 1: What is the peak amplitude (in mm) of the leftward-moving Gaussian wave?
3. Animation 1: What is the peak amplitude (in mm) of the combined wave?
4. Animation 1: How does your answer to #3 depend on your answers to #1 and #2?
5. Animation 2: Consider the interference of two sine waves of equal wavelength traveling in the same direction. What is their phase difference when they constructively interfere?
6. Animation 2: What is their phase difference when they destructively interfere?
7. Animation 3: Consider the interference of two sine waves of equal wavelength traveling in opposite directions. Measure the speed of both waves. Are they the same?
8. Animation 3: Is the superposition of these waves leftward-moving, rightward-moving, or stationary?
9. Animation 3: Is the superposition also a sinusoidal wave?
10. Animation 3: Does the amplitude of the superposition change with time? Guess the formula for the amplitude as a function of time.
11. Animation 4: Consider two rightward-moving waves with slightly different frequencies. What is the frequency of the amplitude envelope of the superposition?
12. Animation 4: What is the beat frequency?
13. Watch this this. [You'll need sound.]

Part B: Standing Waves

Please go Waves on String and follow the instructions below.

1. Set the simulation as follows: "Manual", "Fixed End", Damping = 1 tick to right of "None", Tension = "High". Leave "Rulers", "Timer", and "Reference Line" unchecked.
2. Sharply jerk the red-handled wrench up and down (once), so that that the green ball it holds is yanked above the gold line and returns to it.
3. Was the pulse that you created inverted, when it was reflected from the vise on the right?
4. Set the simulation to "Loose End". Jerk the wrench up and down (once) as before.
5. Was your pulse inverted, when it was reflected by the pole and ring?
6. Click "Restart". Set your simulation to "Pulse", "Fixed End", and "Slow Motion". Check "Rulers" and "Timer". Set Damping to "None", Pulse Width = 0.50 s, Amplitude=0.75 cm.
7. Click the green pulse button and immediately start the timer. Stop the timer the instant that the pulse has completed 5 round trips. Record this time T. [Use the restart button to practice this as much as you need to.]
8. Measure the distance L in cm between the pulse generator (on the left) and the vise (on the right).
9. Obtain the speed v (in cm/sec) of your pulse using v = 5(2L)/T
10. Note that the wavelengths of a wave on the string that are allowed, if both ends of the string are held in place, are given by
    
    λ=2L/n    where n = 1, 2, 3, 4,...
    
    
11. Note that the wavelengths of a wave on the string that are allowed, if only one of the ends of the string is held in place, are given by
    
    λ=4L/n    where n = 1, 3, 5, 7,...
    
    
12. Create the following table.
    
    

    Both Ends Fixed
    n	λ (cm)	f = v/λ (Hz)	Standing Wave Observed?
    1
    2
    3
    4

13. Fill out the table except for the right-most column.
14. Click "Restart". Set simulation as follows: "Oscillate", Amplitude=0.10 cm, Damping=None, Tension=High, "Normal", "Fixed End". "Timer" and "Rulers" unchecked. "Reference Line" checked.
15. Set Frequency to that in the n=1 row of your table.
16. Click "Restart" again. After the string acquires a noticeable amplitude (max amplitude at least half way to reference line), quickly set the frequency of the oscillator on the right to zero. Do this so that the green ball in contact with the top of the oscillator lands on the gold dashed line.
17. Do you observer a standing wave (i.e. one in which nodes are stationary, and anti-nodes move from max to min amplitude)?
18. Repeat for all rows of your table.
19. Create the following table.
    
    

    One End Fixed
    n	λ (cm)	f = v/λ (Hz)	Standing Wave Observed?
    1
    3
    5
    7

20. Fill out the table except for the right-most column.
21. Click "Restart". Leave all simulation settings the same, except change use "Loose End".
22. Set Frequency to that in the n=1 row of the table above.
23. Click "Restart" again. After the string acquires a noticeable amplitude, quicky set the frequency of the oscillator on the right to zero. Do this so that the green ball in contact with the top of the oscillator lands on the gold dashed line.
24. Do you observer a standing wave (i.e. one in which nodes are stationary, and anti-nodes move from max to min amplitude)?
25. Repeat for all rows of your table.
26. Which case (both ends fixed or one end fixed) is analogous to a piano string?
27. Which case (both ends fixed or one end fixed) is analogous to a wind instrument (consisting of a tube with one end closed, the other open)?

Lab for 11-16-23

The Bernoulli Equation

Objective: Test the Bernoulli Equation.

Click on the image above.

1. Set simulation as follows: Units = Metric, Friction: Not checked, Flux meter: Not checked, Flow Rate = 5000 liters/second, Fluid Density = 1000 kg/m³, Dots: Checked.
2. Use the yellow handles to shape the tube so that it looks like this:
   
   
3. Create the following table:
   

   Flow Rate= 5000 L/s,   Density = ρ = 1000 kg/m3
   Location	P (Pa)	v (m/s)	Flux (L/s/m2)	0.5ρv2	P + 0.5ρv2
   Entrance
   Bulge
   Exit

4. Use the ruler to find the depth of the center of the tube. You will take all the measurements for your table at this depth. Your measurements will be at three horizontal locations: Entrance (the point at which the fluid enters from the pipe on the left), Bulge (at the central bulge), Exit (the point at which the fluid exits into the pipe on the right).
5. Use the meters for speed, pressure, and flux to fill out the first 3 data columns of the table.
6. Create another table like that above, but set the Flow Rate to 9000 and the Fluid Density to 1420 kg/m³
7. Take measures as you did before, and fill out the first 3 data columns of your second table.
8. Calculate the values of the right-most columns in both of your tables. [Be sure use pressure in Pa, not kPa. 1 kPa = 1000 Pa].
9. Was the flow rate the same everywhere in the tube?
10. Was the flux the same everywhere in the tube?
11. Was the velocity the same everywhere in the tube?
12. Is P + 0.5ρv² the same at the same depth at different horizontal locations in the tube?
13. Does the Bernoulli Equations appear to be confirmed by your data?
14. Calculate the difference between the greatest and lowest values of P + 0.5ρv². Define the error as this difference divided by the average of the the greatest and lowest value, i.e.
    
    Percentage Error = (Greatest - Lowest) / ((Greatest + Lowest)/2) x 100
    

Lab for 11-30-23

Thermodynamics of Gases

Objective: Determined the relationship between pressure, volume, and temperature of gases.