Working with Spreadsheets

Objective:

• To get familiar with electronic spreadsheets by using them in some simple applications.

Materials:

• Computer with EXCEL software

Procedure:

• Turn on the computer and load the Microsoft Excel software by clicking on Start, move the mouse over Programs them move the mouse over Microsoft Excel and then left click.
• Create a simple spreadsheet that calculates the values of the following function: f(x) = A sin(Bx + C). Initially choose values for of A = 5, B = 3 and C = π/3. Place these values at the right side of the spreadsheet in the region reserved for constants. Put the words amplitude, frequency, and phase next to each as an explanation for the meaning of each constant. Place column headings for "x" and "f(x)" near the middle of the spreadsheet, enter a zero in the cell below "x", and enter the formula shown above in the cell below "f(x)". Be sure to put an equal sign in front of the formula. Create a column for values of x that run from zero to 10 radians in steps of 0.1 radians. Use the copy feature to create these x values. Similarly, create in the next column the corresponding values of f(x) by copying the formula shown above down through the same number of rows.
• Once the generated data looks reasonable, copy this data onto the clipboard by highlighting the contents of the two columns and choosing edit, then copy, from the menu bar. Print out a copy of the spreadsheet and also print out the spreadsheet formulas. Be sure that your rows and columns are numbered and lettered.
• Minimize the spreadsheet window and run the Graphical Analysis program by opening the PhysicsApps icon and then double-click the Graphical Analysis icon. Once the program loads, click on the top of the x column and choose edit, paste, to place the data from the clipboard into the graphing program. A graph of the data should appear in the graph window. Put appropriate labels on the axes of the graph.
• Highlight the portion of the graph for analysis and choose analyze and curve fit from the menu bar to direct the computer to find a function that best fits the data. From the list of possible functions, give the computer a hint as to what type of function you expect your data to match. The computer should display a value for A, B, and C that fit the sine curve that you are plotting. Make a copy of the data and graph by selecting file, then print.
• Repeat the above process for a spreadsheet that calculates the position of a freely falling particle as a function of time. This time your constants should include the acceleration of gravity, the initial velocity, initial position, and the time increment. Start off with g = 9.8m/s^2, v0 = 50 m/s, x0 =1000 m and Δt = 0.2 s. Print out the spreadsheet. Copy the data into the Graphical Analysis program and obtain a graph of position vs time. Fit this data to a function (y = A + Bx + Cx^2) which closely matches the data. Interpret the values of A, B, and C. Get a printout of this graph with the data table.

Conclusion:

The point of this experiment was to implement our knowledge of physics by predicting a trend based off of given data and then graphing the data to observe the resulting curve. Analysis was indicative of a parabolic shape that was based on a position vs time graph and then by evaluating derivatives, the velocity and acceleration could be observed accurately.

Graphical Analysis

Objectives:

1. To gain experience in drawing graphs and in using graphing software.

Materials:

1. Windows based computer with Graphical Analysis software
2. LabPro interface
3. LoggerPro software
4. Motion detector
5. Rubber ball
6. Wire basket

Procedure:

Part I:

1. Gather the required materials and then proceed to turn on the computer. Once started, access the PhysicsApps folder by double-clicking on the folder icon. Subsequently, load the Graphical Analysis software within the folder by double-clicking on its respective icon.
2. Once the program is operational, open the file functionplot by clicking on File (located at the top left-hand corner of the screen), Open, then double-click the Physics folder. If properly executed, it should show a graph of a function and the data used to create the graph. Read the instructions that appear in the text window to see how to enter your own function.
3. Decide upon and choose a mathematical function of your liking to plot. Make sure that the function has the following components:
• a title
• x and y axis labels
• units
4. Make sure that the graph you created has the appearance that you want. Obtain a printout of the graph for each member of your group (excluding data table) by selecting File, then Print.

Part II:

1. Connect the lab pro to the computer and the motion detector to the DIG/SONIC2 port on the LabPro. Load the LoggerPro software in the PhysicsApps folder. Click on File, then Open, and open the mechanics folder. Within that folder, open the graphlab file. Take a position vs. time graph for a falling ball with the motion sensor.
2. When you finally achieve a nice curve, select the appropriate data range and perform a fit to the data. Confirm that the distance fallen adheres to the following relationship: d*α*g*t^n where g = 9.8 m/s2, is the acceleration due to gravity. Obtain a printout of your graph (with the fit) for each member of your group (without the data table).
3. Use dimensional analysis and unit analysis to verify the equation.

Results:


Figure 1



Figure 2

Dimensional Analysis:

d α gt²

v=(dx)/(dt)

g=(dv)/(dt)
therefore:

 v=m/t

 g=m/t^2

 d=g*t^2=[m/t^2]*t^2=m

Unit Analysis:

v=m/t

g=m/t^2

d=[m/t^2]*t^2]=m

Conclusions:

     During the lab, I learned how to use the LoggerPro software in order to capture the velocity of a falling object, accurately detect the acceleration of gravity to be roughly around -9.8 m/s^2, and to analyze a graph to estimate values for velocity through curve fitting a linear fit around a parabolic position vs. time graph (the slope is equal to the velocity).

Drag Force on a Coffee Filter

Purpose:   To study the relationship between air drag forces and the velocity of a falling body.

Equipment:  Computer with Logger Pro software, lab pro, motion detector, nine coffee filters, meter stick

Introduction:
When an object moves through a fluid, such as air, it experiences a drag force that opposes its
motion.  This force generally increases with velocity of the object.  In this lab we are going to
investigate the velocity dependence of the drag force.   We will start by assuming the drag force,
FD, has a simple power law dependence on the speed given by
1)  FD  =  k |v|^n , where the power n is to be determined by the experiment.
This lab will investigate drag forces acting on a falling coffee filter.  Because of the large surface
area and low mass of these filters, they reach terminal speed soon after being released.
Procedure:
NOTE: You will be given a packet of nine nested coffee filters.  It is important that the shape of this
packet stays the same throughout the experiment so do not take the filters apart or otherwise
alter the shape of the packet. Why is it important for the shape to stay the same? Explain and
use a diagram.

1.  Login to your computer with username and password. Start the Logger Pro software, open the
Mechanics folder and the graphlab file. Don’t forget to label the axes of the graph and create an
appropriate title for the graph.  Set the data collection rate to 30 Hz.
2. Place the motion detector on the floor facing upward and hold the packet of nine filters at a minimum
height of 1.5 m directly above the motion detector.  (Be aware other of nearby objects which can
cause reflections.)  Start the computer collecting data, and then release the packet.  What should the
position vs time graph look like?  Explain.
Verify that the data are consistent. If not, repeat the trial.  Examine the graph and using the mouse,
select (click and drag) a small range of data points near the end of the motion where the packet
moved with constant speed.  Exclude any early or late points where the motion is not uniform.
3.  Use the curve fitting option from the analysis menu to fit a linear curve (y = mx + b) to the selected
data. Record the slope (m) of the curve from this fit.  What should this slope represent? Explain.
Repeat this measurement at least four more times, and calculate the average velocity.  Record all
data in an excel data table.
4.  Carefully remove one filter from the packet and repeat the procedure in parts 2 and 3 for the
remaining packet of eight filters.  Keep removing filters one at a time and repeating the above steps
until you finish with a single coffee filter. Print a copy of one of your best x vs t graphs that show the
motion and the linear curve fit to the data for everyone in your group (Do not include the data table;
graph only please).
5.  In Graphical Analysis, create a two column data table with packet weight (number of filters) in one
column and average terminal speed (|v|) in the other.  Make a plot of packet weight (y-axis) vs.
terminal speed not velocity (x-axis).  Choose appropriate labels and scales for the axes of your
graph.  Be sure to remove the “connecting lines” from the plot.  Perform a power law fit of the data
and record the power, n, given by the computer.  Obtain a printout of your graph for each member of
your group. (Check the % error between your experimentally determined n and the theoretical
value before you make a printout – you may need to repeat trials if the error is too large.)
6.  Since the drag force is equal to the packet weight, we have found the dependence of drag force on
speed.  Write equation 1 above with the value of n obtained from your experiment.  Put a box around
this equation.  Look in the section on drag forces in your text and write down the equation given there
for the drag force on an object moving through a fluid.  How does your value of n compare with the
value given in the text?  What does the other fit parameter represent?  Explain.

Conclusion:

The point of this experiment was to determine through observation and experimentation the effect of drag using the formula k|v|^n by evaluating the value of n with the LoggerPro curve fit analysis. There were several factors to which the experiment may or may not go accordingly; the fact that the coffee filters are very fragile and subject to physical modification and, of course, human error for collecting data, dropping the filters accurately, and fitting the data with a curve fit inaccurately or with a non-optimum value. The result of the trials my group conducted hinted us to believe that the n value was approximately 2. The surface area of the filters is also a dependence factor for the drag calculations, which is why it is very important to keep the filters in pristine condition. In conjunction with the former, we concluded that the more filters you have, the greater the drag because there is more surface area to inhibit an expedited fall.

Balanced Torques and Center of Gravity

Objective:  To investigate the conditions for rotational equilibrium of a rigid bar and to determine the center of gravity of a system of masses.

Materials: Meter stick, meter stick clamps (knife edge clamp), balance support, mass set, weight hangers,
unknown masses, balance.
                                               
Procedure:
1. Balance the meter stick in the knife edge clamp and record the position of the balance point.
2. Select two different masses (100 grams or more each) and using the meter stick clamps and weight hangers, suspend one on each side of the meter stick support at different distances from the support. Adjust the positions so the system is balanced. Record the masses and positions. Sum the torques about your pivot point O and compare with the expected value.
3. Place the same two masses used above at different locations on the same side of the support and balance the system with a third mass on the opposite side. Record all three masses and positions. Calculate the net torque on this system about the point support and compare with the expected value.
4. Replace one of the above masses with an unknown mass. Readjust the positions of the masses until equilibrium is achieved, recording all values. Using the equilibrium condition for rotational motion, calculate the unknown mass. Measure the mass of the unknown on a balance and compare the two masses by finding the percent difference.
5. Place about 200 grams at 90 cm on the meter stick and balance the system by changing the balance point of the meter stick. From this information, calculate the mass of the meter stick. Compare this with the meter stick mass obtained from the balance.
6. With the 200 grams still at the 90 cm mark, imagine that you now position an additional 100 grams mass at the 30 cm mark on the meter stick. Calculate the position of the center of gravity of this combination (two masses and meter stick). Check your result by actually placing the 100 g at the 30 cm mark and balancing this system. Compare the calculated and experimental results.

Experiment Questions:
What point in the meter stick does this correspond to?

Is it necessary to include the mass of the clamps in your calculations? EXPLAIN!

Should the clamp holding the meter stick be included as part of the mass of the meter stick?  EXPLAIN!

Where should the point of support on the meter stick be to balance this system?

Vector Addition of Forces

Objective: To study vector addition by:

1. Graphical means
2. Using components.

A circular force table is used to check results.

Materials:

1. Circular force table
2. Masses
3. Mass holders
4. String
5. Protractor
6. Four pulleys.

Circular Force Table, Pulleys,
Masses, Mass Holders, and String

    

Procedure:

1. Your instructor will give each group three masses in grams (which will represent the magnitude of three forces) and three angles. Choose a scale of 1 cm = 20 grams, make a vector diagram (figure 3) showing these forces, and graphically find their resultant. Determine the magnitude (length) and direction (angle) of the resultant force using a ruler and protractor (figure 4).
2. Make a second vector diagram and show the same three forces again and find the resultant vector by components (figure 5). Show the components of each vector as well as the resultant vector on your diagram. Draw the force (vector) you would need to exactly cancel out this resultant.
3. Mount three pulleys on the edge of your force table at the angles given above. Attach strings to the center ring so that they each run over the pulley and attach to a mass holder as shown in the figure below. Hang the appropriate masses (numerically equal to the forces in grams) on each string. Set up a fourth pulley and mass holder at 180 degrees opposite from the angle you calculated for the resultant of the first three vectors. Record all mass and angles. If properly done, the ring in the center of the circular force table should be in equilibrium. Ask your instructor to check your results before going on.
4. Confirm your results by simulation at the website:http://phet.colorado.edu/en/simulation/vector-addition. Add the vectors and obtain a resultant. Draw a diagram showing how you verified the result.
5. In your conclusions, summarize what you learned and discuss any sources of error.

Results:



Figure 1



Figure 2

Figure 3
Vector Diagram

   

Figure 4

Adding Vectors by

Ruler and Protractor



 



Figure 5
Adding Vectors by
Components

Figure 6






Figure 7

 



Given the masses that represent the magnitude of vectors (100, 200, and 160) and the angles (0, 71, and 144) of inclination for those vectors respectively, figure 3 illustrates the resultant vector and the negative of the resultant vector. These vectors are necessary because they allow us to counter-balance the force table's mass vectors and place the vectors in equilibrium as seen in figure 1.

To calculate the magnitude and angle of the resultant vector:

R_x = (100cos0) + (200cos71) + (160cos144) = 36.1

R_y = (100sin0) + (200sin71) + (160sin144) = 283

θ = 82.7
R = (283^2 + 36.1^2)^(1/2) = 285.3

According to the data we collected and the vectors calculated, the approximation of the vectors we visualized online (figures 6 and 7) were close to our exact answers.

Conclusion:

During our experiment we learned that in order to place three forces in equilibrium a fourth force was necessary to counter-balance the other three. The counter-balancing force was exactly equal to the negative of the resultant of the sum of the first three forces (calculated as vectors).

Any source of error that was present was a result of human miscalculation.

Acceleration of Gravity on an Inclined Plane

Objective:

1. To calculate the acceleration of gravity by observing the motion of a cart on an incline.
2. To acquire further computer literacy with respect to collecting and analyzing data.

Materials:

1. Windows-based computer with LoggerPro software installed
2. Motion detector
3. Ballistic cart
4. Aluminum track
5. Wood blocks
6. Meter stick
7. Small carpenter level

LabPro Interface

Aluminum Track, Ballistic Cart,
and Wood Block





Motion Detector

Procedure:

1. Connect LabPro to the computer with the LoggerPro software and the motion detector to the DIG/SONIC2 port on the LabPro. Turn on the computer and load the LoggerPro
   software by double clicking the corresponding icon located within the PhysicsApps folder. Locate and open the file named graphlab; it will be used to set up the computer for collecting the velocity and acceleration data in order calculate the lines of best fit, thereby calculating the acceleration of gravity using the formula "gSinθ=(a_1+a_2)/2". Double-click on the Mechanics folder and then proceed to open the file by double-clicking it.
2. Incline the end of the aluminum track by using the wood block as a wedge. Place the wood block roughly at the 50cm mark on the track. Use the level to make sure that the track is level. Once level, calculate the angle of inclination using basic trigonometry (calculate the heights of the track at both ends, subtract the smaller value from the bigger value, and then take the inverse sin of the result as seen in Figure 1).
3. Place the motion detector at the upper end of the track facing down toward the lower end. Start with the cart at the lower end of the track and gently push the ballistic cart toward the motion detector until it is just outside 50cm from the motion detector. Make sure to stop the cart before it crashes into the track to prevent damage to the cart or track.
4. Start data collection after the cart leaves your hand and observe both the position and velocity graphs simultaneously by having two windows open (one above the other). Select suitable scales for both vertical and horizontal axes to best show the motion of the ballistic cart. Properly label the graphs with titles, units, and other relevant information. Repeat the trial until the graph looks like a consistent curve.
5. Once an appropriate curve has been acquired, use the LoggerPro software to take a virtual snapshot of the slope of the curve v vs. t by selecting a range of times that represents the motion of the ballistic cart going up the incline. Choose the "Analyze/Curve Fit" option to fit the selected portion of the curve to a linear function of time. Repeat this process for when the ballistic cart is returning down the incline.
6. Complete at least two more trials for the same angle of inclination. For each trial, take your two values of the slope and plug them into the formula: gSinθ=[(a_1+a_2)/2]. Doing this allows us to disregard the force of friction and isolate gravity as the only force affecting the acceleration of the ballistic cart.
7. Repeat the experiment for a larger value of theta by either using a larger block of wood or upending the wood block you have (if materials allow) to elevate the aluminum track even further. Doing this assures that the angle of inclination increases.
8. Print out copies of the two graphs from the trials in order to show the set of data from the position and velocity graphs. Show the time intervals used and the slope of the two different velocity curves on the graphs.

Figure 1

Experiment Question:

1. What type of curve do you expect to see for x vs t and v vs t? Explain.

For the position curve (x vs t), a parabolic shape is the expected result because the ballistic car travels up the incline and then hits an apex at which point it descends again. Since the position function is parabolic in shape, then the velocity function must be represented by a linear function because the derivative of the position function is the velocity function.

Results:

Trial 1

Trial 2

Trial 3

Trial 4



Trial 5

Trial 6

Trials 1, 2, and 3 (as labeled in the captions of the pictures) were for the angle:

"θ=ArcSin(9.55cm/228cm)."

Trials 4, 5, and 6 (also labeled by captions) were for the angle:

"θ=ArcSin(18.65cm/228cm)."

The values of gravity for trials 1, 2, and 3 were as follows:

Trial 1: g= 8.63 m/s^2

Trial 2: g= 8.48 m/s^2

Trial 3: g= 8.52 m/s^2

The values of gravity for trials 4, 5, and 6 yielded:

Trial 4: g= 9.08 m/s^2

Trial 5: g= 9.29 m/s^2

Trial 6: g= 9.20 m/s^2

Conclusion:

Based off of the results produced by our group, the only logical conclusion that can be deduced is that the higher the angle of inclination, the more accurate the calculation of gravity will become. That is, the experimental gravity will ultimately become closer and closer to the actual quantity of gravity (g=9.80m/s^2) as the angle of inclination increases.