Physics 201



Physics 211

Lab Manual

PCC Rock Creek

Fellman

Table of Contents

General Instructions for Lab Reports…………………………………………………………………… 3

Uncertainty Analysis and Propagation………………………………………………………………… 4

Lab 1 Conversions and Measurement.………………………………………………………………. 6

.

Lab 2 Position, Velocity and Acceleration………………. ……………..…………………………. 8

Lab 3 Vectors and Two-Dimensional Motion……………………………………………….…… 10

Lab 4 Tension and Friction………………………………………………………………………………… . 11

Lab 5 Circular Motion and Centripetal Force………………………………………………….…..12

Lab 6 Conservation of Energy………..………………………………………………………………….. 13

Lab 7 Center of Mass……………….……………………………………………………………………….. 14

Lab 8 Momentum and Collisions.……………………………………………………………………..…15

Lab 9 Angular Momentum………………………………………………………………………………….. 17

Make-Up Lab Conservation of Energy in Rolling Motion………….…………………….…. 18

Sample Lab Report A………………………………………………………………………………………….. 20

Sample Lab Report B………………………………………………………………………………………….. 24

General Instructions for Lab Reports

For every lab completed, each person will be responsible for handing in a lab report. These reports will be due one week from the day the exercise is completed, at the beginning of class. Reports may be typed or handwritten as long as they are neat, easy to follow, and contain the following elements:

Purpose - What you hope to accomplish through the exercise, in your own words.

Procedure - Brief procedural notes in your own words. It is not acceptable to simply write “We followed the instructions in the lab handout.” Instead, your report should outline the steps you followed in enough detail so that it would make sense to a person who has never seen the lab instructions.

Data - All original data in a well organized format.

Uncertainties - The range of possible values associated with every measurement you take. (Ex: 3.5 cm ( .1 cm) These uncertainties are usually due to precision limitations of measuring equipment and will also propagate through calculations. (See uncertainty handout.) Remember: Every measurement we take in the lab has an uncertainty associated with it.

Calculations - Include formulas, sample calculations, and results of all calculations.

Graphs - (When appropriate) Computer printouts or graphs done by hand.

Diagrams - (When appropriate) Free body diagrams, electric circuit diagrams, etc.

Sketches of Equipment - (Always appropriate) Also include a basic diagram of the experimental setup if applicable.

Comments - Your thoughts and observations throughout the process.

Results - Clear statement of results.

Conclusion - The most important part!!! Almost anything will do as long as it shows that you thought about it. Examples: What did you learn? Did the results make sense? If not, what are the possible reasons? Relate the concept to an everyday experience. Relate it to something in the text. (Get the idea?) The conclusion should be at least a paragraph.

** Note that with the exception of the purpose (at the beginning of the report) and the conclusion (at the end), these elements should not each be contained in a separate section of the report, but instead will flow naturally throughout the report. Please see the sample lab reports at the end of this booklet **

Once again, neatness is VERY important. A well organized, easy to read report with moderately good results will receive a higher grade than a report with excellent results that are hidden in scribble.

Uncertainty Analysis and Propagation

Every measurement has a range of uncertainty associated with it. This uncertainty is usually a result of precision limitations of the instrument used to make the measurement. Any calculations done using a measurement will also have a degree of uncertainty. This is a measure of how confident you are in the result of your calculation. The propagation of the uncertainties through various calculations has to be carefully considered.

One way to proceed with the concept of uncertainty propagation is called the “Worst Case Calculation”, and is shown in the following two examples.

Example #1

Length = 93.10 cm ( .05 cm

Width = 3.540 cm ( .003 cm

Area = L ( W = (93.10 cm)(3.540 cm) = 329.6 cm2

Worst cases:

L & W both largest possible value:

Area = (93.15 cm)(3.543 cm) = 330.0 cm2 (+.4 cm2)

L & W both smallest possible value:

Area = (93.05 cm)(3.537 cm) = 329.1 cm2 (-.5 cm2)

Report final result as: 329.6 cm2 ( .5 cm2

Example #2

Mass = 32.32 g ( .01 g

Volume = 18.8 cm3 ( .1 cm3

Density = M / V = 32.32 g / 18.8 cm3 = 1.72 g/cm3

Worst cases:

Largest M / Smallest V:

Density = (32.33 g) / (18.7 cm3) = 1.73 g/cm3 (+.01 g/cm3 )

Smallest M / Largest V:

Density = (32.31 g) / (18.9 cm3) = 1.71 g/cm3 (-.01 g/cm3 )

Report final result as: = 1.72 g/cm3 ( .01 g/cm3

On the other hand, when your data consists of a number of measurements to be averaged for a final result, the uncertainty can be reported as the standard deviation.

Example #3

You are measuring the range of a projectile. The experiment has been repeated 5 times, yielding 5 different distances: 3.41 m, 3.69 m, 3.33 m, 3.57 m, and 3.50 m.

First find the average:

_

x = ( xi = (3.41+3.69+3.33+3.57+3.50)m = 3.50 m

n 5

Now find the standard deviation:

_

st. dev. = ( (xi - x )2 = .078 m2 = .14 m

(n-1) 4

_ _

x xi (xi - x ) (xi - x )2

3.50 m 3.41 m - .09 m .0081 m2

3.50 m 3.69 m +.19 m .0361 m2

3.50 m 3.33 m - .17 m .0289 m2

3.50 m 3.57 m +.07 m .0049 m2

3.50 m 3.50 m .00 m .0000 m2

.078 m2

(The standard deviation may also be obtained by using your calculator’s standard deviation function.)

Report final result as: 3.50 m ( .14 m

*******************************************************************

Notice the appropriate use of significant figures and decimal places in the three examples above:

*Each final result has no more significant figures than any measurement involved in the calculation.

*The smallest decimal place of the uncertainty matches the smallest decimal place of the measurement or result.

Lab 1 Conversions & Measurement

The purpose of this exercise is to familiarize the beginning physics student with unit conversions, measurement techniques, and measuring devices.

Part A Conversions

1. Measure the height of one of your lab partners in inches. Convert this value to meters, centimeters, millimeters, and kilometers. (Don’t forget to report and convert the uncertainty in the measurement as well.)

2. Convert the speed limit of 55 mph to km/h. (No, you can’t run out to your car to check your speedometer.)

3. We will learn that the speed of light in a vacuum is 3.0 x 108 m/s. How many miles per hour is this?

4. Calculate the number of seconds that you have been alive.

5. A physics textbook lists the age of the universe as 5.0 x 1017 s. How many years is this?

Part B Measurement

Use the most accurate method possible for each step. Don’t forget about the uncertainty of the measurements in each case.

6. Measure the thickness of a page in the textbook. How certain are you of this measurement?

6. Measure the thickness of 10 pages. Is this measurement ten times as great as the thickness of one page? Does the ink have a measurable thickness?

7. Measure the length and diameter of a cylinder. What is the volume of the cylinder?

8. Measure the mass of the cylinder. Density is the amount of mass per unit volume. What is the density of the cylinder?

9. Measure the mass of a grain of rice.

10. Measure the mass of 10 grains of rice. Are 10 grains of rice 10 times the mass of one grain of rice? Do the measurements agree within the range of uncertainty? How would you go about finding the density of rice?

11. Measure the length of your table.

12. Measure the time it takes for one of your lab partners to walk the length of the table. Speed is the distance traveled divided by the time it took to travel that distance. What was the average speed of your lab partner in meters per second?

13. Convert this speed to miles per hour. Is this a reasonable value?

In this section you will gain experience with the Science Workshop program and various data taking devices.

15. Set up the air track and air cart, and observe the motion of the cart when the air is turned on. What advantage do you think this apparatus will provide in our studies of motion?

16. Set up the photogate to measure the velocity of the air cart. Make sure that you enter the correct length of the object passing through the photogate. Experiment with the range of velocities you can achieve. Record three velocities: one slow, one medium, and one fast.

15. Why must you enter the length of the object that passes through the photogate? What other information does the photogate utilize to calculate velocity?

16. Now hook up the smart pulley. By spinning it with your finger, observe how it can record position and/or velocity (or even acceleration which we will get to in a few days). Comment on how position and velocity values change as you spin it at different rates.

17. Observe and record (with drawings) all of the ways in which data can be displayed using this program. Practice printing out a graph.

18. Relying upon the vast experience you have gained using the Science Workshop program today, write an outline of the general procedure that must be followed to take data using this program. Begin with turning the computer on, and list the steps that would help a person doing this for the first time.

Lab 2 Position, Velocity & Acceleration

Part A Position, Velocity, and Acceleration Graphs

When using the motion sensor, begin by adjusting the trigger rate to 5 Hz in order to obtain smooth, readable graphs. Also, become familiar with the procedure for displaying only the desired portion of your graph on the optimal scale.

1. Set up the motion sensor to graph position. When it is recording position, it acts as the origin, with all distances in front of it being positive. Before you start recording any data, sketch four position vs. time graphs that you would expect to obtain for the following situations:

a) standing still

b) walking at a constant slow speed away from the detector

c) walking at a constant fast speed away from the detector

d) walking at a constant slow speed toward the detector

2. Now create position graphs using the motion sensor for each of the four cases. Do they match your predictions? If not, describe how you would move to make a graph that looks like your prediction.

2. Now set up the motion sensor to graph velocity. Before you start recording any data, sketch four velocity vs. time graphs that you would expect to obtain for the previous situations, in addition to four new ones:

e) walking at an increasing speed away from the detector

e) walking at an increasing speed toward the detector

f) walking at a decreasing speed away from the detector

g) walking at a decreasing speed toward the detector

4. Now create velocity graphs using the motion sensor for each of the eight cases. Do they match your predictions? If not, do you now understand why?

5. Sketch constant acceleration graphs for all eight situations (most importantly, decide for which situations the acceleration would be positive and for which it would be negative). Verify the sign of the acceleration in each case using the motion sensor.

Note: please include your predictions in your report as well as representations of the computer graphs. I may be a little skeptical if anyone managed to predict everything correctly.

Part B Reaction Time

6. Have one of your lab partners hold a meter stick at one end, letting it hang vertically downward. Now have another person in your group hold their thumb and index finger approximately 2 inches apart at about the midpoint of the meter stick. When the person holding the meter stick unexpectedly drops it, record the distance the meter stick falls before the other person catches it.

6. Use this distance along with the accepted value for gravitational acceleration (g) to calculate the reaction time.

7. Repeat the process so that everyone in your group gets to measure their reaction time. If you want to, write your name and reaction time on the board so that we can see who is the fastest!

Part C Acceleration Due To Gravity

In the previous section, we used the accepted value for gravitational acceleration at the earth’s surface. In this section, we will pretend that we don’t know its value, and so we are going to do an experiment to determine it.

9. Set up the photogate to measure velocity. As you drop the picket fence through the photogate, the computer will record the velocity of each solid panel as it passes the sensor. To arrive at a value for the acceleration, obtain a graph of velocity vs. time, and read the average slope of the graph.

10. Calculate the percentage error between this value and the accepted value in your textbook.

11. Would dropping a solid colored panel have worked? What information from the experiment did the software utilize to create the graph?

12. Give a detailed description of how you would calculate the acceleration by hand using this information.

Lab 3 Vectors and Two-Dimensional Motion

Part A Vector Addition

The purpose of this section is to explore three different methods of adding vectors: geometrically, adding by components, and using a force table. The first two methods are explained in your text. The force table method involves masses on strings that create tension vectors. The value of each mass is proportional to the magnitude of the tension vector, and the angle of the vector is simply read off the edge of the force table. Today we will add vectors using all three methods, to show that the results are the same.

1. First, make sure the force table is level. With the center pin of the force table in place, set up two tension vectors with an angle between them of 135( . One should have a mass of 250 grams, and the other a mass of 450 grams. Find the tension vector (magnitude and direction) that will balance the system so that the pin can be removed without affecting the center ring. When the force table is balanced in this way, the three vectors add to zero (i.e. the resultant is equal to zero). You have now completed one method of vector addition.

2. Now add the same three vectors geometrically. Remember that your vectors must be drawn to scale. Also, your graph should fill most of a page in order to achieve the best results.

3. Finally, add the same three vectors by components. Your answers from all three methods should agree within the range of uncertainties.

Part B Projectiles

The purpose of this section is to explore an actual example of two-dimensional motion. CAUTION!!!! These projectiles can be dangerous! Make sure that the range is clear before loading the apparatus. Also, please use only the plastic balls unless specifically asked to do otherwise. There have been complaints from our downstairs neighbors in the past.

4. Use a piece of paper and carbon paper to measure the average range of the ballistic launcher aimed in the horizontal direction. There are three different launch positions, make sure that you are using the same one each time. Get at least five data points, average them, and find the standard deviation.

4. What other information do you need to calculate the initial velocity? Record this data as well.

5. In treating this as a textbook problem, what factor(s) do we ignore in our calculation?

6. Calculate the initial velocity of the projectile. Don’t forget about uncertainties. Verify your result with me before proceeding.

7. Calculate the initial velocities for the other two launch positions in the same manner. Again, check your results with me.

8. Predict which one will go farther, the steel ball or the plastic ball. Test your prediction (only once please!). Was once enough to tell the difference? Speculate on the reason for the difference in the range of the two balls.

9. By varying the launch angle, find the angle that launches the plastic ball the farthest.

10. What angle is predicted by theory? Why does this differ from the actual result that you found?

Lab 4 Tension and Friction

Part A Tension and Weight

1. First make sure the air track is level. Use the air track, smart-pulley, and computer interface to measure the acceleration of the air cart pulled by the weight of three different masses:

m = 20 g, 30 g, and 40 g .

2. Sketch a generalized free body diagram of the hanging mass. Remember: free body diagrams are sketches of ONE body and ALL forces acting on THAT body.

2. Apply Newton’s 2nd law to the hanging mass. Use the resulting equation to solve for the tension in each of the three trials.

3. Now sketch a generalized free body diagram of the cart. Where is the normal force coming from in this case?

4. Apply Newton’s 2nd law to the horizontal motion of the cart. Use the resulting equation to solve for the mass (M) of the cart for each of the three trials. Average these results and find the standard deviation.

5. Now measure the actual mass of the cart. How do the measured and calculated values compare?

6. Now, without actually setting it up, calculate the acceleration you would expect if the air cart were pulled by a mass of 25 g. Need a hint? Consider the equations from steps 3 and 5. These were results of applying Newton’s second law to the motion of both objects. Combine these two equations and two unknowns to arrive at a formula for the acceleration.

7. Now, verify your answer by actually trying it out.

Part B Tilted Air Track

9. Tilt the air track using wood blocks so that the pulley end of the track is highest.

10. Hang just enough mass from the string to hold the cart near the center of the inclined plane without motion (in equilibrium). What is the sum of the forces on the cart at this point?

11. Give the cart an initial velocity by gently pushing it down the track, and measure its acceleration using the computer interface. Before the brief push, what was the sum of the forces on the cart? After the brief push, what is the sum of the forces on the cart? What should the acceleration be? (Hint: This question requires no calculation.)

12. Sketch a free body diagram of the cart.

Part C Friction

13. Place the friction block flat on your table, and slowly increase the tension in the string until the block just begins to slide. At this point, the tension in the string is equal to fs,max, the maximum static frictional force. Use this fact to calculate the static coefficient of friction, (s. Draw a free body diagram of the block.

14. Again, place the friction block flat on your table. This time, carefully find the tension in the string that causes the block to slide along the table at constant velocity after a brief push. At this point, the tension in the string is equal to fk, the kinetic frictional force. Use this fact to calculate the kinetic coefficient of friction, (k. Draw a free body diagram of the block.

Be sure to include ALL free body diagrams in your report.

Lab 5 Circular motion, Centripetal Force & Springs

1. First set up and level the rotational apparatus. Attach the rotational drive motor, and notice how the speed of rotation can be controlled using this motor.

2. While the platter is spinning at a constant rate, measure the time it takes to make ten complete rotations. Divide this time by ten to obtain the period of motion.

3. In order to calculate the linear speed of a point on the edge of the platter, what other measurement do you need? Take the appropriate data, and calculate this speed.

4. Now, verify that you have calculated this correctly by using the smart-pulley to measure the linear speed of the edge of the platter directly. Do this by holding the rim of the smart-pulley in contact with the lower edge of the platter.

5. Now attach the centripetal force accessory to the system. Tie a string to the cart, run the string over the pulley and down through the hole in the center. Attach the weight hanger to the hanging end of the string. The weights attached here will provide the centripetal force for circular motion.

Cautionary Statements: When you attach weights to the trolley, make sure that they are taped down very securely!! (Otherwise things will be flying all over the room!) As a further precaution, do not position yourself so that your face is in the plane of the rotation at any time. Finally, please do not exceed a reasonable speed of rotation.

6. Begin with roughly 20 g on the weight hanger, and 30 g on the trolley. Slowly increase the speed of rotation until the trolley holds steady at the radius of the platter. The best way to find this speed is to hold the hanging weights lightly in your hand with the trolley at the desired radius as the speed is increasing, and eventually release the weights when they cease to feel heavy (but don’t let them fly up too fast either). Also, make sure that the hanging weights do not swing back and forth.

6. Record the range of speed that results in the most stable equilibrium.

7. Using the appropriate parameters from your system, verify the fact that centripetal force = mv2/r (within the range of uncertainty, of course).

8. If you quadruple the total mass of the trolley, by what factor will the equilibrium speed change? Verify your answer.

9. In part 9, would there have been another way to reach equilibrium without changing the speed?

10. Set up another combination of masses, calculate the speed required for equilibrium, and test your result.

11. Discuss the interrelationships between centripetal force, mass, speed and radius in relation to the behavior of this particular system (i.e. what happens when you increase or decrease their values?)

12. With the setup we have, is it possible to take measurements for a different radius as well? Why or why not?

The Force Applied by a Spring

14. Use whatever method you choose to make an informative investigation of the force applied by a spring to other objects. In other words, how does the force change with the extension of the spring?

14. Make a graph of the force exerted by the spring vs. the distance the spring was extended. Don’t forget about uncertainties. Does your data appear to fall in a straight line? What does this imply about the behavior of a spring? (How are force and extension related?)

15. Find the slope of the line. What do we call this number?

Lab 6 Conservation of Energy

Part A Gravitational Potential Energy

1. Drop a racquetball from the height of one of the counters, and measure the height of the bounce. Do this at least five times to get an average of the bounce height.

2. What is the initial potential energy of the ball? What is the potential energy at the height of the first bounce?

3. How much mechanical energy was lost? Identify two ways in which mechanical energy could have been dissipated.

Part B Conservation of Energy in Vertical Motion

In this section, we will be using the ballistic launcher again. However, it will be set up to launch vertically, and we will be using superballs to cut down on the noise.

4. Set up the photogate so that it measures the velocity of the ball just as it leaves the launcher. Using the first launch position, record the vertical launch velocity of the superball. (Do it several times to get an average.) How much kinetic energy does the ball have when it is launched?

4. How much kinetic energy does the ball have as it reaches the top of its trajectory?

5. What must be the increase in potential energy between these two points?

6. Now measure the height difference between these two points (average several trials), and calculate the corresponding change in the potential energy of the superball. Does this value agree with your answer for the previous step?

7. Repeat this process for the second and third launch positions. This time, calculate the expected height before you actually measure it.

8. Do you need to know the mass of the ball to calculate the height?

9. How did the calculated and measured heights compare in each case? Was there a trend in your results? Give a possible explanation for any discrepancy.

Part C Conservation of Energy on an Incline

11. Set up the air track with one end tilted up. Compress the bumper of the air cart against the bottom end of the track, and then release it quickly so that it travels a good distance up the track before starting back down.

11. Use the photogate to measure the velocity of the cart shortly after it is released, and also record the highest position reached by the cart.

12. How much kinetic energy does the cart have as it is passing through the photogate?

13. How much kinetic energy does the cart have at its highest position on the track?

14. What must be the increase in potential energy between these two points?

15. Think of a way to measure the difference in potential energy between these two points. Do the values agree within the range of uncertainty?

Part D Elastic Potential Energy

17. In part C, when the cart was initially held compressed against the end of the air track, what type of energy did the system possess? In part B, when the ball was sitting in the ballistic launcher, ready to be launched, what type of energy did the system possess?

17. Each time the ball left the launcher in part B, what did the initial elastic potential energy become? Consider again your data from part B. Use this data along with some additional measurements to calculate the spring constant for the spring in the ballistic launcher. You should end up with three values to be averaged for a final result.

Lab 7 Center of Mass

Part A Meter Stick

1. Place the meter stick on the balancing stand in order to locate its center of mass. Record this value. Recall that the center of mass is that point of a rigid body where all of its mass can be considered to act.

2. Without actually setting it up, suppose that a 50 g mass were suspended from the meter stick at the 95 cm mark, and calculate the new center of mass of the system.

3. Now test your calculation by placing the stand at the center of mass position you just calculated. Does it balance? If not, how far from the true balance point was it? Your results should definitely fall within the range of experimental uncertainty. If they don’t, find out why and correct your setup.

4. Add an additional hanging weight of your choice to the system at a position of your choice. Find the new center of mass by calculation and experiment, and compare them. Once again, the results should agree.

5. Until this point, you have most likely been calculating the center of mass by measuring all distances from the 0 cm end of the meter stick. Interestingly, distances can be measured from any position on the meter stick. Try doing the calculation from part 4, but this time measuring all distances from some arbitrary position on the meter stick. (Note that some distances will now be negative.)

6. Do the calculation one more time, measuring all distances from the original center of mass of the meter stick found in part 1. Why is this the easiest way to do the calculation?

Part B People

7. Stand with your heels and back against a wall and try to bend over and touch your toes. Explain why this doesn’t work.

7. Measure the minimum distance of your heels from the wall for which you can touch your toes. Compare this distance with others in the room. On average, do men or women need to stand farther from the wall? What does this imply about our centers of mass?

Lab 8 Momentum and Collisions

Part A Estimation and Calculation of Momentum

An essential part of science is the investigator’s ability to estimate quantities without precise measurement. One important aspect of this ability is used when an experimenter designs an experiment to investigate some physical phenomenon. In everyday life this skill is also quite valuable, so we will be practicing the art of estimation while determining the momentum of various objects that are in motion at different speeds. There is no right or wrong answer, but the use of logic and reasonable assumptions in the estimation process is important for your result to be close to the answer you would obtain from an exact measurement. For the following, estimate both the mass and speed, and report the resulting momentum in scientific notation.

a) a bug crawling across the sidewalk

b) a butterfly flying past you

c) a bowling ball rolling down the lane

d) a fastball thrown by a major league pitcher

e) an average bicyclist at medium speed

f) a car at highway speed

g) a 747 at cruising speed

h) a supertanker cruising from the Middle East full of oil

Part B Momentum and Kinetic Energy in Collisions

The first objective of this section is to qualitatively observe several types of collisions. We will then verify quantitatively the equations for velocities resulting from various types of collisions. We will be considering both elastic and inelastic collisions between aircarts having various initial velocities and masses.

1. Qualitative Observation

For each of the following collisions between roughly equal masses, predict the outcome by drawing a diagram of the carts both before and after the collision, using vectors to illustrate their velocities in each case. Then test your predictions by observing each actual collision. Make sure the air track is level.

Elastic collisions mass 1 mass 2

a) velocities: -----( 0

b) velocities: -----( (-----

c) velocities: -----( -(

Inelastic collisions

d) velocities: -----( 0

d) velocities: -----( (-----

e) velocities: -----( -(

Now repeat several of the above collisions, this time giving the carts different masses by adding weights. Make sure you indicate the mass differences on your diagrams. Don’t forget to draw your predictions first, then test your predictions by observation.

2. Quantitative Measurement

You will be setting up each of the following collisions, adding weight to the air carts when necessary. For each one, use the photogates to measure the velocities of each cart both before and after the collision. Remember that the photogates do not recognize direction, so you must include the appropriate signs for velocity. Then, for each collision, use the appropriate equations from your text to calculate the expected final velocities, and compare these to your experimental values.

a) A moving aircart collides elastically with another initially at rest, and having twice as much mass.

b) A moving aircart collides elastically with another initially at rest, and having half as much mass.

c) The same situation as part a), but this time a completely inelastic collision.

d) The same situation as part b), but this time a completely inelastic collision.

For which of these four collisions would we expect kinetic energy to be conserved?

For which of these four collisions would we expect momentum to be conserved?

Lab 9 Angular Momentum

1. The purpose of this first part is to experience the effects of a changing moment of inertia on angular velocity. Hold one steel disk in each hand, and sit on a rotating stool. Start to spin with your arms extended, them bring your arms in toward your chest and notice the difference in your angular velocity. Describe the effect, and explain the reasoning behind it.

2. This part is fun - everyone should try it! Sitting on a rotating stool again, take the bicycle wheel and get it spinning as fast as you can, holding it so that the momentum vector is pointing straight up. The stool should stay still when you lift your feet up. Now, keeping your feet up off the ground, turn the wheel (as quickly as you can) so that the momentum vector is pointing straight down. Explain in words and pictures (complete with momentum vectors) what happens in terms of conservation of angular momentum.

3. For the rest of this exercise, we will be using the centripetal force apparatus without the track or trolley. Make sure the platter is level. Using the rotating platter alone, apply a torque to it by wrapping string around one of the center spools, then running the string over the smart pulley and hanging weights (50 grams or less) from the loose end. Measure the tangential acceleration using the computer interface. Use this tangential acceleration and the radius of the spool to calculate the angular acceleration of the system.

4. Calculate the torque that is applied to the disk by the force of the hanging weights: torque = (force) x (lever arm), where the lever arm is the radius of the spool around which the string was wound.

5. Using the angular acceleration found in part 3 and the torque found in part 4, calculate the rotational inertia of the platter using Newton’s 2nd law for rotational motion. (Of course, you would never forget about uncertainty.)

6. Now, remove the metal ring from the box. Using the appropriate formula from your text, calculate the rotational inertia of the ring.

7. Recognizing the fact that moments of inertia are additive, calculate the total rotational inertia that would result from adding the ring to your system.

8. Using Newton’s 2nd law for rotational motion, calculate the angular acceleration that should result from adding the ring to your system. Convert this into a tangential acceleration that can be measured with the smart pulley.

9. Now add the ring to your system, measure the resulting acceleration, and compare it to the value that you just calculated.

Make-Up Lab: Conservation of Energy in Rolling Motion

1. In order to observe rolling motion we need an inclined surface. Raise one end of your table by putting three stacking blocks under each leg at that end. Now measure the time it takes for the solid ball to roll the entire length of the table, starting from rest. This timing can be tricky, but a good method will yield surprisingly accurate results. Make sure that the ball rolls nearly straight down the table, and that no bumping or slipping occurs. Most importantly, average many different trials with various people operating the stopwatch. The standard deviation will be your uncertainty.

2. We are interested in the translational speed of the ball at the bottom of the table. Using the fact that it experienced constant linear acceleration as it rolled, and taking into consideration the length of your table, calculate its speed at the bottom. (Hint: think back a few chapters.)

3. Now we will verify that mechanical energy was conserved. The two points at which we will consider the energy of the ball are the top of the table and the bottom.

At the top: What is its gravitational potential energy?

What is its translational kinetic energy?

What is its rotational kinetic energy?

What is the sum of the mechanical energy at the top?

At the bottom: What is its gravitational potential energy?

What is its translational kinetic energy?

What is its rotational kinetic energy?

What is the sum of the mechanical energy at the bottom?

Your results should reflect that fact that mechanical energy was conserved (within the range of uncertainty).

4. We could have used conservation of energy to derive an expression for the speed of the ball at the bottom of the incline, and then simply verified this speed. See the board for this example. Using this as a guide, derive an expression for the speed of the hoop (tin can) at the bottom of the incline. Note that the moment of inertia is different for a hoop, and therefore the velocity expression will be slightly different also. Check your result with me before proceeding

5. Now measure the time required for the hoop to roll the length of the table in exactly the same manner used for the ball. Use this time to determine the speed at the bottom, and compare it with the speed predicted in step 4.

6. Repeat step 4 (derive expressions for final velocity) for the other two objects that have been provided. You don’t need to test these with the stopwatch, but do check your answers with me.

7. You now have expressions for the final speeds of each of your four objects. Summarize these in a table, and refer to it in answering the following questions:

Do any of the final speeds depend on the mass of the object?

Do any of them depend on the radius?

List their final speeds in order of magnitude, greatest first.

In a four-way race down the incline, list the order in which they will finish.

8. Since racing all four at the same time proves nearly impossible (although you may try if you like), race them two at a time to verify their order of finishing.

Sample Lab Report A:

Lab 4: Tension and Newton’s 2nd Law

Purpose: The purpose of this lab is to study the effect of tension in different situations. First we will consider the tension caused by a hanging mass that is connected to another mass resting on a flat surface. Since the two masses are connected using a string of negligible mass running over a frictionless pulley, and one of the masses is resting on a frictionless surface, we will be able to neglect friction. During the second part, we will again neglect friction, but this time the hanging mass is connected to a mass resting on an inclined plane. We will attempt to find the sum of the forces acting on the cart which is on the inclined plane.

Procedure, Data, Calculations, Diagrams:

Part A: Tension and Weight

1. Using the computer interface, we measured the acceleration of the air cart pulled by three different hanging masses. The air cart represented in the drawing by m1, is resting on a frictionless surface and is connected to the hanging mass m2, by means of a frictionless pulley. The pulley is also interfaced with the computer and will be used to measure the acceleration of m1. We measured the acceleration for three different hanging masses (m2) of 20, 40, and 60 grams. The following table is the result of these tests:

2. Here is a free body diagram for our hanging mass, m2:

The only forces acting on the hanging mass are gravity, which is pulling the hanging mass toward the ground, and the tension of the string which is pulling the mass upward. The tension in the string is present because of the mass it is attached to on the flat, frictionless plane.

3. Newton’s second law tells us that the sum of the forces acting on an object in any one direction is equal to its mass multiplied by its acceleration in that direction. Since both forces acting on the hanging block are along the vertical axis, we can apply Newton’s second law to it as follows:

(sum of forces down) = (mass)(acceleration down)

(m2g-T) = (m2)(a)

Then, we can rearrange it to solve for the tension:

T = m2g – m2a = m2(g-a)

Since we used three different masses which resulted in three different accelerations, we have to solve for three different values of T:

T(20g) = (.020 kg)[(9.8m/s2) – (.795 m/s2)] = .180N

T(40g) = (.040 kg)[(9.8m/s2) – (1.447m/s2)] = .334N

T(60g) = (.060 kg)[(9.8m/s2) – (2.006m/s2)] = .468 N

4. Next we will draw a free body diagram for the air cart:

In this case, the cart is resting on a cushion of air,

Thus the air cushion is where the normal force

is coming from. Since there is no motion in the

vertical direction, it is safe for us to assume that

the normal force, FN, is equal to and in the

opposite direction as gravity. Therefore these

two forces cancel each other out. At the same time,

since there is no friction, any amount of force acting

in the horizontal direction will produce an acceleration

for the cart.

5. Applying Newton’s second law to the cart tells us that the sum of the forces acting in the horizontal direction will be equal to the cart’s mass times the cart’s acceleration in the horizontal direction. Since the only force acting in the horizontal direction is the tension, T, of the string, we can set up a relationship between T, m1 and a:

(sum of the forces in the x direction) = T = (m1)(a)

← m1 = T/a

Now we are ready to predict a value for m1 for each of our three trial cases:

Averaging these results for m1, we get a value of 230 g.

6. We measured the actual mass of the cart on the digital scale to be

221 g, so our average was only off by about 4%.

Part B: Tilted Air Track

7. Now we will tilt the air track so that the pulley end of the track is highest:

8. Since the cart is no longer on a level plane, gravity will now have some effect in pulling the air cart towards the ground, or the lower end of the air track. Therefore, to balance the cart we must apply a force in the opposite direction, which can be accomplished by hanging a mass from the other end. We had to hang 15 g on the string for the cart to remain in equilibrium. At this point, since there is no motion in any direction, the sum of the forces acting on the cart is zero.

9. Now we will give the cart an initial velocity by gently pushing it down the track. Since the sum of the forces is zero before the push, it must also be zero after the push, and therefore the acceleration should be zero also. Using the computer interface, we measured an acceleration for the cart of .004 m/s2, which is very close to zero. We would also expect the cart to have zero acceleration (constant velocity) after the push in consideration of Newton’s first law, which states that an object in motion will continue in straight line motion if no force acts.

10. Now we will sketch a free body diagram for the cart. I have included the original direction of the force due to gravity, but only with a dotted line as it is resolved into the directions of the tilted x and y axes for our calculations.

Conclusion:

Looking at the two separate parts of this lab, I find a common link: Newton’s second law of motion, which tells us that the force acting on an object in any direction is equal to the object’s mass times the object’s acceleration in that direction. Sure, it sounds pretty straightforward, but as this lab has shown, we run into different kinds of obstacles when applying this law. So what I have learned by doing this lab is that in applying Newton’s second law, we must also consider the angle at which the object is moving, as well as any kinds of frictional forces that may be resisting the object’s motion, not to mention all the uncertainties involved in the motion of everyday objects that we have not even considered in this lab.

Sample Lab Report B:

Introduction to Circuits

Purpose: To explore the relationship between voltage, resistance and current in a circuit.

Procedures & Comments:

1. Using a multimeter, measure and record the voltage of two different batteries.

| |Battery 1 |Battery 2 |Battery Pak (1&2 combined) |

|Voltages |1.502 V |1.513 V |3.010 V |

If the leads are reversed when measuring voltage, the multimeter will display the same magnitude but will indicate that the voltage is negative.

2. A. From their color bands, determine and record the resistance and the uncertainty rating for five different resistors.

B. Measure and record the resistance with the multimeter.

Resistor Color Guide

|Band Color | |Band Color | |

|Black |0 |Green |5 |

|Brown |1 |Blue |6 |

|Red |2 |Purple |7 |

|Orange |3 |Gray |8 |

|Yellow |4 |White |9 |

Uncertainty: Silver = (10% ; Gold = ( 5.0%

Resistor: 1. Red, Purple, Black, Silver

2. Red, Red, Yellow, Silver

3. Red, Purple, Red, Silver

4. Orange, Orange, Red, Gold

5. Brown, Green, Yellow, Silver

|Resistor |Determined Resistance, ( |Determined Uncertainty, %|Measured Resistance, ( |Actual Difference, % |

|1 |27X10^0 |10 |30.0X10^0 |10.0 |

|2 |22X10^4 |10 |23.2X10^4 |5.17 |

|3 |27X10^2 |10 |29.0X10^2 |6.90 |

|4 |33X10^2 |5 |33.2X10^2 |0.60 |

|5 |15X10^4 |10 |15.2X10^4 |1.32 |

All of the measured values fall within the range given by the color bands.

3. Using two resistors of the same order or magnitude (the third band of the same color), connect them in series and measure and record the resistance of the combination. Repeat twice again with different resistors.

Equivalent Resistance = R1 + R2

|Resistors |Measured Resistance, ( |Calculated Resistance, ( |

|3 & 4 |6.23X10^3 |6.22X10^3 |

|2 & 5 |3.83X10^5 |3.84X10^5 |

|4 & 5 |1.55X10^5 |1.55X10^5 |

4. Using two resistors of the same order or magnitude (the third band of the same color), connect them in parallel and measure and record the resistance of the combination. Repeat twice again with different resistors.

1 / Equivalent Resistance = 1 / R1 + 1 / R2

|Resistors |Measured Resistance, ( |Calculated Resistance, ( |

|3 & 4 |1.55X10^3 |1.55X10^3 |

|2 & 5 |9.18X10^4 |9.18X10^4 |

|4 & 5 |3.25X10^3 |3.25X10^3 |

5. Rules for adding Equivalent Resistances and Capacitances in Series and Parallel Circuits:

Series Parallel

Resistance: R=R1+R2+R3+… 1/R=1/R1+1/R2+1/R3+…

Capacitance:1/C=1/C1+1/C2+1/C3+… C=C1+C2+C3+…

6. A series circuit acts like a one-lane highway and therefore all of the traffic must pass along the same path, which resists and slows down flow.

A parallel circuit acts like a freeway and therefore the traffic

can always take the path of least resistance, which allows it

to flow freely.

7. Construct a circuit consisting of a battery and a resistor and measure the current.

Resistor 3

| |Voltage, V |Resistance, ( |Current, A |

|Measured |3.00 V |2.90X10^3 |1.03X10^-3 |

|Calculated |V / R = I |1.03X10^-3 |

Measured and calculated answers agree.

8. Construct a circuit consisting of a battery and two resistors, with roughly the same order of magnitude, in series and measure the current.

Resistors 3 & 4

|Measured Across |Battery |Resistor 3 |Resistor 4 |

|Current, A |4.80X10^-4 |4.80X10^-4 |4.80X10^-4 |

|Voltage, V |3.00 |1.40 |1.60 |

In series the current must be the same through all elements. The voltage measured across resistor 3 and 4 equaled 1.40 V and 1.60 V respectively. The sum of these two voltages equals 3.00 V, which is the same as the voltage of the battery. In series the sum of the voltages through all of the elements must equal the power source.

9. Construct a circuit consisting of a battery and the two resistors used in procedure 8 in parallel and measure the current through the battery, the first resistor and the second resistor. Then measure the voltage across each resistor.

Resistors 3 & 4

|Measured Across |Battery |Resistor 1 |Resistor 2 |

|Current, A |1.93X10^-3 |1.03X10^-3 |9.00X10^-4 |

|Voltage, V |2.99 |2.99 |2.99 |

In parallel the voltage must be the same through all elements. The current measured across resistor 3 and 4 equaled 1.03X10^-3 A and 9.00X10^-4 A respectively. The sum of these two currents equals 1.93X10^-3 A, which is the same as the current through the battery. In parallel the sum of the currents through all of the elements must equal to the current through the power source.

Conclusion: The resistance of a resistor can be determined easily by deciphering its color bands. These determined resistances, although not perfect, are quite accurate. Using the multimeter to measure voltage, resistance and current is straight forward, but one must take special consideration when measuring current not to damage to meter. For series circuits, the equivalent resistance is equal to the sum of the resistances of the resistors; and the reciprocal of the equivalent capacitance is equal to the sums of the reciprocals of the capacitances of the capacitors. And for parallel circuits just the reverse is true. The reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the resistances of the resistors; and the equivalent capacitance is equal to the sum of the capacitances of the capcitors. In a series circuit the current is the same through all of the elements in the circuit; and the sum of the voltages through the elements in the circuit are equal to the voltage of the power source.

And for parallel circuits, again just the reverse is true. The voltage is the same through all of the elements in a circuit; and the sum of the currents through the elements in the circuit are equal to the current through the power source. Whenever possible all circuit lab work should be done on the breadboard. It is easier to handle and is a much more efficient way to do experimentation.

Sample Lab Report A:

Lab 4: Tension and Newton’s 2nd Law

Purpose: The purpose of this lab is to study the effect of tension in different situations. First we will consider the tension caused by a hanging mass that is connected to another mass resting on a flat surface. Since the two masses are connected using a string of negligible mass running over a frictionless pulley, and one of the masses is resting on a frictionless surface, we will be able to neglect friction. During the second part, we will again neglect friction, but this time the hanging mass is connected to a mass resting on an inclined plane. We will attempt to find the sum of the forces acting on the cart which is on the inclined plane.

Procedure, Data, Calculations, Diagrams:

Part A: Tension and Weight

6. Using the computer interface, we measured the acceleration of the air cart pulled by three different hanging masses. The air cart represented in the drawing by m1, is resting on a frictionless surface and is connected to the hanging mass m2, by means of a frictionless pulley. The pulley is also interfaced with the computer and will be used to measure the acceleration of m1. We measured the acceleration for three different hanging masses (m2) of 20, 40, and 60 grams. The following table is the result of these tests:

7. Here is a free body diagram for our hanging mass, m2:

The only forces acting on the hanging mass are gravity, which is pulling the hanging mass toward the ground, and the tension of the string which is pulling the mass upward. The tension in the string is present because of the mass it is attached to on the flat, frictionless plane.

8. Newton’s second law tells us that the sum of the forces acting on an object in any one direction is equal to its mass multiplied by its acceleration in that direction. Since both forces acting on the hanging block are along the vertical axis, we can apply Newton’s second law to it as follows:

(sum of forces down) = (mass)(acceleration down)

(m2g-T) = (m2)(a)

Then, we can rearrange it to solve for the tension:

T = m2g – m2a = m2(g-a)

Since we used three different masses which resulted in three different accelerations, we have to solve for three different values of T:

T(20g) = (.020 kg)[(9.8m/s2) – (.795 m/s2)] = .180N

T(40g) = (.040 kg)[(9.8m/s2) – (1.447m/s2)] = .334N

T(60g) = (.060 kg)[(9.8m/s2) – (2.006m/s2)] = .468 N

9. Next we will draw a free body diagram for the air cart:

In this case, the cart is resting on a cushion of air,

Thus the air cushion is where the normal force

is coming from. Since there is no motion in the

vertical direction, it is safe for us to assume that

the normal force, FN, is equal to and in the

opposite direction as gravity. Therefore these

two forces cancel each other out. At the same time,

since there is no friction, any amount of force acting

in the horizontal direction will produce an acceleration

for the cart.

10. Applying Newton’s second law to the cart tells us that the sum of the forces acting in the horizontal direction will be equal to the cart’s mass times the cart’s acceleration in the horizontal direction. Since the only force acting in the horizontal direction is the tension, T, of the string, we can set up a relationship between T, m1 and a:

(sum of the forces in the x direction) = T = (m1)(a)

← m1 = T/a

Now we are ready to predict a value for m1 for each of our three trial cases:

Averaging these results for m1, we get a value of 230 g.

6. We measured the actual mass of the cart on the digital scale to be

221 g, so our average was only off by about 4%.

Part B: Tilted Air Track

7. Now we will tilt the air track so that the pulley end of the track is highest:

11. Since the cart is no longer on a level plane, gravity will now have some effect in pulling the air cart towards the ground, or the lower end of the air track. Therefore, to balance the cart we must apply a force in the opposite direction, which can be accomplished by hanging a mass from the other end. We had to hang 15 g on the string for the cart to remain in equilibrium. At this point, since there is no motion in any direction, the sum of the forces acting on the cart is zero.

12. Now we will give the cart an initial velocity by gently pushing it down the track. Since the sum of the forces is zero before the push, it must also be zero after the push, and therefore the acceleration should be zero also. Using the computer interface, we measured an acceleration for the cart of .004 m/s2, which is very close to zero. We would also expect the cart to have zero acceleration (constant velocity) after the push in consideration of Newton’s first law, which states that an object in motion will continue in straight line motion if no force acts.

13. Now we will sketch a free body diagram for the cart. I have included the original direction of the force due to gravity, but only with a dotted line as it is resolved into the directions of the tilted x and y axes for our calculations.

Conclusion:

Looking at the two separate parts of this lab, I find a common link: Newton’s second law of motion, which tells us that the force acting on an object in any direction is equal to the object’s mass times the object’s acceleration in that direction. Sure, it sounds pretty straightforward, but as this lab has shown, we run into different kinds of obstacles when applying this law. So what I have learned by doing this lab is that in applying Newton’s second law, we must also consider the angle at which the object is moving, as well as any kinds of frictional forces that may be resisting the object’s motion, not to mention all the uncertainties involved in the motion of everyday objects that we have not even considered in this lab.

Sample Lab Report B:

Introduction to Circuits

Purpose: To explore the relationship between voltage, resistance and current in a circuit.

Procedures & Comments:

10. Using a multimeter, measure and record the voltage of two different batteries.

| |Battery 1 |Battery 2 |Battery Pak (1&2 combined) |

|Voltages |1.502 V |1.513 V |3.010 V |

If the leads are reversed when measuring voltage, the multimeter will display the same magnitude but will indicate that the voltage is negative.

11. A. From their color bands, determine and record the resistance and the uncertainty rating for five different resistors.

B. Measure and record the resistance with the multimeter.

Resistor Color Guide

|Band Color | |Band Color | |

|Black |0 |Green |5 |

|Brown |1 |Blue |6 |

|Red |2 |Purple |7 |

|Orange |3 |Gray |8 |

|Yellow |4 |White |9 |

Uncertainty: Silver = (10% ; Gold = ( 5.0%

Resistor: 1. Red, Purple, Black, Silver

2. Red, Red, Yellow, Silver

3. Red, Purple, Red, Silver

4. Orange, Orange, Red, Gold

5. Brown, Green, Yellow, Silver

|Resistor |Determined Resistance, ( |Determined Uncertainty, %|Measured Resistance, ( |Actual Difference, % |

|1 |27X10^0 |10 |30.0X10^0 |10.0 |

|2 |22X10^4 |10 |23.2X10^4 |5.17 |

|3 |27X10^2 |10 |29.0X10^2 |6.90 |

|4 |33X10^2 |5 |33.2X10^2 |0.60 |

|5 |15X10^4 |10 |15.2X10^4 |1.32 |

All of the measured values fall within the range given by the color bands.

12. Using two resistors of the same order or magnitude (the third band of the same color), connect them in series and measure and record the resistance of the combination. Repeat twice again with different resistors.

Equivalent Resistance = R1 + R2

|Resistors |Measured Resistance, ( |Calculated Resistance, ( |

|3 & 4 |6.23X10^3 |6.22X10^3 |

|2 & 5 |3.83X10^5 |3.84X10^5 |

|4 & 5 |1.55X10^5 |1.55X10^5 |

13. Using two resistors of the same order or magnitude (the third band of the same color), connect them in parallel and measure and record the resistance of the combination. Repeat twice again with different resistors.

1 / Equivalent Resistance = 1 / R1 + 1 / R2

|Resistors |Measured Resistance, ( |Calculated Resistance, ( |

|3 & 4 |1.55X10^3 |1.55X10^3 |

|2 & 5 |9.18X10^4 |9.18X10^4 |

|4 & 5 |3.25X10^3 |3.25X10^3 |

14. Rules for adding Equivalent Resistances and Capacitances in Series and Parallel Circuits:

Series Parallel

Resistance: R=R1+R2+R3+… 1/R=1/R1+1/R2+1/R3+…

Capacitance:1/C=1/C1+1/C2+1/C3+… C=C1+C2+C3+…

15. A series circuit acts like a one-lane highway and therefore all of the traffic must pass along the same path, which resists and slows down flow.

A parallel circuit acts like a freeway and therefore the traffic

can always take the path of least resistance, which allows it

to flow freely.

16. Construct a circuit consisting of a battery and a resistor and measure the current.

Resistor 3

| |Voltage, V |Resistance, ( |Current, A |

|Measured |3.00 V |2.90X10^3 |1.03X10^-3 |

|Calculated |V / R = I |1.03X10^-3 |

Measured and calculated answers agree.

17. Construct a circuit consisting of a battery and two resistors, with roughly the same order of magnitude, in series and measure the current.

Resistors 3 & 4

|Measured Across |Battery |Resistor 3 |Resistor 4 |

|Current, A |4.80X10^-4 |4.80X10^-4 |4.80X10^-4 |

|Voltage, V |3.00 |1.40 |1.60 |

In series the current must be the same through all elements. The voltage measured across resistor 3 and 4 equaled 1.40 V and 1.60 V respectively. The sum of these two voltages equals 3.00 V, which is the same as the voltage of the battery. In series the sum of the voltages through all of the elements must equal the power source.

18. Construct a circuit consisting of a battery and the two resistors used in procedure 8 in parallel and measure the current through the battery, the first resistor and the second resistor. Then measure the voltage across each resistor.

Resistors 3 & 4

|Measured Across |Battery |Resistor 1 |Resistor 2 |

|Current, A |1.93X10^-3 |1.03X10^-3 |9.00X10^-4 |

|Voltage, V |2.99 |2.99 |2.99 |

In parallel the voltage must be the same through all elements. The current measured across resistor 3 and 4 equaled 1.03X10^-3 A and 9.00X10^-4 A respectively. The sum of these two currents equals 1.93X10^-3 A, which is the same as the current through the battery. In parallel the sum of the currents through all of the elements must equal to the current through the power source.

Conclusion: The resistance of a resistor can be determined easily by deciphering its color bands. These determined resistances, although not perfect, are quite accurate. Using the multimeter to measure voltage, resistance and current is straight forward, but one must take special consideration when measuring current not to damage to meter. For series circuits, the equivalent resistance is equal to the sum of the resistances of the resistors; and the reciprocal of the equivalent capacitance is equal to the sums of the reciprocals of the capacitances of the capacitors. And for parallel circuits just the reverse is true. The reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the resistances of the resistors; and the equivalent capacitance is equal to the sum of the capacitances of the capcitors. In a series circuit the current is the same through all of the elements in the circuit; and the sum of the voltages through the elements in the circuit are equal to the voltage of the power source.

And for parallel circuits, again just the reverse is true. The voltage is the same through all of the elements in a circuit; and the sum of the currents through the elements in the circuit are equal to the current through the power source. Whenever possible all circuit lab work should be done on the breadboard. It is easier to handle and is a much more efficient way to do experimentation.

-----------------------

m1

m2

Hanging mass m2 20g 40g 60g

Acceleration of m1 (in m/s2) .795 1.447 2.006

aa

m2

T

m2g

FN

m1

T

m1g

m2 T(m2) a(m2) m1=T/a

20 g .180 N .795 m/s2 226 g

40 g .334 N 1.447 m/s2 231 g

60 g .468 N 2.006 m/s2 233 g

m1

m2

FN

T

m1

m1gsin(

m2gsin(

m1g

m1

m2

Hanging mass m2 20g 40g 60g

Acceleration of m1 (in m/s2) .795 1.447 2.006

aa

m2

T

m2g

FN

m1

T

m1g

m2 T(m2) a(m2) m1=T/a

20 g .180 N .795 m/s2 226 g

40 g .334 N 1.447 m/s2 231 g

60 g .468 N 2.006 m/s2 233 g

m1

m2

FN

T

m1

m1gsin(

m2gsin(

m1g

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