Berger,+Dember,+Rabin,+Tucker

= =

Inertia Mass
 * Group Members:** Alyssa Berger, Ross Dember, Rebecca Rabin, Jessica Tucker
 * Period:** 4
 * Date Completed:** November 19, 2010
 * Date Due:**
 * Note: Although everything is uploaded from rabinre account, we all worked together in class.**


 * Purpose/Objective:** Find the mass of a Rubix cube only using its inertia.


 * Hypothesis with Rationale:** The greater the mass, the less vibrations it has per time. We think this because it takes more force to change a heavier objects state of motion and the mass and inertia are directly related.

a) Materials 1. Inertial Balance 2. Known Masses 3. Stopwatch 4. Clamp 5. Rubiks Cube
 * Procedure:**

b) Set-Up and Methods 1. To begin, clamp the inertial balance onto a table. 2. Place a paper towel in the balance tray to help the mass remain still. If the mass slides while vibrating, you will get poor data. 3. Place the 50-gram mass in the inertial balance and begin the vibrations. 4. With a stop watch, calculate the time it takes for the inertial balance to complete ten periods. media type="file" key="Movie 10.mov" width="300" height="300" 5. Repeat step 4, at least 3 times and record your results. 6. To calculate the individual period, divide the total time by 10 7. Repeat steps 3-6 using 100, 200, and 300 gram masses. 8. Repeat steps 3-6 using a rubiks cube.


 * Graphs:**


 * Data and Calculations Tables in Excel:**

mass of Rubix cube y=0.0011x+0.3202 .442=0.0011x+0.3202 x= 110.73 grams
 * Calculations to find the mass of the Rubix cube:**

Percent Error: ((110.73-101.41)/101.41)x100 9.19%

2**. Did an increase in mass lengthen or shorten the period of motion?** It lengthened the period of the motion.
 * Follow-Up Questions:**
 * 1. Did gravitation play a part in this operation? Was this measurement process completely unrelated to the "weight" of the object?** Yes, gravitation played a big part in this operation. The objects that had a larger mass weighed more, therefore, there is more of a gravitational pull downward. This way, the balance would not move as fast. Thus periods are related to the weight of the object.
 * 3. How do the accelerations of different masses compare when the platform is pulled aside and released?** The heavier the objects had a slower acceleration, while the light objects had a faster acceleration.
 * 4. Do you imagine that the period would have been different if the side arms were stiffer? Would this change lengthen or shorten the period of motion?** The period would be lengthened if the arms were stiffer because it would not move as feel, so it would take the object longer to complete one full movement.
 * 5. Is there any relationship between inertial and gravitational mass of the object?** Yes, the larger the object the heavier it is. This means that the larger objects will have a larger gravitational pull. Because of this larger gravitational pull the large objects are not changing direction as quickly as the smaller objects. Thus inertia is inversely related to gravitational mass.
 * 6. Why do we almost always use graviation instead of inertia as a means of measuring mass of an object?** Gravitation is easier to measure because all you have to is weigh the object instead of testing it on an inertia balance. Also, gravitation is present more often than inertia.
 * 7. How would the results of this experiment be changed if you did this experiment on the moon?** All of the objects would have shorter periods, however; these results would all be proportional to each other.

Yes, the purpose of discovering the mass of an object only using its inertia was satisfied. We were able to find the mass of the Rubix cube after we had created a vibration vs. mass Excel graph, and inserted the line of best fit. This allowed for us to plug the period of the unknown mass, .442 seconds/vibration, into the equation of the line, and ultimately satisfied our purpose when we found the unknown mass to be 110.73 grams. Yes, our hypothesis was correct. Our hypothesis is correct because the mass and inertia are directly related, which ultimately proved that the greater the mass, the less vibrations per time, and the less the mass, the more vibrations per time.
 * Evalution/Conclusion:**
 * Part 1:**
 * Was the purpose satisfied and provide specific evidence from the experiment to justify your claims.**
 * Was your hypothesis correct?**

The error could have occurred when counting the periods of the inertial balance, or the reaction time of the person holding the stop watch. The person counting the balance potentially could have incorrectly counted the individual periods, which then would have ultimately effected the time of the individual period. Also, when the balance hit 10 periods, the stop watch person could have stopped the time too late. Error also could have been seen in the mass of the Rubix cube, considering if one variable of the lab was incorrect, then it would have affected the ultimate result for the project.
 * Part 2: Errors**
 * Where and why did the error occur?**

To address the errors within the lab, we could have done more than three trials, maybe even ten, to get a more precise time for each individual period, which could have made our mass of the Rubix cube more exact. Also, if the process was mechanized**,** there would be no reliance on human reaction time, thus the error would be non-existent. Since the use of machinery is not allowed, we should have either waited for a longer time or counted more vibrations**,** as the amount of error on the reaction time would have been less, and the results would have a bigger sample of time**.**
 * Part 3:**
 * i) How would you change the lab to address the errors?**

= **Newton's Second Law** = **Group Members:** Alyssa Berger, Ross Dember, Rebecca Rabin, Jessica Tucker **Period:** 4 **Date Completed:** December 3, 2010 **Date Due:**

**Purpose/Objective:** What is the relationship between system mass, acceleration and net force?

**Hypothesis with rationale:** In the comparison between the system's mass, acceleration, and net force, using Newton's Second Law of Motion f=ma, the resulting relationship between mass and acceleration will be indirect whereas the acceleration and net force will be direct.

**Procedure:** a) Materials 1. Dynamic Cart with Mass 2. Dynamics Cart 3. track 4. Photogate timer 5. Data Studio 6. Super Pulley with Clamp 7. Base and Support Rod 8. String 9. Mass hanger and mass set 10. Wooden or metal stopping block 11. Mass blanace 12. Level

b) Set-Up and Methods 1. Clamp pulley to track 2. Place track with dynamic cart on table allowing pulley to hang off edge 3. Attach pulley wire photogate port 4. Attach photogate port to laptop and open up date studio 5. Tape piece of string to dynamic cart, threading the string through the pulley on the opposite end of the track

Part 1: Relationship between Force and Acceleration 1. First you want to place a small amount of weight onto your cart to make the mass of the system larger, for example, we used 40g. 2. Then you want place a hanging off the end of the string that is not attached to the cart. The mass of this hanging weight should be significantly lower than the mass of your cart. Our first mass was 5g. 3. Place the USB into your computer and open DataStudio. Make sure that you are looking at the velocity vs. time graph because the slope of that line will give you your acceleration. 4. You then want to get the acceleration of the cart by letting go of the cart while the hanging mass is hanging off the side of table. Make sure that your string is thread through the pinwheel before you let go of the cart. 5. Look at the velocity-time graph in DataStudio. Once you have completed your first trial, click the "fit" tab and then choose "linear." This will give you the slope of your line, which in this case is also your acceleration. 6. Repeat these steps at least three times, or until you get consistent results. 7. You then add to the mass of the hanging weight, but take away that same mass from the cart in order to keep the total mass of system constant. 8. Repeat steps 1-5 until you have sufficient data to make a graph.
 * Procedure**:


 * Trials from Part 1:**
 * Data Acquired from Part 1:**

(.005kg)+(.542kg)= .547 kg [mass of hanging (kg) + mass of cart (kg)]= total mass of system (kg)
 * Example of how we calculated the total mass of the system:**

(0.547 kg)*(.062 m/s2)= 0.033914 N [total mass of system (kg) * acceleration (m/s2)]= force (N)
 * Example of how we calculated the force of the system:**


 * Graph of force versus acceleration:**

Percent Error:

__|Experimental Value-Theoretical Value|__ x100 Theoretical Value

__|0.547-.588|__ x100 0.547 Percent error = 7.49%

Part 2: Relationship between Mass and Acceleration 1. Place a large amount of weight onto your cart, for example 2000g, and a small amount of weight onto your hanging mass, for example 20g 2. Place the USB into your computer and open DataStudio. Make sure that you are looking at the velocity vs. time graph because the slope of that line will give you your acceleration. 3. You then want to calculate the acceleration of the cart by letting of it while the hanging mass is hanging off the side of the table. Make sure that your string is thread through the pinwheel before you let go of the cart. 4. Look at the velocity-time graph in DataStudio. Once you have completed your first trial, click the "fit" tab and then choose "linear." This will give you the slope of your line, which in this case is also your acceleration. 5. Repeat these steps at least three times, or until you get consistent results. 6. You then take away mass from the cart, but leave the hanging mass constant. 7. Repeat steps 1-6 until you have sufficient information to make a graph.
 * Procedure**:

(2.502kg)+(.02kg)= 2.522 kg [mass of cart (kg) + mass of hanging (kg)]= total mass of system (kg)
 * Trials from Part 2:**
 * Data acquired from part 2:**
 * Example of how we calculated the total mass of system:**

**Example of how we calculated the theoretical acceleration:** (.02kg*9.8 m/s^2)/ (2.522 kg)= .077716098 [mass of hanging (kg) * gravity (m/s^2)]/ [total mass of system (kg)]= theoretical acceleration


 * Graph of mass versus acceleration:**

Percent Error: __|Experimental Value-Theoretical Value|__ x100 Theoretical Value __|(-1)-(-1)|__ x100 -1 Percent error = 0%


 * Analysis Questions**-


 * 1a. If linear: What is the slope of the trendline? To what actual observed value does the slope correspond? How does it compare to this actual observed value (find the percent error between the two)? Show why the slope should be equal to this quantity. What is the meaning of the y-intercept value?** In the first graph, the slope is equal to .588, and it represents the mass of the cart. Right below the graph, the percent error is shown to be 17.49%. The slope should be equal to the mass of the system since the equation for force is F=ma, thus the value of acceleration is multiplied by the constant mass of the system (the system) to derive at the force. However, in this case there is friction present so the mass will be affected a little bit. The y-intercept value is .0137. This is the amount of friction that is present when we conducted our experiment.


 * 1b. If non-linear: What is the power on the x? What should it be? What is the coefficient in front of the x? To what actual observed value does the coefficient correspond? Find the percent error between the two. Show why this value should be equal to this quantity.** In the second graph, the x is the power of -1, which it should be. It should be -1 as mass and acceleration are inversely related when there is a constant force. The constant force is depicted by the coefficient, which is .196. As F=ma, it is also true that a=F/m, thus .196x-1 gives the acceleration as mass is equal to the x. The percent error is shown under the graph, and it is equal to 0%.

Friction would slow down the cart making there less acceleration. A bigger force would be needed to create the same acceleration so you would have to add mass to hanging block to increase the force. Our slope was too big, so yes friction can be a source of error in this lab. F=ma T-f=ma .049-f=.542(.062) f = -.685N
 * 2. What would friction do to your acceleration? Would you need a bigger or smaller force to create the same acceleration? Was your slope too big or too small? Can friction be a source of error in this experiment? Redo the calculation of acceleration WITH friction to show its effect.**

The purpose/objective of this lab was to prove Newton’s second law of motion, which required comparing the relationship between a system’s mass, acceleration, and net force. Based on our accurate hypothesis (Using Newton's Second Law of Motion f=ma, the resulting relationship between mass and acceleration will be indirect whereas the acceleration and net force will be direct), we were able to successfully satisfy this. Our success can be best illustrated through our graphs. In part 1, force vs. acceleration, our graph will show that as the force goes up so does the acceleration thus creating a direct relationship. In part 2, mass vs. acceleration, our indirect relationship is shown through the graph because as our mass goes up our acceleration goes down. Therefore, we have proved that our hypothesis was correct.
 * Conclusion:**

= **Coefficient of Friction** = **Group Members:** Alyssa Berger, Ross Dember, Rebecca Rabin, Jessica Tucker **Period:** 4 **Date Completed:** December 10th, 2010 **Date Due:** December 13th, 2010

**Objectives:** -To measure the coefficient of static friction between surfaces -To measure the coefficient of kinetic friction between surfaces -To determine the relationship between the friction force and the normal force

**Hypothesis:** The relationship between the friction force and the normal force is direct. This should be true because of the equation. If normal force increases the friction force will increase as well thus showing a direct relationship.

**Procedure** a) Materials 1. Force Meter  2. USB Link  3. Wooden block  4. Miscellaneous masses  5. String  6. Aluminum track  7. Clamp

b) Set-Up and Methods Part A: Measuring Coefficient of Friction on a Flat Surface  1. Mass the wooden block.  2. Clamp the surface board to the table top.  3. Place the block on the surface and put 500-g on top of it.  4. Tie a short (15 cm) string to the block at one end, and to the force meter on the other.  5. Plug the force meter into your computer. Choose Data Studio, and "Create Experiment". A force-time group will automatically open.  6. Go to SETUP and check **//Force - Pull Positive// and uncheck ****//Force - Push Positive// . Then on the group display, click the y-axis label to chane the name to ****//Force - Pull Positive// . **  ** 7. Leaving the string slack, press the button "ZERO" on the sensor. **  ** 8. Press START on Data Studio, and gently pull the block with the force sensor **  a. Be sure to pull with a very slow constant speed once it starts to move.  b. HOld the string parallel to the board. 9. Highlight the straight line part and click S. Record the MEAN as the value for Tension at Constant Speed. 10. Highlight the maximum point and record that vlue as teh Maximum Tension. 11. Repeat twice more with the same mass. 12. Repeat Steps 8-11 adding more mass each time (best to make large changes)

Static Friction (with tension force):
 * Data: **


 * Mass (kg) |||||| Maximum Tension (N) ||= Avg MaximumTension (N)/Friction || Weight/Normal || Coefficient ||
 * || 1 || 2 || 3 ||  ||   ||   ||
 * 0.651 || 1.3 || 1.1 || 1.1 || 1.17 || 6.38 || 0.182868846 ||
 * 1.651 || 3.1 || 3.3 || 3.3 || 3.23 || 16.18 || 0.199837658 ||
 * 2.651 || 5.4 || 5.1 || 5 || 5.17 || 25.98 || 0.198872457 ||

Kinetic Friction (with tension force):


 * Mass (kg) |||||| Mean Tension at Constant Speed (N) || Avg Tension (N)/Friction || Weight/Normal || Coefficient ||
 * || 1 || 2 || 3 ||  ||   ||   ||
 * 0.651 || 1.3 || 1 || 0.9 || 1.066666667 || 6.3798 || 0.167194374 ||
 * 1.651 || 2.9 || 2.9 || 3.1 || 2.966666667 || 16.1798 || 0.183356201 ||
 * 2.651 || 4.6 || 4.7 || 4.6 || 4.633333333 || 25.9798 || 0.178343688 ||

Sample Calculations: FBD Static Friction with tension: __Normal Force__ ∑ y =N-W N-W=ma ma=0 N=W W=mg W=1.651(9.8) W= 16.18 N=16.18 Newtons __Friction__ ∑ x =ma ma=0 T-f=0 T=f T=3.23 f=3.23 Newtons __Coefficient__ f=µN 3.23=16.13µ µ= .20 __Percent Difference__

__abs(class average – experimental average)__ *100 (class average - experimental average)/2

__abs(.2114 – .1985)__ *100 (.2114 + .1985)/2 Percent difference= 6.29%

Kinetic Friction with Tension

∑ y =N-W N-W=ma ma=0 N=W W=mg W=1.651(9.8) W= 16.18 N=16.18 Newtons __Friction__ ∑ x =ma ma=0 T-f=0 T=f T=2.97 f=2.97 Newtons __Coefficient__ f=µN 2.97=16.13µ µ= .18 __Percent Difference__ Percent Difference= 1.51%


 * Analysis:**

The graph shown above depicts the relationship between normal force and friction force through the relationship of wood on aluminum. Friction force and normal force are directly related through the equation. Within the graph above, the r^2 value for static friction is .9987., and the r^2 value for kinetic friction is .9983, which portrays our results to be mainly accurate. The slope of the static friction line represents µ of static friction as being .1985. According to the results solved by hand, the µ of static friction is equal to .20, which displays our solutions from the slope compared to the solved coefficient of friction to be extremely similar, with a minute source of error. The slope of the kinetic friction line is .1792, which shows the coefficient of friction for kinetic friction. According to the results solved by hand, the coefficient of friction equals .18, which also depicts our solutions as being primarily accurate, with hardly any error present. As seen above, the percent difference of the class results versus our results for the static friction is equal to about 6.29%, and the results for the kinetic friction is equal to 1.51%. Clearly, less error was seen in the kinetic friction trials of the lab, but these numbers are relatively low, portraying our results as being very close to fully accurate.

Part B: Measuring the Coefficient of Friction on an incline 1. Attach a protractor to the track, using the nut-screw assembly that slides onto the track 2. Secure a 200-g mass to the block with masking tape, and place at the raised end of the track 3. Slowly lift the end of the track until the block just begins to slide down the aluminum surface. Record the angle at which this occurs. Have each member of your group conduct this step 2 times, recording each angle measurement, then take the average 4. Place the track on an incline by clamping it to a ring stand. Make the angle just slightly less than the angle measured in Step 3. The block should NOT slide down on its own. When you nudge the block just slightly, it should continue down the ramp at constant speed. Have each member of your group conduct this step 2 times, recording each angle measurement, then take the average

Data Static Friction Kinetic Friction

Calculations

Static Friction FBD



Normal Force

__∑__ y =ma ma=0 N-sin( θ)*W=0 N=mg*sin( θ )

Friction Force __∑__ x ﻿= ma ma=0 f-cos( θ )*W=0 f=mg*cos( θ)

Coefficient

µ=f/N µ= µ= tan(θ) µ= tan(8.75 µ=.154

Percent Difference



Percent Difference= 25.4%

Our value of the static friction from Part A very much differs from our value found in Part B with regards to static friction. As calculated above, the percent difference of our results from Part A of the coefficient of friction for static friction versus the results from Part B is 25.4%.

Kinetic Friction:

Normal Force ∑ y =ma ma=0 N-sin( θ )*W=0 N=mg*sin(θ)

Friction Force ∑ x =ma ma=o f-cos( θ )*W=0 f=mg*cos(θ)

Coefficient

µ=f/N µ= µ=tan(θ) µ= tan(8.1625) µ= .1434

Percent Difference

Percent Difference= 22.2%

Our value from Part A also differ from our values found in Part B, in terms of the kinetic friction. As seen above, the percent difference of our results from Part A of the coefficient of friction for kinetic friction versus the solution from Part B is 22.19%.

- Since the friction force is equal to the tension force, the amount of tension that is put on the system will affect the magnitude of the the force of friction. Also, the surface that the system is sliding across will affect the friction force. If the surface is slippery, wet, unclean, dry, or at all different throughout the experiment, your results will not be accurate because the friction force will be greatly affected. - The normal force and the friction force both affect the coefficient of friction. The normal force, weight and the coefficient of friction are all directly related, so when there is a larger weight, there will be a larger normal force meaning that the coefficient of friction will also be larger. The friction force is indirectly related to the coefficient of friction, so the more friction the system has, the smaller the coefficient of friction will be. - The coefficient of static friction is larger than the coefficient of kinetic friction. This is because the friction force for static friction is larger than the friction force for kinetic friction. Because the friction force and coefficient of friction are inversely related, the coefficient of static friction should be smaller than the coefficient for kinetic friction because the static friction has a stationary system meaning that there will be more of a friction force to hold it in place. - The coefficient of friction should not change when putting the track on an incline. This is because the same materials are used so there is the same amount of friction and the same amount of normal force. Since these are the two major components that affect the coefficient of friction, if they stay the same the coefficient of friction should stay the same as well.
 * Discussion Questions:**
 * 1. Why does the slope of the line equal the coefficient of friction? Show this derivation.** The slope of the line is equal to the coefficient of friction (µ) due to the relationship between our resulting graph and the equation f=µN (f represents friction, represents the coefficient of friction, and N represents normal force). After observing our graph we can come to the conclusion that friction lies along the y-axis while normal force lies along the x-acis. Therefore using these observations and plugging the values into the linear equation y = mx + b, m or the slope of the linear equation will be equivalent to the coefficient of friction.
 * 2. Look up the coefficient of friction between wood and aluminum. Discuss if your measured results fall within the range of theoretical values. Be sure to cite your source!** The coefficient of friction between wood and aluminum (wood and any clean metal) is 0.2 - 0.6. Our measured result for the coefficient of static friction was .2 and therefore fell within the range of theoretical values. Our measured result for the coefficient of kinetic friction however was .18 and therefore did not fall within the range of theoretical values. Source: [|http://www.engineeringtoolbox.com/friction-coefficients-d_778.html]
 * 3. What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction?**
 * 4. How does the value of coefficient of kinetic friction compare to the value for the same material's coefficient of static friction?**
 * 5. Did putting the track on an incline significantly change the coefficient of friction? Why or why not?**


 * Conclusion:**

Our hypothesis, that the relationship between normal force and friction force is direct, was correct, proven by the positive, linear slope of the both lines. The slopes also prove that static friction is greater than kinetic friction. In Part A, there was more force needed to pull the block at constant speed than to maintain it, and in Part B, the angle to start the block to move at constant speed was greater than the one for kinetic friction. Also, as shown in the table, as the friction force increased, the normal force increased at a relatively constant interval.

As shown in the calculation section, the percent difference from the class in Part A was minimal: 6.29% and 1.51% for static and kinetic, respectively. However, our results in Part B were much different than those in Part A (25.4% and 22.2%). These errors came from multiple sources. In Part A, it is difficult to ensure that the wood is being pulled at constant speed, just as in the Part B, it is difficult to decide if the wood is sliding at constant speed. Also, despite our best efforts, it is plausible that the string was not pulled completely parallel to the ground, meaning there was some unaccounted for force in the experiment. In Part B, it was important to steadily raise the slide, as it would sometimes lean towards a side, which would lead to more gravitational pull. Furthermore, unlike Part A, the measurements in Part B were not taken by computer, so trying to eyeball the angle did not ensure accuracy. All of these errors occurred since subjectivity was needed, such as measuring the angle and properly determining what was constant speed. Although we made sure each person took on a different role to try to even out possible errors and despite the fact each trial was done multiple times, there were still errors in the experiments.

To eliminate error, a device such as a constant motion vehicle in Part A would have taken away human error in that aspect, as there would definitely be a constant speed to the block. In part B, a mechanism that could steadily rise the slide would have helped in eliminating some error, as well as more accurate measuring tools.

Understanding the relationship between friction force and normal force can be applicable in numerous situations. For example, if someone on ski patrol is pulling an injured individual, he would need to know what force to apply on the rope to ensure that he was not pulling too fast and injuring the individual. Knowing the coefficient of friction between the snow and sled would assist the patrolman in deciding what force to apply and how fast to go. Furthermore, if someone wanted to create a slide that would have an object maintain speed and not accelerate (without being pushed), to ensure a small impact on arrival, knowing the static friction coefficient would be useful, in order to put it at the perfect angle. If said object had to be pushed a little, knowing the kinetic friction would allow for a smaller angle to be used and would also get a constant speed.

Acceleration Down an Incline **Group Members:** Alyssa Berger, Ross Dember, Rebecca Rabin, Jessica Tucker **Period:** 4 **Date Completed:** December 17, 2010 **Date Due:** December 20, 2010

**Objective:** How does the acceleration of an object down an incline depend on the angle of the incline?

**Hypothesis:** The larger the angle of the incline, the faster the acceleration of the object down an incline will be. This is due to the equation a = g(SIN q - m COS q ) derived from the equations of //Newton's Second Law//.

**Procedure:** a) Materials 1. Wooden Block  2. Aluminum Track  3. Ring stand  4. Clamp  5. Meter stick  6. 2 Photogates  7. Picket fence  8. Tape  9. Photogate Stands  10. Pulley  11. String  12. Masses  13. Mass Hanger

b) Set Up and Methods **PART A: Acceleration Down an Incline**

1. Attach a small clamp to the track and place it on a ring stand rod. Set up the track with one end raised higher than the other. Carefully measure the exact height. 2. Tape picket fence to block so that the small black lines are at the top. Be sure that no tape is on the bottom of the block. Find the mass of the block/fence. You may want to tape a 200 or 500-g mass to the block to give it more mass if it is too light. 3. Measure the thickness of one of the small black lines on the picket fence. BE PRECISE! 4. Set up the height of the Photgate Timer so that the small black lines will run through the laser. This can be anywhere along the length of the track, as long as the block is moving all the way through. 5. Open “Data Studio” and select “Photogate – Picket Fence”. Click on Settings, “Constants”, and enter the measurement you made in step 2 under “Band Spacing”. 6. Move the block to the end of the track. 7. Release the block and record the acceleration through the gate. Repeat this measurement several times. Record all the values in your Data Table. 8. Change and re-measure the height, and repeat the acceleration measurements. Do this for at least 5 different heights.

Data of acquired acceleration:

Data Table: This data chart shows all of the information we collected in this lab. The height and length were measured using a meter stick and then in order to get the angle we

took the sine of the height/length. This information gave the angle in radians. We then used data studio to get our acceleration by completing each angle with three

trials. These three trials were then averaged together to get our average acceleration. We then realized that when using data studio, we had not measured the

correct part of the picket fence measuring device. This caused our accelerations to be half of they needed to be, so in our excel, we multiplied the accelerations by

two and that is the acceleration we used in our graph.

This graph shows the relationship between the sine of the angle and the acceleration. As you can see, they have a direct relationship: as the sine goes up, so does that angle and vice versa. This causes the positive slope and straight line that the graph shows. Our coefficient of x which is 9.7156 is the acceleration of gravity on the sine of the angle. The y-intercept of -1.8744 is the -µ times the acceleration of gravity times the cosine of the angle. This information comes from the equation .


 * Calculations**:

Percent Error: __|theoretical value-experimental value|__ x100 theoretical value __|9.81-9.7156|__ x100 9.81 1.12%

Coefficient of Kinetic Friction (taken from previous lab):

∑ y =N-W N-W=ma ma=0 N=W W=mg W=1.651(9.8) W= 16.18 N=16.18 Newtons __Friction__ ∑ x =ma ma=0 T-f=0 T=f T=2.97 f=2.97 Newtons __Coefficient__ f=µN 2.97=16.13µ µ= .18

Theoretical Acceleration

**PART B: Acceleration Up an Incline**

9. Attach a string to the end of the block. Run the string over a pulley and attach a mass hanger to the other end. 10. Determine the mass needed in order for your block to move along a 0.5 m portion of your incline in exactly 1.50 seconds. 11. YOU MUST SHOW YOUR CALCULATIONS TO YOUR TEACHER BEFORE CONTINUING FURTHER. 12. Set up two photogate timers, one where you will begin the timing and the other exactly 0.5 m up the track. 13. Open “Data Studio” and select “Photogate Timing”. 14. Hang the calculated amount of mass on the hanger. 15. Press start on Data Studio and read the results on “Time Between Two Gates”. Repeat at least 5 times. If you are not within 2% error of both your mass // and // your time, then you did something wrong. Go back and fix your errors!

**Calculations**:

For percent error of **time** Part B, we are using the most precise time found through the trials, which occurred in trial three, 1.52 seconds.
 * Data**:
 * Percent Error Part B:**

**__PERCENT ERROR FOR TIME:__** __|theoretical value-experimental value|__ * 100 theoretical value =|1.52-1.50|/(1.52)*100 =1.32%

__**PERCENT ERROR FOR MASS:**__

__[theoretical value-experimental value|__ * 100 theoretical value

**Discussion Questions:**

The graph above displays the relationship between the angle on the incline and acceleration. The acceleration on the graph was obtained through Data Studio, as the slope of a velocity vs. time graph portrays acceleration and we took the sin of the angle to show the incline angle versus the acceleration. The slope of this graph displays g, or 9.8, shown through the equation. The y- intercept of the graph shows the friction of the graph divided by mass. This number is negative, which it should be as you adjust the incline angle.
 * 1. Discuss your graph. What does the slope mean? What is the meaning of the y-intercept?**

Our measured value for g is approximately 9.716 m/s^2, which happens to be lower than the true value of g, 9.8 m/s^2. Friction definitely could have caused the minimal difference between the calculated value versus the true value of g. The y- intercept on our graph displays the friction, and when this number is altered, the g value is altered as well, proving that the friction has a definite affect on the measured value of g.
 * 2. If the mass of the cart were doubled, how would the results be affected?** If the mass of the cart were doubled then the acceleration would be doubled as well thus resulting in a steeper slope of the graphed line. For example, if the mass of that cart was .35 instead of .175 then the acceleration would have to be
 * 3. Consider the difference between your measured value of g and the true value of 9.80 m/s2. Could friction be the cause of the observed difference? Why or why not?**


 * 4. How were your results in Part B? Why was the expectation that your results be within 2% considered to be reasonable when in other labs we allow much larger margins of error?** Our percent difference for the time calculated in Part B was 1.32%. This result was very close to accurate. With the calculated hanging mass of .069 kg, in trial 3, we received a time of 1.52 seconds, while we were supposed to equal 1.5 seconds, which is a very close result.


 * Conclusion:**

As shown in our graph, the hypothesis was correct. The sine of an angle increases as an angle increases, and since the graph has a positive slope, it is understood that as the sine of an angle increases, the acceleration increases. As all other variables remained the same, it is apparent that the increase in angle was directly related to the increase in acceleration.

The error in both parts of the lab was minimal. In part A, the error was 1.12% as our value for //g// was about 9.72, which was not that far from the accepted theoretical value of 9.81. For the second part, the error was 1.33%, which also showed that our results were fairly accurate. Error for the first part could have come from multiple sources, other than simple measurement and human mistakes. When the weight was dropped, it did not always go straight down, which means its force was also going left and right, which would give the block less of an acceleration when going up the incline. The error could have been reduced by ensuring the weight was dropped straight down. For Part B, the errors were the same as the prior part since most of our assumed values came from Part A.

Knowing the effect of an angle on the acceleration has important uses. For example, if one were to be going sledding on a snow day, he or she would most likely believe that greater acceleration would correlate with more fun, so using the ideas of the lab, one would find the steepest hill they could. Furthermore, a groundskeeper in baseball would deviate the size of a pitching mound depending on the home team's starting pitcher. If the team had a fireballer starting, the groundskeeper would want to put the mound at a higher height, so the angle would be greater, thus the pitcher's fastballs would have an even greater acceleration.