Group2_4_ch4

toc Group Members: Garrett Almeida Lauren Kostman Ryan Luo

=Gravity and the Laws of Motion Lab (Galileo's Ramp)= 11/18/11 Lauren-A Ryan-B,D Garrett-C

__**Objectives:**__
 * Find the value of acceleration due to gravity.
 * Determine the relationship between acceleration and incline angle.
 * Use a graph to extrapolate extreme cases that cannot be measured directly in the lab.
 * Determine the effect of mass on acceleration down an incline.

__**Hypothesis:**__ In this lab, one of the objectives is to find the value of acceleration due to gravity. We learned that acceleration due to gravity in free fall is 9.8 m/s 2, so we assume that the acceleration due to gravity in this lab should be 9.8 m/s 2 , too. Our group also needed to determine the relationship between the acceleration and the incline angle. We feel that the acceleration is directly proportional the incline angle, so if the angle increases, then the acceleration also increases. Lastly, we must determine whether or not the mass has an effect on acceleration on a down incline. We hypothesize that mass does not have an effect on acceleration and the acceleration should be constant down the incline because acceleration is directly proportional to the total forces being exerted on the ball that is rolling down the incline. Therefore, if acceleration changes, it is because of a change in the forces, and not mass.

__**Materials and Method:**__ The first step to this lab is setting the height of the ramp (attached to a stand) to .15 meters high using a meter stick for precision. Before performing the lab activity, we weighed our metal ball, which was .0282 kilograms. We also determined the distance from one end of the ramp to the other end of the ramp, which was 1.22 meters. From there, we rolled the metal ball down the ramp 5 times per each of the 5 heights we used (added .15 m to the height for each test) in order to make sure the results were accurate, timing each drop. Upon completing all of the trials and gathering results, the data was put into an excel spreadsheet, after which we calculated our information. We then compared our results with the results of the class and drew conclusions on whether or not our data tells us of the answers to our objectives.

__**Video:**__ media type="file" key="Movie on 2011-11-18 at 12.02.mov" width="300" height="300"

__**Trial Data:**__ __**Class Results Data Table:**__ __**Graph: Acceleration vs. Sin(theta)**__ __**Link To Excel Spreadsheet:**__

Sample Calculations for average time at a height of .15m: Sample Calculations for Acceleration (Trial 1 - the table is above): Sample Calculations of Average Accelerations for the set of five trials above: Sample Calculations to find sin of theta (using trig equations) for the first five trials: Sample Calculations for Percent Error: Sample Calculations for Percent Difference (Average class data- individual our data): __**Analysis:**__ In our acceleration versus sin (theta) graph (located above), the equation of the line of our graph is y = 7.5805x - 0.3707. The slope of the equation above, 7.5805, is what our group got as the acceleration due to gravity. Ideally, the acceleration due to gravity would be 9.8 m/s 2. This error could have been a result of human error when timing the ball and the force of friction; it's possible that we began timing a second too quickly or slowly, which would've impact our results and made them less accurate. Furthermore, it is possible that our measurements for height and distance were not exactly precise, which would also harm the results. These factors could have been at fault for the disparity between the ideal acceleration of gravity and the experimental acceleration of gravity. The r 2 value in our graph was about .98, which means that the data is pretty close to being exactly on the line of best fit, yet since it's not 100%, this suggests that there is some error present, as we assumed. In the best case scenario, the equation of our line would have a slope of 9.8 (g). The "b" value in the equation of our line is related to the force of friction, but is not the magnitude of friction itself. The percent error between the experimental value for free fall acceleration and the theoretical value for free fall acceleration was 22.65%. This does not show a strong correlation between our experimental value and our theoretical value (acceleration of gravity), however the percent difference, just 7.78%, shows that we have a strong correlation between our data and the class average. The class data on the table below shows that there is no relationship between masses and gravity; for example, the ball with the smallest amount of mass was __not__ the slowest. Regardless of a groups mass, all balls traveled at around the same acceleration. In addition, the masses did not affect the acceleration of the ball when it traveled down the ramp .15 cm above the ground. Also, the acceleration values that the class got were relatively near the acceleration of gravity, 9.8m/s 2, which allows us to assume they are both precise and accurate (to a certain degree). In the free body diagram below,we notice that the weight (w1) is the force causing the ball to roll down the ramp. This is because the normal force and the weight (w2) make the ball balanced. It is the weight (w1) that is not balanced and thus causes the ball to roll down the ramp. Using the formula in Newton's Second Law, F=ma, one can calculate the acceleration, which in this case was 1.176 m/s 2. This, however, was not too close to our class' experimental average acceleration, .84m/s 2. Friction plays a minor role in this process, however gravity and weight prove to be much stronger forces.
 * __Calculations:__**

__**Discussion Questions:**__ //**1. Is the velocity for each ramp angle constant? How do you know?**// The velocity for each ramp is not constant. This is because the time and acceleration values for each incline were different. Because acceleration is the rate of change of velocity per unit of time, the velocity was changing, not constant. The time value would get smaller and the acceleration value would increase as the incline got steeper. Also, while going down the ramp, the ball changed its velocity. Therefore at each ramp, the velocity values were different and the velocity changed as the ball traveled down the ramp.

//**2. Is the acceleration for each ramp angle constant? How do you know?**// The acceleration for each ramp angle was not constant; the acceleration is only constant for trials done on the same angle. This is because our calculations above show that the acceleration increased as the ramp got steeper. Using the kinematics equation shown in the sample calculation of acceleration above, I was noticed that the acceleration values became bigger as the angle became steeper.

//**3. What is another way that we could have found the acceleration of the ball down the ramp?**// We could have used a device such as a radar gun to measure the balls final velocity. Then we could have plugged this value into another kinematics equation to find the acceleration. The equation is a=(V f -V i )/t. Also, we could have used a motion sensor that we used on a previous lab to give us the balls velocity relative to the time. The slope of this graph would be equivalent to the acceleration of the ball.

//**4. How was it possible for Galileo to determine g, the rate of acceleration due to gravity for a freely falling object by rolling balls down an inclined plane?**// It was possible for Galileo to have determined the acceleration due to gravity for a free falling object by rolling balls down an inclined plane. This is because he would have calculated the acceleration of the ball down the ramp at many different angles. Galileo would have had to assume that the rolling the ball down the ramp did not cause friction. With more angles and more acceleration calculations, Galileo could examine the correlation between time and acceleration. He then could have extrapolated this information to determine what the acceleration would be for dropping the ball vertically, or at 90 degrees.

//**5. Does the mass of an object affect its rate of acceleration down the ramp? Should it affect the motion of an object in free fall in the same manner?**// The mass of an object does affect its rate of acceleration down the ramp. This is because the class data shows a correlation between the different masses of the balls tested and the acceleration. As the mass increases, the acceleration tends to increase. This is noticeable in the class data table. The .28 gram ball only moved at a velocity of .72 m/s^2 and the .535 gram ball moved at a velocity of 1.04 m/s^2. This example serves to demonstrate this trend. The reason that we see this trend is because friction impacts the smaller objects more than it does for the larger balls. This would make the acceleration value smaller for the smaller balls.

__**Conclusion:**__ Upon completing this lab, we found that our hypothesis of the incline angle is the only one out of our three that is correct. We believed that acceleration due to gravity is 9.8 m/s 2 ; our acceleration is not 9.8 m/s 2, but the average is 7.58 m/s 2. Also, mass did affect acceleration, contrary to what we believed. However, acceleration is inversely proportional to mass, unlike how acceleration is directly proportional to net force. We supported our hypothesis knowing the relationship between the acceleration and the incline of the angle, which are directly proportional to each other. We found this by increasing the angle in each experiment and timing each run to see if the time it took for the ball to roll down the ramp is shorter, which it was. In other words, the higher the incline (or closer to a 90 degree angle), the faster the acceleration, and the closer it gets to 9.8 m/s 2. Overall, while we were not entirely accurate with our hypotheses, we completed our objectives, and learned from both our results and our mistakes. A relative, real life application of this experiment could be if someone wanted to know how much time they had for a ball to roll down a ramp in a sports events. For example, if a little kid were to go bowling and rolled the bowling ball down a ramp, he (or his parents) would be able to determine how fast the ball would go, and how much time it would take to reach ground level. Also, the kid may want to use different weights of bowling balls, some a little heavier than others, so due to the results from this lab, he would be able to know that the weight of the ball wouldn't impact the acceleration, but rather the incline angle of the ramp he was using would. Therefore, completing such a lab can be very useful to everyday activities and is very applicable to other subjects.

While we did complete our objectives and calculated our results, the results are not completely accurate and include some error. We believed our acceleration due to gravity would be 9.8 m/s 2, yet the acceleration we found is 7.58 m/s 2. This yielded a percent error of 22.65%, which is strangely high. However, we did not take into account what the force of friction and normal should have been during the experiment. If friction and normal exerted a force that neutralized 2.22 m/s 2 from the acceleration of gravity, then the acceleration due to gravity would be what our prediction was (9.8 m/s 2 ). Unfortunately, we do not know how to calculate the force of friction and normal as of yet, so this result cannot be determined so far. Another factor that contributed to an elevation of our percent error and the percent difference of the class is the fact that we all took an average of all our times. Had our times all been the same, our result would have matched. This occurred because people do not have the same reaction time as each other, so the push of the stopwatch could have been delayed and our varying results are the outcome. To combat this, we can have a machine that uses a radar gun and which starts timing the ball from the start to the end of the route without any delays or preemptive button pushes. Additionally, each group could have performed several trials using different masses, repeatedly. In other words, if each group had a ball with the same mass, the results could've been compared for both precision and accuracy, and //then// different masses could be used to determine whether or not the weight of the ball wold change the acceleration.

=Newton's Laws Lab= 12/2/11 Garrett-A Lauren-B,D Ryan-C Period 4/5

__**Objectives:**__ What is the relationship between system mass, acceleration, and net force? (1. Determine the relationship between system mass and acceleration; 2. Determine the relationship between acceleration and net force). Graphs will be used to test these relationships. __**Hypothesis:**__ 1.) As the acceleration increases, so will the net force, since they are directly proportional. 2.) As the acceleration increases, the mass will decrease, since it is inversely proportional to the acceleration. __**Method and Materials:**__ Set up the "Drop and pull" system by placing a track on a desk (flat surface) and tightly clasped the pulley onto the surface using a table clamp. Then, place the cart on the track with a string attached it it; on the other end of the string hang a second, lighter mass. Before obtaining the data, record the weight of the cart and each of the masses, including the round masses and the long log-shaped bar masses. For the first part of the lab, place different masses onto the hanging mass, which ultimately changes the net force; but, don't change the mass of the cart, it remains constant. Perform several trials (a minimum of three) to assure precision for each new mass. Using a USB link and Photogate timer, measure the acceleration, which appears on a graph on Data Studio. For the second part of the lab, add mass (the long bars) onto the cart, which changes its mass, while the weight of the hanging mass (net force) remains the same. Complete several trials per each new mass to make sure the results are consistent. Use the same use of technology (Data Studio) to obtain the acceleration. __**Equipment Setup:**__ __**Procedure (video):**__ Part 1: Acceleration vs. Net Force media type="file" key="part 1- a vs ef.mov" width="330" height="330" Part 2: Acceleration vs. Mass media type="file" key="part 2- a vs m.mov" width="330" height="330" __**Data and Graphs:**__ __Acceleration vs. Net Force__ In the data above, the total mass, the hanging mass, the cart mass, the net force, the trial number, the experimental acceleration, the average acceleration, and the theoretical acceleration are all shown. In this particular case, the total mass is the same because in we are trying to keep the mass constant for the entire **system**. This means that we can alternate the cart and hanging masses as long as the total mass remains the same. Keeping the system mass the same will allow us to compare the net force with the acceleration. This is exactly what we did in order to create the graph that was made above.

__Acceleration vs. Mass__ In the data above, the total mass, the hanging mass, the cart mass, the net force, the trial number, the experimental acceleration, the average acceleration, and the theoretical acceleration are all shown. The net force is constant in this set of data so that we can see the correlation between mass and acceleration. Essentially, the net force is the constant and the mass (total mass) would be the variable.

Link to Excel Spreadsheet -

__**Sample Calculations:**__ __Acceleration vs. Net Force__ Percent Error for Acceleration (Trials 10-12)= [(Theoretical-Actual)/Theoretical]*100 % Error= [(.555-.501)/.555]*100 % Error= 9.74%

Percent Error for Slope (Graph)= [(Theoretical-Actual)/Theoretical]*100 % Error= [(1.89-1.7621)/1.89]*100 % Error= 6.61%

__Acceleration vs. Mass__ Percent Error of Exponent of "x"= [(Theoretical-Actual)/Theoretical]*100 % Error= [(

Percent Error of "x" Coefficient= [(Theoretical-Actual)/Theoretical]*100 % Error= [(

__Net Force Calculation (Trials 10-12)= mass*acceleration:__ ∑F= .030*9.8 ∑F=.294 N

__Average Acceleration:__

__**Analysis:**__ 1.) The slope of the Acceleration vs. Net Force trend line is linear, and its value is 1.7621. This slope corresponds to the acceleration of the system. The slope of the graph (acceleration) can be obtained by doing the inverse of the mass of the system, or 1/m. This can be concluded by taking the equation ∑F=m*a and divide both sides by "m" and obtain (∑F)/m=a, or a=(1/m)*∑F. For our experiment, the mass of our system is .530 kg. Therefore, to find the theoretical slope of our graph, you would divide 1 by .530 and obtain the answer of 1.89. To find the percent error of our slope, you would then do [(theoretical-actual)/theoretical]*100, so for our slope the percent error would be [(1.89-1.7621)/1.89]*100=6.61% error. The y-intercept value is friction divided by the mass of the system, so to find the friction force of the system, you would take the y-intercept and multiply it by the system mass. However, friction is generally ignored, so this does not matter in our experiment.

2.) If friction is calculated into the experiment, the value for acceleration will go down, and you will need a larger force to create the same acceleration. These observations can be seen in this equation if force was accounted for: a=(1/m)*∑F, where ∑F is now the force of the hanging weight minus friction. As a result, our equation in accordance to the graph becomes a=(9.8*.03)(1/.530)-(.0146/.530)=.527 m/s 2 . Using this information, we can conclude that the slope we have on our graph for Acceleration vs. Net Force is too big. Also, this shows that friction can be a source of error in our experiment, because without friction being calculated into our experiment, our calculated acceleration becomes too big.

__**Conclusion:**__ Our first hypothesis was supported by our results; we believed that as the acceleration increased, so would the net force. Our results from this lab prove this to be correct, for as apparent on the acceleration versus net force graph, the line of best fit has a positive slope, and as the acceleration gets larger, so does the magnitude of the net force. Our second hypothesis stated that as the acceleration increases, the mass (of the cart) will decrease. This too was proven accurate when we collected our results, for on our graph one can see the parabolic line of best fit, which decreases from left to right; in other words, as the acceleration gets smaller, the mass gets larger. Both of these hypotheses were based off the equation: net force= mass x acceleration, ∑F=m*a, which shows that acceleration and net force are //directly// proportional, whereas acceleration and mass are //inversely// proportional. This explains why as the acceleration increases, the net force does, but the mass does not.

Four different percent errors were calculated for this lab, ranging from a low 6% error, to a high of 64% error. The percent error between the actual acceleration and the experimental acceleration was about 20.9%. In the acceleration versus net force graph, we calculated the slope percent error, which came out to be 6.89%. The equation for this graph is in the standard y=mx+b form; when applied to this lab, the equation means a=(1/total mass)*(hanging mass*g). In the perfect situation, the acceleration would have come out to be .0924 m/s 2, albeit with 20.9% error, it was .0732 m/s 2. The theoretical slope was 1.89, however our experimental value was 1.76, which gave us the 6.61% error. In the second graph, acceleration versus mass, the slope, or value of "A", without any error would equal the net force; ours, however, did not equal the net force, for it had an extremely high percent error of 64%. Lastly, when comparing the exponential value of "B", which would ideally equal -1, to that of our graph, we had a percent error of 11%. There are several factors that could have led to these errors, the first of which relates to the cart's acceleration. It's possible that when letting the car go, an outside force (such as a hand) pushed it a little bit, giving it an increased acceleration than it would have had on its own. Therefore, the magnitude of the acceleration may not be completely accurate. Furthermore, when the cart's data appeared on Data Studio, we highlighted and only used the middle section of the line, disregarding the very beginning and very end in case there was an issue with the start or finish; however, we could have left __off__ too much, or left __on__ too much, which would have impacted our results and increased the percent error, as well. If one were to redo this experiment in the future, there are several mistakes that can be avoided. If a longer track were used, there would be more points on the Data Studio graph to represent acceleration, which would enable us to have much more solid, concrete results. Also, we only performed each trial three times; although three times is better than just one, doing at least five trials would have enabled us to have more data to incorporate into our results and analysis, and most likely give more accurate results.

This lab is extremely applicable to everyday life. Pulley systems, particularly when it comes to construction, are essential towards safety. For example, when in an elevator, there is almost always a sign that states the maximum weight capacity. This is because if the weight of the elevator (it's net force) is too large, it will cause an extremely low acceleration, since the two are inversely proportional, and thus it will be unable to move (safely). This concept was tested in this lab because different masses were added onto the hanging mass, and as a result changed the overall acceleration. Due to the close relationship between a hanging mass and acceleration, builders, engineers, and other construction workers must consider the physics behind their work plans before putting them into real life situations.

=Coefficient of Friction Lab= Ryan- task A Garrett- task B and D Lauren- task C 12/5/11

__**Objective:**__
 * 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

We hypothesize that the coefficient of static and kinetic friction will be between 0 and 1. Also, we hypothesize that the the coefficient of friction will increase as mass increases because they are directly related. Finally, we hypothesize that as the as the normal force increases, the friction force will increase as well.
 * __Hypotheses:__**

__**Methods and Materials:**__ To begin, we set up a track in which a cart was attached to a string. We were then able to add masses to the cart. The masses were used to see differences in the tension force between trials. Then we hooked a force meter to the string and put the USB link into our computer to measure the tension. We had to create an experiment using Data Studio to track the Tension. We had to change the setting on the y-axis to Force-Pull positive first in order to get accurate results. We began trials with different masses. We pressed the zero on the sensor so that the experiment would start at zero on the y axis. We pulled the string parallel to the track, moving the cart at a constant speed. We then recorded the maximum tension (which is the high point of the graph) for static friction and the mean (which is the leveled portion of the graph after the maximum tension) to get the kinetic friction.

__**Procedure (videos):**__ media type="file" key="7.mov" width="300" height="300"

media type="file" key="7 video 2.mov" width="300" height="300" __**Free Body and Motion Diagrams:**__ __**Data Table:**__ __**Graph:**__

__**Excel Link:**__

__**Calculation Derivations:**__

__**Sample Calculations:**__ Average Tension- Solving for µ- Percent Difference- Static: (class average= .156) Kinetic: (class average= .110)

__**Discussion Questions:**__
 * 1) Why does the slope of the line equal the coefficient of friction? Show this derivation.
 * 2) The slope of the line on the graph can be represented by the equation y=µ*N. The "y" value is the friction force, and the "x" value is the normal force. Therefore, the equation can be turned into the standard y=mx+b form. The "m", or slope, variable, therefore is represented by µ, as derived from the equation above.
 * 3) Look up the coefficient of friction between your material and the aluminum track. Discuss if your measured results fall within the range of theoretical values.
 * 4) According to [|this online source], the coefficient of static friction between plastic (form the cart) and metal (from the ramp) is approximately .300 (ranges from .200-.400). Our value, however, was .140, which is half the magnitude of the actual one. This website also states that the coefficient for kinetic friction between these two materials is about .200 (ranges from .100-.300). Similarly to our value for the coefficient of static friction, our value for kinetic friction was also much lower than the actual value. This could be due to possible errors within our lab performance, yet our results still seem plausible and not too far from the actual ones.
 * 5) What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction?
 * 6) The magnitude of the force of friction was affected by both the weight of the cart (which is equal to the normal force) and the coefficient of friction (µ). The magnitude of the coefficient of friction was affected by the force of friction, and also by the weight of the cart.
 * 7) How does the value of coefficient of kinetic friction compare to the value for the same material's coefficient of static friction?
 * 8) The value of coefficient of kinetic friction (.111) is smaller than the value of the same material's coefficient for static friction (.140). This makes sense because a greater force is required to //prevent// something from sliding (static), for example, than to allow it to slide down (kinetic). This is why the magnitude of the coefficient of static friction is of a higher value than that of kinetic friction. The results from the website linked above support these results.

__**Conclusion:**__

Our hypotheses at the beginning of the lab were that the coefficient of kinetic friction would be between 0 and 1, that the coefficient of friction will increase as mass increases, and that as normal force increases, so will the friction force. To determine whether our hypotheses were correct or incorrect we performed an experiment in which we measured the maximum and mean tension from pulling a mass with a string. The tension that we measured was equivalent to the static and kinetic frictions because acceleration was 0. If we set up the equation, it becomes T-f=m*a or T-f=0. The answer then becomes T=f. After carry out multiple trials with different masses, we created a static friction vs. normal graph and a tension vs. normal graph. The linear graph that resulted demonstrates that as normal force increases, so does the friction force. This proves the third hypothesis to be correct. Our coefficients of static and kinetic friction (.140 and .111 respectively) proves our first hypothesis correct. Finally, our results indicate that a grater mass requires more tension, resulting in a greater value of friction. This proves our second hypothesis.

Though we did have correct hypotheses, we witnessed some error in our experiment. When we compare our static and kinetic friction values to that of the classes with percent error, we notice that we have a 10.3% difference for static friction and a .91% difference on kinetic friction. One potential source of error during this experiment was that we could have angled the string. This would have provided tension. This is because the cart should be moved at an ideal angle of 0 degrees. Also, the pull may not have been steady moving the cart at a non constant speed. We can fix these errors by having a machine drag the string straight and at a constant speed because the human hand may not be that reliable.

The results that we got from this lab can be applied to that of the real world as well. The lab allowed us to calculate the coefficient of friction between two surfaces. This can be applied to any two surfaces that touch. For example, a the bottom of a sled sliding down a hill.