Group1_4_ch4

Group 1: Robert Kwark, Kosuke Seki, Noah Pardes Period 4 | Date Assigned: 11/18/11 | Date Due: 11/21/11toc =Gravity and the Laws of Motion= Task A: Noah Pardes Task B: Kosuke Seki Task C: Robert Kwark


 * 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 effect of mass on acceleration down the incline.


 * Hypothesis:** We believe that the acceleration due to gravity is -9.8m/s/s. This is the value that we have always used in our physics class to represent the acceleration due to gravity. By doing this experiment, we will be able to understand why this is so. Our acceleration, however, will be directly proportional to the incline angle. As the incline angle increases, there is less normal force to counteract the effect of gravity, making our acceleration faster. By doing this, we will be able to graph this relationship and use this data to extrapolate the effects on a much larger scale. Acceleration will increase with increases in mass. The force of friction will have a higher effect on less mass; therefore, smaller masses will have a lower acceleration due to friction while larger masses will have a little bit more acceleration b/c they aren't affected as much.


 * Materials:** The materials we used were metal balls, a balance, a ramp, a stopwatch, a ruler, and a protractor.

media type="file" key="Movie on 2011-11-18 at 12.03.mov" width="300" height="300"
 * Method/Materials:** Set up a ramp to a height of 15 centimeters, using a ruler in order to get an exact height. Drop a .66kg metal ball from the top of the ramp and time how long it took to get to the bottom with a stopwatch. Try to be as synchronized as possible by having the person drop the ball in one hand and start the timer in the other. When the ball hit the bottom, the person must stop the timer, getting an exact time for the ball's descent. Repeat this process a total of five times to get consistent results and to try to negate the effects of human error. Next, Change the height of the ramp from .15m (15cm), to .20m (20cm), to .30m (30cm). to .40m (40cm), and finally to .60m (60cm). At each unique height, Drop the ball five times and use a stopwatch to get the exact time of the ball's journey down the ramp.

Mass of ball: 0.066 kg __Overall Experimental Data__
 * Data:**

__Acceleration of Ball vs. Angle of Ramp (Sinø)__

//Slope: y = 8.5508x - 0.2893// The slope of this graph is the acceleration of the ball due to gravity. In an ideal world, the acceleration due to gravity is 9.8m/s^2. However, as we can see from our slope, our acceleration due to gravity was 8.55 m/s^2. This is due to human error, such as difference in timing and friction.
 * Analysis of Graph:**

//r^2 = .996// From this, we can tell that our data was very accurate, because our r^2 value is 99.6% accurate. The only reason it is not perfect is because of very minute errors.

Average Time (Trial 1):
 * Sample Calculations:**

To find Acceleration: 1.66 seconds at a height of .15m (Trial 1)

Average Acceleration (Trial 1)

To find sinø: 1.18m long ramp and a height of .15m (Trial 1) sinø=.1271

Percent Error: Percent Difference from Class:

__Class Data__ Class Avg : 8.218 m/s^2

This diagram shows the forces acting upon the metal ball as it rolls down the incline. The force of gravity is acting upon the ball to cause it to roll down the ramp. It is not the whole force, but just a part of it; the x-component of the force of gravity. This is because the y-component is negated by the normal force. Friction is also present, but plays a minor role, as gravity is a much stronger force here.
 * Free-body Diagram:**

Trial 1 (at .15m high) This value was somewhat close to our experimental value of 0.844 m/^2 at a height of .15m.

1. Is the velocity for each ramp angle constant? How do you know? No, the velocity for each ramp angle changes. We know this because we found that there was an acceleration through our calculations. An acceleration is, by definition, the rate of change of velocity per unit of time (seconds, in our case), which means that the velocity was changing, not constant. Also, the initial velocity of the ball at the top of the ramp was 0 m/s; if it had constant velocity, it would have stayed at 0 m/s. This is obviously not the case, as the ball noticeably gained sped. 2. Is the acceleration for each ramp angle constant? How do you know? The acceleration is constant at each ramp angle because we did 5 tries for each ramp angle, and they all resulted in more or less the same acceleration. However, for different ramp angles, the acceleration was not constant; the steeper the incline, the higher the acceleration. 3. What is another way that we could have found the acceleration of the ball down the ramp? We could have used a radar gun to determine the speed of the ball at the bottom of the ramp to find final velocity, and also timed it at the same time. Then we could have used the formula a=(V f -V i )/t or the formula Vf^2 = Vi^2 + 2ad 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? Galileo would have kept increasing the incline and rolling the ball down the incline until the incline was vertical. Each time he rolled it, he would have calculated the acceleration, and could have extrapolated that information to when the degree of the incline was 90º, which is basically freefall. 5. Does the mass of an object affect its rate of acceleration down the ramp? Should it affect the motion of an object in freefall in the same manner? Yes, the mass does affect the rate of acceleration down the ramp. The class data shows a correlation between mass and acceleration. This is because friction has a larger effect on smaller objects. Mass would not affect the motion of an object in freefall at all, since it would have no other forces other than gravity exerting force on it. When we try to find acceleration due to gravity, mass becomes irrelevant, and thus, it has no affect on the force of gravity.
 * Discussion Questions:**

Though our experiment, we found that the acceleration due to gravity was 8.55 m/s^2, which was very close to our hypothesis. We assumed wrongly that the acceleration of gravity would be 9.8 m/s^2 because we did not take into account the force of friction working against the ball. Our second goal, finding the relationship between acceleration and incline angle, was found to be true. As we made the angle larger, the acceleration also became much larger due to the larger x component of gravitational force. This is supported by our data. The mass also had an effect on our acceleration. In general, as our balls became more massive, the effect of friction didn't affect them as much as the smaller balls. Thus, the acceleration got larger when the masses of the balls became bigger. However, overall, mass did not play a large roll in acceleration, as we learned in the last unit with freefall. We were also able to use a graph in order to extrapolate more extreme cases. As the sin theta gets larger, the acceleration gets larger. Thus, larger angles would produce faster accelerations. Our percent difference from the class was very small, a mere 4%. This shows that we were very close to the average of the class, which means that many of us received similar results; this consistency mainly shows that friction was present and lowered our acceleration due to gravity, which would have been very close had friction not been present. We achieved a percent error of 12.8%, which is relatively good. Most of this error occurred b/c we did not account for friction, which would have a significant impact on accelerating bodies. However, there were also many sources of error in our data. Our timing was not perfect, though we did get close. We did not take very accurate measurements, and assumed that the data on the ramp was correct. Thus, our measurements were off by about a centimeter or two. We could have fixed this by actually measuring the distance. In addition, when we placed the ball on the ramp, sometimes the dropper may have added a bit of force to the ball, making the time it took to go down the ramp decrease. We could fix the timing error by using a digitized timing device at the end of the ramp to correctly measure time. In order to decrease friction, we could have bought a ramp with lower friction.
 * Conclusion:**

=Newton's Second Law Lab (12/2/11)= A: Kosuke B: Robert C: Noah D: All

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

Hypothesis: We believe that with a higher net force, there will be a higher acceleration of the system. Also, with a higher mass, there will be a lower acceleration of the system. Basically, acceleration is directly proportional to net force and inversely proportional to mass.

Materials: Use a Dynamics Cart with Mass, Dynamics Cart, track, Photogate timer, USB link, Data studio, Super Pulley with table clamp, String, Mass hanger and masses, and a Mass balance.

Procedure:

Attach the photogate to the pulley, and then plug the USB link to your computer. Open Data Studio, and select “Digital Input”. Remove the position-time and acceleration-time displays. We will use the velocity-time graph, and use the slope to find acceleration. Attach one end of the string to the cart, and the other to the mass hanger. Place the cart on the track, and hang the string over the pulley so that the mass hanger attached to the string is hanging freely. Make sure that the track is level and that the string is parallel to the track.

To test acceleration vs. Net force, start off with 25g (two 10g plates, one 5g plate) on the cart and just the hanger (which weighs 5g). Press record on Datastudio, and let the cart go; make sure to stop it before the hanger reaches the ground or the cart hits the pulley system. Do this three more times and record the slopes of the lines as well as the masses. Then do this process again 4 more times with the 5g plate on the hanger, 10 g plate on the hanger, 20g (two plates) plate on the hanger, and 25g plate (all three plates). Use the plates from the cart and put them on the hanger (so when you put 25g on the hanger, you would have no additional mass on the cart).

To test acceleration vs. Mass, we must keep the net force constant, meaning that the weight on the hanger must be the same for all trials. So we put all three blocks on top of the cart (one 495 g, one 493 g, one 250 g). Then put two 20g plates on the hanger (for 25g total). Press record on Datastudio, and let the cart go. Record the slopes of the lines and the mass of the system. Take the 250 g block off, do the experiment, then record the results. Take off the 493 g block off, record the results. Then switch the 495 g block with the 250g block, and take results. Finally, take the 250g block off the cart cart (so there are no weights on the cart) and record the results.

Video of Accel vs. Net force: media type="file" key="Movie on 2011-12-02 at 11.48.mov" width="300" height="300" Video of Accel vs. mass: media type="file" key="Movie on 2011-12-02 at 11.48.mov" width="300" height="300"

Data: This graph shows the correlation between net force and acceleration. As net force increased, it is clear that acceleration also increased. Therefore, it is directly proportional. This cart shows the relationship between the average acceleration and the mass of the cart. As the mass of the cart increased, the average acceleration decreased. Thus, the relationship is inversely proportional.

Graphs: Document:

Sample Calculations: __Net Force__ __Theoretical Acceleration__ __% Error__
 * Note: All calculations are based on Set 1 of the Net Force vs. Acceleration Graph*

Analysis: 1) Theoretical Acceleration (as seen above): Percent Error between Theoretical and Experimental acceleration (as also seen above): 2) The slope of the Acceleration vs. Net Force trend line is linear, and its value is 1.78. This slope corresponds to the acceleration of the system. The slope of the graph can be obtained by finding 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 (1/m)*∑F=a. For our experiment, the mass of our system is .537 kg. Therefore, to find the theoretical slope of our graph, you would divide 1 by .537 and obtain the answer of 1.86. To find the percent error of our slope, you use the percent error equation seen above. So for our slope the percent error would be [(1.86-1.78)/1.86]*100=4.3% error. The y-intercept value represents friction divided by the mass of the system. From this, we can find the friction force of the system by taking the y-intercept and multiplying it by the system mass. However, friction is generally ignored, so this does not matter in our experiment.

3) This graph is hyperbolic. As the x increases, the y decreases. However, the rate at which it decreases is decreasing as time goes on. As long as the net force is the same throughout, the mass is inversely proportional to the acceleration. Since we get the inverse of the mass, the power on the x should be -1. That makes sense when going back to Newton’s Second Law equation, making mass x and acceleration y: Therefore, the coefficient in front of the x is the net force. Since we’ve discussed earlier that the net force is the weight of the hanging mass, and that it is the same in this experiment, we can find percent error between the weight of the actual and theoretical weights. The actual weight was 0.025, and the graph tells us it was 0.1919. The percent error for the A coefficient is: This is a HUGE percent error, so it is clear that there are other sources of error acting on our experiment.

4) Friction should decrease the acceleration. A larger force is needed to restore the acceleration. This makes sense because the theoretical acceleration values we had were significantly higher than the actual. That is because in the theoretical, friction, which decreases acceleration, is ignored. With this in mind, the theoretical values of acceleration should change when friction is added. With this information, we can conclude that the slope on our Acceleration vs. Net Force graph is much too large. Also, this shows that friction can be a huge source of error in our experiment because without accounting for friction in the experiment, our calculated acceleration would be much larger than it should be.

Conclusion: After completing this lab, our hypothesis was proven to be correct. Through the use of a pulley system, we first tested the situation where the mass of the system, which included both the mass of the cart and the hanging mass, were kept constant. After tests, it was proven that with more mass on the hanging mass, the acceleration would be greater, despite the mass of the system remaining the same. The only thing acting on the hanging mass was the force of weight (m*g). Therefore as more mass is added to the hanging mass, the net force of the system increased. As the force increased, the acceleration did as well, proving that force is directly proportionate to acceleration. The graph for this part of the lab behaves as we had predicted in our hypothesis. This means that it shows the relationship y = x, being a straight line with a positive slope. On the other hand, in our second pulley situation, we tested what would happen if the mass of the hanging mass stayed constant, and only the mass of the cart changed. The result of this was as we predicted in our hypothesis, that as the mass of the cart decreased, the acceleration increased. Therefore, as the mass decreases, the acceleration increases and vice versa. The graph for this part of the lab shows the relationship y = 1/x and is a parabolic line with a negative slope. However, the r^2 values for both graphs were quite high, being in the 99th percentile. The acceleration vs. net force graph had a r^2 value of 0.99996, which is almost perfect, while the acceleration vs. mass graph had an r^2 value of 0.99593, which is still very good, but not as good as our other graph.

Our lab overall was pretty not as accurate as it could have been. When calculating the average accelerations to the theoretical accelerations in the first demonstration, our percent errors sometimes got as high as 32%, while also being as low as only 9%. This was because the force of friction would have a larger affect on less massive objects; in general, as the mass increased, the percent error decreased. However, some aspects of our lab were more accurate. For example, in our mass v. acceleration graph, our data points were quite on target sicne the slope of our quadratic line read y = 0.1919x^(-1.085). Ideally, the exponent should be -1 because the slope is the inverse of the total mass. There was a 8.5% error between our -b value and the actual -b value. Although this is still relatively high, it is not a very high percent error compared to the rest of our results. There are several factors that could have led to errors in the lab, the first of which relates to the cart's acceleration. It is possible that when letting the car go, an outside force (such as someone's hand) pushed it slightly. This would give it an increased acceleration than it would not 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 taken off too much, or taken on too much, both of which would have impacted our results and the percent error. If we were to redo this experiment in the future, there are several mistakes that can be avoided. To start, 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 more concrete and reliable results. Also, we only performed each trial three times. Even though three times is better than just one, doing at least five trials would have been ideal, as it would have enabled us to have more data to incorporate into our results and analysis, and most likely have give more accurate results.

The pulley system scenario is very applicable in everyday life. An example of pulleys we see everyday is at the gym. More specifically, with exercise equipment. The more mass being lifted, the greater the weight force is on the weights. This means that more force is needed to lift these masses. Also, this is important to know for safety reasons in the gym. The smaller the mass, the faster it will accelerate. So someone who is working with lighter weights should be careful that they do not simply let go of the weights, as they will come crashing down at a fast acceleration.

=Coefficient of Friction= Part A: Robert Kwark Part B: Noah Pardes Part C: Kouske Seki Part D: All

What is the coefficient of static friction between surfaces? What is the coefficient of kinetic friction between surfaces? What is the relationship between the friction force and the normal force.
 * Objective:**

Both of the coefficients should be values from 0 to 1. We also know that the static friction coefficient will be higher than the kinetic. Because of the smooth surfaces involved, we hypothesize that both are between .1 and .4. Based on knowledge of the coefficient of friction, the relationship between the friction force and the normal force as proven through this lab should be f = µ * N
 * Hypothesis:**

First, find the mass of the cart. Attach a 15 cm string between a cart and a block on one side. Then, attach a 15 cm string to the force meter on the other side. Connect the force meter, via USB port, to a computer. Data studio will then automatically record your data. Once all of that is set up, pull the block with the force sensor. Try to have as constant of a speed as possible, and make sure the string is horizontal.
 * Materials/Method:**

__Static Friction Chart__ __Kinetic Friction Chart__
 * Data:**



__Excel Spread Sheet__


 * Analysis:**

__FBD__

__Static vs. Kinetic Friction__ //Class Average:// Static = 0.161 Kinetic = 0.113
 * The class averages for the coefficient of static friction and kinetic friction were .161 and .113, respectively (as seen above). Ours, however, were a little bit different. Our coefficient for static fiction was about .135, so our percent difference is about 16.15%, which is relatively large. However, the coefficient for kinetic friction was much closer. With a calculated coefficient of about .112, our percent difference there was .885%.

//Percent Difference (static friction):// //Percent Difference (kinetic friction)://

__Sample Calculations:__ **[We should also have sample calculations for friction and coefficient of friction]** (m = 1.085 kg)

//Normal force:// //Average Max Tension:// __Discussion Questions:__ 1) //Why does the slope of the line equal the coefficient of friction? Show this derivation.// 2) //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. Be sure to cite your source!// 3) //What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction?// 4) //What variables affected the magnitude of the coefficient of friction?//
 * The coefficient of static friction between plastic and the metal track was .25 to .4. While our static friction was .135, the class average was not much higher than that, being at .161. The kinetic friction is supposed to be a range from .1 to .3. Our kinetic friction coefficient, at .112, lies in this range. (http://www.tribology-abc.com/abc/cof.htm)
 * The mass of the cart affected the force of friction. This is because friction is dependent upon the product of mu and the normal force. The normal force is, of course, dependent on the mass because it is the force the surface needs to push back to compensate for the weight of the object in question. The coefficient of friction is also dependent on the weight and the magnitude of the friction force itself.
 * The coefficient of kinetic friction is smaller than the coefficient for static friction. This is because with static friction, you have to overcome the object's inertia in order to set it into motion, as opposed to kinetic friction, where it is already in motion.

__Conclusion__: Through our experiment, we have shown that our hypothesis was correct. We observed that the static friction coefficient was higher than the kinetic, as well as it being between the range of .1 and .4. The static friction coefficient was .135, while our kinetic was at a lower .112. Our percent difference was variable between the two. It ranged from very close to somewhat. The static friction's percent difference was 16.15%, while the kinetic friction's percent difference was a mere .885%. This shows that our data is accurate. However, despite accurate, there are measures that we could have taken to be even more precise. The force needed to start moving the block was completely unknown to us while pulling, so it was easy to overestimate how much force was needed in order to accelerate it.This skewed our max tension, and thus, our static friction coefficient. It could explain why our value was lower than the class average. When pulling, the string wasn't perfectly parallel to the metal ramp, and thus, we didn't get a perfect measure of the tension on the string. (add more later). We could have started pulling with a very small force and gradually increased the pull until movement was achieved. That way, we could have had much more accurate results. For issue of parallel, we could have used a laser as a guide in order to get a more accurate angle. That way, some of the force wouldn't act on the y axis, which would be both wasteful and inefficient.