Group5_4_ch4

Group 5: Gabby Leibowitz, Maxx Grunfeld, Tim Hwangtoc

=**__Gravity and the Laws of Motion__** = Task A: Tim Hwang Task B: Gabby Leibowitz Task C: Maxx Grunfeld Task D: Gabby Leibowitz

**__Purpose with Rationale__**

1. Find the value of acceleration due to gravity 2. Determine the relationship between acceleration and incline angle 3. Use a graph to extrapolate extreme cases that cannot be measured directly in the lab 4. Determine effect of mass on acceleration down the incline
 * Objective: **

**Hypothesis:** It is hypothesized that the acceleration due to gravity is -9.8 m/s/s without air resistance because it will never change in value as previously experimented and determined in a free-fall lab. In addition, the steeper the incline, the more the object will accelerate because it mirrors a free-fall situation. Finally, the acceleration of an object depends inversely upon the mass. This is because the equation of a net force is mass times acceleration, so the larger the mass the smaller the acceleration and vice versa.

__**Materials and Methods** __ In this experiment, one begins by setting up a ramp with a height of 15 cm. One can measure the height of the ramp by using a ruler and measure the angle by using a protractor. Before one starts the experiment, the mass of the ball is found using an electric balance. Our particular ball had a mass of .535 kg. After the mass is identified, one member of the group releases the ball from the top of the ramp while another member uses a stopwatch to determine how many seconds it takes for the ball to roll to the bottom of the ramp. This step is repeated five different times in order to get an accurate average. After this, the height is changed in order to illustrate the relationship between the acceleration of the ball and the incline angle of the ramp. Our group used the heights of 25 cm, 44 cm, 8 cm, and 35 cm in order to get a good idea of how the ball will accelerate with an extremely steep incline and then an extremely low incline. Each time the height is changed, one drops the ball and times how long it takes to reach the bottom of the ramp. For each height, this step is repeated five times. After this data is recorded, by use of kinematics, one can determine the acceleration of the ball at each particular height.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Picture of our lab set up at two different heights

media type="file" key="MOVIE.mov" width="300" height="300" <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Video of lab

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Observation and Data Table** __ <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">This data shows the multiple trials our group performed while conducting this experiment. It is shown that our group chose 5 different heights to set the ramp at, and then tested rolling the ball down the ramp 5 times per each height. We recorded the time it took the ball to reach the bottom of the ramp for each trial and then, by the use of kinematics equations, found the acceleration per each trial, and eventually the average acceleration for each height. The average acceleration provided us with a fairly accurate estimate of what the acceleration would be at each height. Finally, we were able to find the angle by use of sin theta calculations.

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Graph** __ __<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Analysis** __ <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">As shown in our graph, our experimental value of our ball's acceleration due to gravity was 8.139 m/s/s. Our equation is a representation of acceleration vs. sin of theta. The theoretical value of the acceleration due to gravity is 9.8 m/s/s. The reason why we did not set our y-intercept equal to zero is because gravity was not the sole force acting upon the ball; friction was involved. We calculated a percent error of 17% between our experimental and theoretical values. We also calculated a percent difference of 23.3% from our class average, possibly due to the friction present, human error, or inaccurate measures of height. Our R^2 value is 0.969, proving that our results were extremely accurate and only slightly off, a result of minute errors that occurred during our procedure, such as the present force of friction or simply human error. <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">This is a free body diagram representing our experiment. The two forces acting on the ball are weight and normal force. Weight always points downward toward the center of the Earth. The normal force acting on the ball is the push upward from the ramp below the ball. The x-component of weight causes the ball to roll down the ramp; the y-component is balanced out with the normal force.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">In our calculations of average acceleration using the kinematics equation, we received an answer of 1.067 m/s/s. However, when we did it using Newton's Second Law of Motion, or the equation F = ma, we obtained results of 2.45 m/s/s. Although they are not the same, they are relatively close, showing that we were not too far off in the accuracy of our results.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**__Class Data Table__** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">The class data illustrates that, regardless of the masses, everyone had similar acceleration values, close to the theoretical acceleration due to gravity value of .988 m/s/s. This proves that mass does not affect the acceleration of an object.

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Sample Calculations** __ <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">(With a height of .15 m)

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Average Time <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: 0px; overflow-x: hidden; overflow-y: hidden;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Acceleration

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: medium;">Average Acceleration <span style="font-family: 'Times New Roman',Times,serif;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Final Velocity <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Sin of Theta <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Percent Error <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: medium;">Percent Difference: <span style="font-family: 'Times New Roman',Times,serif; font-size: medium;">Percent Difference = ((average experimental-experimental)/average experimental)*100 <span style="font-family: 'Times New Roman',Times,serif; font-size: medium;">Percent Difference = ((8.21-8.14)/8.21)*100 Percent Difference = .85% __**<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Discussion Questions **__
 * 1) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Is the velocity for each ramp angle constant? How do you know?** The velocity for each ramp angle is not constant. This is because as the ball moved down the ramp at each given angle, it accelerated and was therefore changing its velocity.
 * 2) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Is the acceleration for each ramp angle constant? How do you know?** The acceleration for each ramp angle is not constant either. We know this because for each given angle of the ramp, the ball’s acceleration changed. We noticed that the greater the angle, the greater the acceleration.
 * 3) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**What is another way that we could have found the acceleration of the ball down the ramp?** Another way we could have found the acceleration of the ball down the ramp is by using the formula for acceleration as opposed to the kinematics equation we used. We would have needed to find final velocity first, and then we could have just as easily found our acceleration.
 * 4) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**How was is 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 may have experimented similarly to how we did, calculating the acceleration at several different heights. Because the angle of the ramp’s incline could not be greater than 90 degrees, his calculations must have matched accordingly and he saw that the acceleration never was higher than 9.8 m/s/s.
 * 5) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**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?** No, the mass of an object does not affect its rate of acceleration down the ramp. There is no real correlation between the ball’s mass and acceleration, as shown in the class data. In freefall, gravity is the only force acting upon the object, so the acceleration always remains the same at 9.8 m/s/s. Therefore, the mass of the object does not affect its acceleration at all.

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Conclusion** __ <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">We hypothesized that the acceleration due to gravity would be 9.8m/s^2 as a result of the fact that throughout the year, we have proved this through experimentation of objects in free-fall multiple times. After performing this experiment 5 times, each time with a different height, our acceleration due to gravity was 8.138m/s^2, a value very close to our predicted value of 9.8m/s/s. In addition, we believed that there was a relationship between acceleration and incline angle. It was hypothesized that the steeper the incline, the more the object will accelerate because it mirrors a free-fall situation. This proved accurate as well, for the closer our angle got to 90 degrees, the more the object accelerated and the faster it took for the ball to reach the bottom of the ramp. For example, at an angle (sin theta) of .125 degrees and a height of .15 m, the acceleration was 1.038 m/s^2, but at an angle of .367 degrees and a height of .44m, the acceleration was 3.13 m/s^2. Our final hypothesis claimed that the acceleration of an object inversely depended on the mass, using the equation ∑F=ma as support. We reasoned that this shows that the larger the mass, the small the acceleration, and vise versa. However, it was evident through the data collected by our class, who each performed this experiment with a ball of different masses, that mass didn't have an impact on the acceleration as we hypothesized it would. Everyone in class had a relatively similar acceleration, regardless of the mass of their ball. This is because acceleration due to gravity will theoretically //always// be 9.8m/s^2 in free-fall. Our ball had a mass of .535 kg and had a 1.04 m/s^2 acceleration at 15 cm, while a group with a mass of .085 kg and had a .840 m/s^2 acceleration at 15 cm, very close in difference. Furthermore, the average was .84 m/s^2, a value very close to .98 m/s^2. Any difference that did exist in acceleration between the groups is mostly a result of friction present.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">It is this friction present in the experiment which accounts for the sources of error present within our experiment. Our group calculated a percent error of 17%. This percent error is largely a result of the fact that friction existed between the ball and the incline, causing the ball to reach the bottom of the ramp in a greater amount of time than it would under frictionless circumstances. This causes the acceleration to be less than 9.8 m/s/s. Furthermore, it was merely human error that got in the way of maintaining perfect results. It is natural for the person dropping the ball to add additional force by accidentally pushing the ball down the incline, or for the person timing the ball's fall to be a few seconds off, all leading to errors in the acceleration value. Finally, if the ball did not follow a straight path and hit the side of the ramp, which occurred often in our experiment, the acceleration would have decreased for it would have taken longer for the ball to reach the bottom. There are ways in which these sources of error could be prevented. If additional materials were available, a ramp which provides less friction would have presented an acceleration value closer to the expected 9.8 m/s/s, for friction wouldn't have altered the results to such an extent. The only way to limit the human error that interfered with our results would be for the ball to be released on the ramp by use of machine rather than a group member, and for a computer to time when the ball reached the bottom of the ramp rather than a group member pressing a stop watch. However, these two changes aren't very realistic for a in-class lab. Finally, a wider ramp would have reduced the problem our group faced because the ball wouldn't have hit the side of the ramp and therefore, a more accurate time and ultimate acceleration value would be reached. A relevant real-life situation to this experiment would be sledding. If there are two person on a sled, compared to the same sled with only one person, all going down the same mountain and the same trail, they will have equal accelerations.

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 22px;">**__Newton's Second Law__** =

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task A: Maxx Grunfeld <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task B: Tim Hwang <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task C: Gabby Leibowitz <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task D: Tim Hwang

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**__Purpose with Rationale__**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">What is the relationship between system mass, acceleration, and net force?
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Objective: **

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Hypothesis:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">It is hypothesized that acceleration is directly proportional to the net force and inversely proportional to the system mass. This is derived from the formula net force = mass x acceleration. Therefore, an acceleration vs. net force will illustrate a linear relationship while an acceleration vs. mass graph will illustrate a curved line in which acceleration is decreasing as mass increases.

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Materials and Methods** __ <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Before starting the experiment, an electric scale is used to measure the mass of the cart, the weights, and the hanging mass. With use of the USB, Data Studio is opened on one group members laptop which will aid in measuring the acceleration of the cart. The dynamics cart is placed on the track connected to a hanging mass by string and use of a pulley system. The pulley is clamped to the table and the string is placed through the pulley, hanging the mass off the table. The experiment is performed by using 5 different combinations of weights and hanging masses, repeating each combination three times to secure results. During a sample trial, a certain amount of weight is placed on the already 5g hanging mass and another amount of weight is placed in the cart. A large weight is used to keep the cart in place until ready to measure the acceleration. Once all group members are ready, the weight is removed and the cart is released, with Data Studio measuring its acceleration. One repeats this three times with identical weights on both sides. Once these three accelerations are recorded, a different combination of weights is used in order to demonstrate the relationship between system mass, net force, and acceleration. The exact same process is used for each of the trials.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">media type="file" key="vid1111.mov" width="300" height="300" <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">media type="file" key="vid222.mov" width="300" height="300" <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: 19px;"><-- Increasing Mass
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Videos of two trials **

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Picture of lab set up **

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Data** __ <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Graphs:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">This graph shows how net force and acceleration interact in the first data table. Acceleration is directly proportional to net force because as acceleration increases, the net force also increases. This is represented by an increasing, linear graph. <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;"> <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">This graph shows how acceleration and mass interact in the second data table. As mass increases, acceleration decreases so acceleration and mass are inversely proportional. This is represented by a decreasing, curved graph.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**__Sample Calculations__**










 * __<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Analysis __**


 * 1) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Explain your graphs:**
 * 2) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**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?** The graph of Acceleration vs. Net Force is linear. The slope is 1.865, which represents the actual value of 1/total mass. From our data table, we calculated 1.894 for our value of 1/total mass. Between these two values, we received a percent error of 1.53%, showing that our results were very accurate. In our equation, the net force is the x-variable as it corresponds to the equation of a = net force/(1/total mass). (1/total mass) represents the slope. The y-intercept of the equation represents friction.
 * 3) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**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.** The graph of Acceleration vs. Mass is non-linear and shows a curved pattern instead. The graph shows a power fit. The power on the x-variable is -1.354, but it should be -1. This corresponds to the inversely proportional relationship between acceleration and mass. The coefficient in front of the x corresponds to the net force, which is also equal to the hanging mass. From our graph, the value of this is 0.2103, or otherwise known as the slope. The theoretical value is 0.294. This value represents the net force for when the hanging mass remains constant at 0.030 kg.The experimental value is very close to the theoretical. From this, we obtained a percent error of 39.8% which shows that our results may have been slightly altered by sources of error.
 * 4) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**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.** If we took friction into account for this experiment, it would decrease our acceleration. Friction opposes the motion of the object and would therefore slow it down, which is why it can be a source of error in this experiment. We would need a greater force to create the same acceleration to cancel out some of the friction. Our slope was slightly too big for both graphs, and friction may very likely be the cause of this. With friction, the equation would have been (shown with the equation of our line y = 1.865x – 0.0705):



<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**__Conclusion__** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">It was demonstrated, through experimentation and analysis of our results, that our hypothesis that the acceleration of the cart would be directly proportional to its net force, but inversely proportional its mass was proved correct by this experiment. In addition, our hypothesis regarding what the graphs would look like was also correct, that the acceleration vs. net force graph would be linear and the acceleration vs. mass graph would be be a graph with an exponentially decreasing slope. Our R^2 value is 99% for the acceleration vs. net force graph and 96% for the acceleration vs. mass graph. These are both extremely precise. Our percent error for the acceleration vs. net force graph show pretty accurate results as well, with the acceleration vs. net force graph at 1.53%. However, our acceleration vs. mass graph has 58% error, which is a little higher than we anticipated. This percentage can be attributed to many sources of error that could have occurred during the experiment. Firstly, the experiment largely depended on the assumption that the string was parallel to the ramp. Since we attached the string by hand, we did not guarantee that the string was exactly parallel to the ramp, which could have led to errors. In addition, outside sources could have effected the acceleration, such as an accidental push by the group member who was responsible for releasing the cart or the fact that the hanging mass often times hit the group before the cart had finished moving. This would give misleading results for acceleration on Data Studio.If this experiment were to be redone in order to limit the sources of error, a protractor should be used to measure the angle of the string and ramp, ensuring that they are exactly parallel. Furthermore, since human error got in the way of 100% accurate acceleration values, there is very little to do to limit some of the problems this experiment poses. If other materials and space were available, the experiment could be set up in a way where the cart is automatically released and the hanging mass is in no contact with the ground. However, with what we were given, the only way to reduce this error is to be more cautious and make sure to only use the middle section of the acceleration recorded on the Data Studio, allowing for no possible push or contact with the group (which occurred only at the beginning or end of the experiment) to get in the way. It is critical to understand the concepts behind this lab because it relates to everyday life. One relevant real-life application would be the engines inside a car which use pulleys for parts such as the alternator and power steering pump. In order to drive, it is necessary for an engineer to understand this concept in order to properly install these engines. Furthermore, construction workers often use a pulley system, especially if they are fixing the roof of a house. They will use this system to lift their tools or materials to the roof, and it is therefore, beneficial for these workers to understand their function.

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 22px;">**__Coefficient of Friction__** =

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task A: Gabby Leibowitz <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task B: Maxx Grunfeld <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task C: Tim Hwang <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Task D: Maxx Grunfeld

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**__Purpose with Rationale__**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">1. Measure the coefficient of static friction between surfaces. <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">2. Measure the coefficient of kinetic friction between surfaces. <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">3. Determine the relationship between the friction force and the normal force.
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Objective: **

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Hypothesis:** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">It is hypothesized that as normal force increases, the friction force increases. In addition, the coefficient of static friction will be between 0 and 1 as we have learned is always the range. This will be the same for kinetic friction, as well. Finally, as the mass increases, so will the coefficient of friction. Both relationships presented in this experiment are hypothesized to be directly proportional.

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Materials and Methods** __

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Before starting the experiment, an electric scale is used to measure the mass of the cart. After recording this data, the cart is placed on an aluminum track held down to the table by use of a clamp. A string is tied to the cart in order to attach it to Force Meter that is also attached to a USB link in one of the group members' laptops. This allows for Data Studio to be run in order to measure the mean tension and maximum tension in the string. A certain amount of mass is placed onto the cart, which is measured by use of the electric scale, as well. Once the experiment is set up, one group member pulls the cart by holding onto the Force Meter and slowly pulling away from the clamp at a constant speed. It is important to keep the string parallel to the ramp. Another group member works with Data Studio in order to find the mean tension and maximum tension. To find the mean tension on Data Studio, the most "straight" portion of the graph is highlighted. By clicking "Net Force", the value appears. To find maximum tension, the highest point on the graph is located. These steps are repeated various times, with the mass in the cart differing for each trial.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">media type="file" key="vidddforlab.mov" width="300" height="300"
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Videos of two one trial: **


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Picture of lab setup: **

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Data** __

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Analysis** __

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Static Friction

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Kinetic Friction

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">FBD: <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Discussion Questions** __


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">1. Why does the slope of the line equal the coefficient of friction? Show this derivation. **<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> The equation of a line is y = mx + b. We set the y-intercept to 0, so in our graph, b = 0. As a result, we are left with the equation y = mx. In this situation, this reflects the equation f = µ*N. The y-axis of our graph represents the force of friction, and the x-axis represents the normal force. Therefore, the m value, or the slope, represents the µ in the equation for friction, otherwise known as the coefficient of friction.


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">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! **<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">The coefficient of friction between the material and the aluminum track, for static friction, should be somewhere between 0.25 and 0.4. The coefficient of static friction we obtained was 0.1737, so we were slightly off. The coefficient of friction between the material and the aluminum track, for kinetic friction, should be somewhere between 0.1 and 0.3. The coefficient of kinetic friction we obtained was 0.1705, so our measured results fall within that range of theoretical values. The website we used to find these theoretical values was [].


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">3. What variables affected the magnitude of the force of friction? What variables affected the magnitude of the coefficient of friction? **<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">There are a few variables that could affect the magnitude of the force of friction. The surface we used to slide the cart affected the force of friction because whether the surface was rougher or smoother, it would determine how much friction is needed to oppose the motion. Also, in this situation, friction = tension, so a greater or lesser force would alter the magnitude of friction. The magnitude of the coefficient of friction would be affected by the friction force itself and the normal force, as seen in the equation f = µ*N. Changing the value of these two forces would affect the magnitude of the coefficient of friction.


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">4. How does the value of coefficient of kinetic friction compare to the value for the same material’s coefficient of static friction? **<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">The coefficient of kinetic friction we obtained was 0.1705. The coefficient of static friction we obtained was 0.1737. The value of coefficient of kinetic friction is slightly smaller than the value for the same material's coefficient of static friction. In the website we used previously, it appeared that static friction should be greater than kinetic friction, and our results show this. This makes sense because kinetic friction is moving, so the friction should be less than an object that is not moving and therefore has static friction.


 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Graph: **


 * 1) [[image:Screen_shot_2011-12-12_at_6.12.47_AM.png]]<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Lower = kinetic; Upper = static

__<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**Conclusion** __ <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">This lab proved our hypotheses accurate. It was originally hypothesized that as normal force increases, the friction force will increase as well. In addition, our group believed that the coefficient of friction would have to be within the range of 0-1. Finally, we predicted that as mass increases, the coefficient of friction will also increase. Through experimentation, the mean tension and maximum tension of the string attached to the cart were found, which allowed us to calculate the static and kinetic friction that existed. The equation fx=max led us to this discovery, and led us to the conclusion that friction and tension were equal. Since the cart was pulled at a constant speed, the system was not accelerating, and therefore, T-f=0, or in other words, T=f. The data we recorded during the experiment allowed for a friction vs. normal force graph to be constructed, exhibiting a directly proportional relationship by showing two linear lines with an increasing slope. It was this slope that served as the coefficient of friction and also proved our second hypothesis accurate. As shown on the graph, the coefficient of kinetic friction is .1075 and the coefficient of static friction is .1737, all values in the range of 0-1. Our last hypothesis was proven correct, which is evident through examination of our data tables. It is obvious that as the mass on the cart increased, a larger tension was required to move the cart. Our R^2 values demonstrate the fact that we received extremely accurate results, with both kinetic and static friction having an R^2 value of 99%. Furthermore, the percent difference found between our results and the results of the class for static friction was 8.075%. For kinetic friction, the percent difference was 4.425%. However, there are many sources of error that could have occurred during this experiment. Firstly, this experiment allows for human error to get in the way in many areas. It is an extremely difficult task for the group member in charge of pulling the cart to make sure to keep the string exactly parallel to the ramp. There is a great chance that this wasn't the case, which is responsible for some of the error that existed. In addition, it was also up to that group member to pull the cart at a constant speed, with no acceleration. However, this is not very realistic, and the accidental acceleration that the cart underwent also throws off our results. These two errors could be minimized if different materials were available. A protractor could be used to ensure that the string is parallel and a mechanical device could be used to pull the cart also to ensure that the cart moves at a constant speed. There are multiple ways in which one could relate this lab to every day applications. Take into consideration dragging a suitcase. It is much easier to drag a suitcase over a flat, hardwood floor, then a patch of grass. This is because the coefficient of friction between the suitcase and the flat surface is much smaller than the coefficient of friction between the suitcase and the grass.