Group4_4_ch4

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 * Group 4:** Nicole Tomasofsky, Matt Ordover, Stephanie Wang

Gravity and the Laws of Motion
A+B: Nicole C: Stephanie D: Matt


 * Objective**
 * After conducting the experiment, it will be possible to find the value of acceleration due to gravity, determine the relationship between acceleration and the incline angle, use a graph to extrapolate extreme cases that cannot be measured directly in the lab, and determine the effect of mass on acceleration down an incline.


 * Hypothesis**
 * From our knowledge about free fall, acceleration due to gravity should be about 9.8 m/s/s.The steeper the incline angle, the higher the acceleration should be because it has a higher initial angle. Extreme cases in the lab would be high angles like very high angles and very small angles or very high or low masses which will be difficult to produce in the lab. The effect of mass should not effect the acceleration down the incline because only gravity is acting upon it.


 * Methods and Materials**
 * First, an adjustable ramp is set up at an incline, using a metal ramp and an adjustable clamp. so it is possible to adjust the height the ramp stands at. Then, the height and distance are measured using a yard stick. The height will be changed, but the distance will never be changed. A metal ball let go of will be freely roll down the incline, at least 5 times, having the time it takes being recorded each time with a stopwatch. This will be completed for 3-5 different heights. With the time, distance, and height known, sin ø can be found as well as the acceleration. Finally a graph relating sin ø and acceleration will be created.

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 * Data:**


 * Personal Results:**


 * Class Results:**


 * Graph:**

The slope is what the acceleration due to gravity is. Through knowledge about free fall, it is known that the acceleration of gravity is 9.8 m/s/s. Our graph indicates that our acceleration of gravity is 5.57 m/s/s. Reasons for error will be discussed in the conclusion. Our b value .059 is related to friction.
 * y = 5.5669x + .059**

The r^2 value is relatively good. It states that our values were 98.6% accurate to the trend line. There of course were errors however, which is why it was not higher.
 * r^2 = .986**


 * Link to Spreadsheet:**


 * Sample Calculations**

**For percent error:**


 * For percent difference: **

Our graph compares sin(theta) to acceleration. Sin(theta) is shown on the x axis, which goes from 0 to 1 because those are all the possible values of sin(theta). Acceleration is shown on the y axis, which goes from 0 to 10. This is the range because the acceleration due to gravity is 9.8 m/s/s, and unless an object is thrown down or pushed down, it will not have an acceleration higher than this. Since we simply dropped the ball down the ramp with no initial velocity, our value for acceleration cannot possible exceed 10 m/s/s. The equation of our graph is y=5.57x + .0 59. The slope represents the acceleration due to gravity. In this case, that would be 5.57 m/s/s. This value represents our experimental value for freefall acceleration. Since the theoretical value for freefall acceleration is 9.8, our percent error is 43.16%. The class average was 7.96, and our value was 5.57. This means that our percent difference was 30.03%. For the class data, there was no correlation between the mass of the ball and it's acceleration. However, for acceleration at 15 cm, mass did matter. The less massive balls had a slower acceleration, and the more massive balls had a faster acceleration. This is due to friction, which is a constant. This means that friction would have a bigger impact on a smaller mass.
 * Analysis:**

Free Body Diagram:



The weight of the ball, especially the x component, in addition to the imbalance between the normal force and weight is what caused the ball to roll down the ramp. Since the height we used to test this was 15 cm, theta is equal to 6.89 degrees. Our calculated acceleration is 1.23 m/s/s. This is different from our experimental acceleration value, 5.57 m/s/s. The difference is probably due to friction, which we did not account for in our calculated result. We learned that friction has a greater affect on less massive objects since it is a friction, and our ball was extremely light, so this makes sense. This proves that the formula F=m x a is a good equation for solving for the acceleration.

//1.) Is the velocity for each ramp angle constant? How do you know?// The velocity for each ramp angle is not constant. The ball is not going at constant speed because it accelerates as it goes down the ramp. The initial velocity for each ramp angle was the same. Since the ball always started at rest, the initial velocity for each ramp angle is 0 m/s. However, the final velocity is not the same for each ramp angle because the time and acceleration was different for each ramp angle.
 * Discussion Questions:**

//2.) Is the acceleration for each ramp angle constant? How do you know?// No, the acceleration is not the same each ramp angle. The steeper the ramp, the faster the acceleration. You also know that the acceleration is not the same for the different ramp angles because distance, initial velocity, and time affect acceleration, when using the equation . The distance and initial velocity are the same for each ramp angle, but the time is different. This means that acceleration also must be different for each ramp angle.

//3.) What is another way that we could have found the acceleration of the ball down the ramp?// We could have used a motion sensor to make a velocity-time graph of the ball's motion down the ramp. Since the slope of a v-t graph equals the acceleration, we could have just found the slope of the v-t graph to get the acceleration.

//4.) How was it possible for Galileo to determine g, the rate of acceleration due to gravity for a free falling object by rolling balls down an inclined plane?// A free falling object would be closest to a ball rolling down a ramp that is set at a 90 degree angle. Galileo would realize this because he could probably see that the larger the incline, the larger the acceleration. Galileo probably conducted an experiment very similar to ours, where he set up a ramp at multiple angles, timed a ball going down the ramp, and calculated the acceleration of the ball. To find g, the acceleration due to gravity, Galileo probably graphed the acceleration vs. sin(theta) . He could then extrapolate this data to find the acceleration of the ball at 90 degrees. This value would be about 9.8 m/s/s, which is the value of gravity.

//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?// No, the mass of an object doesn't affect its rate of acceleration down the ramp. This is obvious from the class data, which shows that a heavier ball doesn't necessarily mean a faster acceleration. For example, a .54 kg ball had an acceleration of 8.14 m/s/s while a .08 kg ball had an acceleration of 9.95 m/s/s. Similarly, mass does not affect the motion of an object during free fall. When an object is in free fall, the only force acting on it is gravity. Although more gravity is acting on objects that are more massive, these massive objects reject the change more, and would therefore fall at the same rate as a less massive object.

We hypothesized that the acceleration due to gravity is 9.8 m/s^2. We based this on previous experiments this year in which we have calculated acceleration due to gravity on objects in free fall. We found our acceleration to be 5.57 m/s^2. The reason this is off is because of error and because it was on an incline and not free-falling. In addition we were also correct that acceleration would change based on the incline. A higher incline means a higher acceleration. Also, a lower incline means a lower acceleration. However, we were wrong that mass does not affect acceleration because we did not take friction into account. Larger balls have less friction and therefore have a slightly higher acceleration. Our percent difference from the class was 30.03%. This means that our results were drastically different from our class. In addition, our percent error was 43.16%. This is extremely high and there are a few possible reasons for this. The most probable cause is simply human error. We probably did not measure the times accurately. We might have started the timer too early or too late. We also might have stopped it to early or too late. We could fix this by using a timing device that automatically stops the time when the ball passes it. Another source of error was the fact that we did not take friction into account. To fix this we could calculate with friction next time or use a smoother ramp and ball to reduce friction. A real life application of this would be snowboarding. A snowboarder could calculate the angle and distance of the ramp so he knows what his speed and acceleration will be. He can then calculate how far he will go off of a jump and how much time he has to do tricks. There are many other real life applications of this including roller coasters and extreme sports.
 * Conclusion:**

Newton's Second Law
A+B: Stephanie C: Matt D: Nicole

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

**Hypothesis**: Since the acceleration is directly proportionate to net force, the acceleration vs. net force graph should look like y = x. Since the acceleration is inversely proportionate to the system mass, the acceleration vs. mass should look like y=1/x. **Method and Materials (Procedure):**

First, place a track on the table. Make sure it is level, and if it is not, place something underneath the lower end to make it even. You can check to see if the track is even by placing a dynamics cart on the track, and seeing if it moves by itself. After this, connect the photo gate timer and base support rod, and clamp them to the table. Make sure the wheel and track are in line. After this, plug in the USB for the photo gate timer into your laptop, and open data studio. Start a new experiment of Acceleration Using a Linear Pulley. Then, check to see if the mass hanger is attached onto the cart. If it is not, attach it with a string. Dangle the mass hanger over the table, with the string resting on the wheel.

Next, place two 10g and one 5g mass on the cart. Let the cart roll along the track and record its acceleration. Do this two more times. Then, take the average acceleration and use that for calculations as the experimental acceleration. Then, repeat the whole experiment four more times, each time moving a mass off of the cart and onto the mass hanger.

For the next part of the experiment, put all of the mass blocks on the cart, and remove all mass disks from the mass hanger. Let the cart make its run on the track, and record it's acceleration. Repeat this three times, and use the average acceleration as the experimental acceleration. Then, repeat this entire thing four more times. Make sure there are no masses on the mass hanger for any of these runs, and make sure to use varying masses on the cart.


 * Picture of Lab set up:**

media type="file" key="Movie on 2011-12-05 at 07.54 **Data:**
 * Video of experiment:**

This table is for the first part of the experiment. It is to compare force and acceleration. We measured the masses using a scale, and got the acceleration using the photo gate timer and data studio. Then, using this data, we found the force.

This table is for the second part of our experiment, which compares mass to acceleration. We performed three trials for each different mass, and took the average to get the most accurate result. For this, we measured the mass using a scale and the acceleration using data studio and the photo gate timer.



**Graphs:**

This graph shows how net force and acceleration are proportional in the first data table. They are directly proportional and we can see this because acceleration gets larger as force gets larger as well. This makes the graph linear.

This graph shows how mass and acceleration are proportional. They are indirectly proportional and we can see this because acceleration gets smaller as mass gets larger. This makes the graph decreasing and curved.

[|fixed newtons first law dec 20.xlsx]

**Average Acceleration Sample Calculation:**

**Net Force Calculation:**

**Percent Error Acceleration:**


 * Theoretical Acceleration:**

**Analysis:1**


 * 1) Explain your graphs:
 * 2) 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 acceleration vs. net force graph is linear. Its slope is 1.7998 and it represents 1/total mass. Our calculated value for 1/total mass was 1.887 which produces a percent error of 4.61%. This means that our results were very accurate. Slope should be equal to this quantity because of the equation (1/m) x ∑F = a. In this equation 1/m is the x variable and therefore the slope. The y-intercept represents the friction on the whole system.**


 * 1) 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 mass vs. acceleration graph is non-linear. The power on the x is -.899, but it should be -1. The coefficient on the x is the net force and is equal to the hanging mass. Ours was .0507, but it should be .005. This produced a percent error of 90.14%, which means that there were some huge sources of error.**


 * 1) 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 friction was taken into account, the acceleration would be smaller. Friction goes in the opposite direction from the system and therefore slows it down. To create the same acceleration you would need a greater force. Our slope was too large and friction could be a cause of this. With friction the new equation would be:**

**Conclusion:**

After completing the lab, our hypothesis was proven to be correct. Through the use of a pulley system, first the mass of the system, which is the entire thing including both masses and string, was kept constant. After tests, it was proven that with more mass on the hanging mass, the acceleration would be greater even though the mass of the entire system did not change. The hanging mass had only the force of weight (m*g) and therefore as more mass is added to the hanging mass, force increases. As the force increases, the acceleration also increases proving that force is directly proportionate to acceleration. This graph, is as our hypothesis predicted, shows the relationship y = x, being a straight line with a positive slope. On the other hand, in our second pulley set up demonstration, it was proven that if the mass of the hanging mass stayed constant, and only the mass of the cart changed, as the mass of the cart decreased, the acceleration increased. Therefore, as the mass decreases, the acceleration increases and vice versa. This graph, as our hypothesis predicted, shows the relationship y = 1/x and is a curved line with a negative slope. However, the r^2 values for both graphs were relatively high. The force v. mass graph had an r^2 value of .99989 which is very good, and the mass v. acceleration graph had an r^2 value of .96302.

Our lab overall was pretty accurate. When calculating the average accelerations to the theoretical accelerations in the first demonstration, all percent errors no higher than 3.25%. Our other results were not as good. For example, in our mass v. acceleration graph, our data points were proven to be not the best because the slope of our quadratic line read y= .132e^(-.878x). The exponent should be -1 because the slope is the inverse of the total mass. There was a 12.2% error between our -b value and the actual -b value. Although this is higher, it is still not a very high percent error. Error in this lab could have come from many things. First, although we double checked our ramp to be level, it seemed at points to move by itself when it was not being pulled by the pulley. There might have been a slight incline of the desk. Also, error could occur if the ramp was not in a perfect straight line, lined up with the pulley. If the ramp was a little bit angled the string would have experienced a pull creating more friction in the pulley where there should not have been a lot. Lastly, if the hanging mass was not perfectly still and still swinging side to side because of a previous trial, the mass would have more than just a weight force on the y axis acting upon it, it would also have forces along the x axis and this could as well alter the results. Therefore, to make this lab more accurate, the ramp could be put on a different counter to ensure that it was level. The ramp could also be double checked to not be at an angle and double checked to make sure the hanging mass was not moving.

This pulley scenario is very applicable to everyday life. An example of pulleys we see everyday is at the gym with exercise equipment. The more mass being lifted, the greater the weight force is on the weights, therefore more force needs to be used to lift these masses. Also this is important to know for safety reasons. The smaller the mass, the faster it will accelerate, so one that is doing light weight lifting should be careful that they do not simply let go of the weights so they come crashing down at a fast acceleration.

** Coefficient of Friction **
Stephanie Wang, Nicole Tomasofsky, Michael Poleway, Max Llewellyn

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 **

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, then attach a 15 cm string to a block on one side, and attach a 15 cm string to the force meter on the other side. The force meter will use Data Studio to instantaneously graph the force. After setting up Data Studio correctly, pull the block with the force sensor, at as constant of a speed as possible, making sure the string is horizontal.
 * Methods/Materials**

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 * Data**






 * Excel Sheet**
 * Free Body Diagram**




 * Sample Calculations**





2. The coefficient of static friction between our material and the aluminum track is between 0.25 and 0.40. We got .1252, so we were a little bit lower than the actual range. This is probably due to some errors in the lab. The coefficient of kinetic friction is between 0.10 and 0.30. We got .1795, which is within this range, so our results for that are very good. We got our information from this website: []


 * Analysis**
 * Compare the slope of line with calculated m s average (% difference).
 * The slope of our line was .1795. The class average for this value is .161. This results in a 11.491% percent difference.
 * Compare your result with the class results.
 * Our results were pretty much in line with the class results. We had a 11.491% percent error with our static friction, but for kinetic friction, we only got a 10.796% percent error. Since this is almost under 10%, this is very good. The class average for kinetic friction was .113, and our value for this was .1795. Since these both fall within the range of values for the coefficient of friction between plastic and aluminum, I think they are both valid results.


 * Conclusion**

Our hypothesis stated that the kinetic friction would be in between 0 and 1 but it would be less than the static friction. The static friction would also be in between the same values, yet it would be a higher value. Static friction is when an object isn't in motion and kinetic friction, or sliding friction, is when an object is moving. We also hypothesized that the relationship between friction force and normal force can be found in the equation f= µ * N. Our hypothesis was correct, and this can be seen in our graphs. By finding the coefficient of friction, you just find the slope of the lines on the friction vs normal force graphs.Static friction turns out to have a coefficient of friction of about .18, while kinetic friction has one of .125. The slope shows that the static friction is obviously larger like we hypothesized. With the coefficient of friction, you could then plug that into the equation with either friction force or normal force to find the other. Normal force has to be larger than friction force in almost all cases because you have to multiply that by the coefficient of friction, which is always a decimal, to get your friction force. This is shown in the graphs and data tables, and all in all, our hypothesis was spot on.

Our percent error was _, which is a pretty decent percent error. All percent errors should be below 10 percent to be considered acceptable. Like in all experiments, the theme is always the same, as there are several errors that we can try to eliminate. In this experiment, we are accounting and trying to find the value for friction, so that's not an error like we usually have for most labs. When pulling the cart, it s possible that I moved it faster at certain points and pulled it with a stronger force. It was important to pull the cart with the same force and at constant speed on every trial, and doing otherwise could've caused different results. With the normal force being different, the friction force and the coefficient of friction would be different as well. If we could keep the same force and speed throughout the whole trial then its possible our error would go down. In addition, when pulling the cart full of weight, its possible that the string was on an angle that we didn't want. If the angle wasn't 180 degrees, then it could have changed the results. We need to make sure its the right angle every time, and to help this, I would put my hand under my other hand to give the string something to rest on. This would give the right angle and wouldn't mess up our results. Furthermore, we needed to make sure the string had slack in the beginning. At times I forgot to slack the string, and that hurt our results. If I did this experiment again, I would make sure every time that the string had slack. All in all, we did a great job in this experiment and tried to eliminate all the possible errors. If we did this again, we would try to make sure there would be absolutely no flaws.