Danielle,+Jae,+Jessica

Moving in a Horizontal Circle
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 * Objective**: To generate the relationships between radius and max velocity when the angle is banked and unbanked and to determine the relationship between the radius and the banking angle.

- The radius and the maximum velocity of a banked turn have a direct square relationship because the equation for velocity is. Therefore if you square both sides of the equation you will get:, which shows if you double the radius for example the maximum velocity will be four times greater. -Radius and the banked angle of a banked turn are inversely proportional due to the equation:, which shows as you increase the angle radius will get smaller and vice versa.
 * Hypothesis**: Radius and maximum velocity of an unbanked turn have a direct square relationship because of the equation [[image:jtequationyi.png width="58" height="18"]]. If you square both sides of this equation, you end up with [[image:jtequationsan.png width="64" height="26"]], which shows for example that if you double the radius the maximum velocity will be four times greater.


 * Procedure:**
 * 1) Calculate the 4 different speeds on the record player to be more accurate by counting the number of rotations in sixty seconds or one minute.
 * 2) Calculate the radians per second of each speed on the record player.
 * 3) Place the penny on the surface at a set velocity and find the radius of the circle at the point that the penny does not move, but if you move it even slightly outward, the penny would go flying media type="file" key="Movie jt1.mov" width="300" height="300"
 * 4) Record data and repeat with each different speed.
 * 5) Take the average of the radii recorded from multiple trials for each speed on the record player. This is the experimental radius for each speed.
 * 6) Use the radians per second and the experimental radius for each speed to calculate the experimental velocities that were used for each speed on the record player.
 * 7) Graph the experimental velocities versus the experimental radii

This data was collected throughout our experiment. We found the radians per second by completing step 2 of our procedure. We then measured the radius of the circle like it says to do in step 4. In order to get the velocity, we multiplied these two numbers together. We then repeated each radius four times and took the average radius and velocity for our graph. These averages are in bold and are the points that were used in our graph.
 * Data**:

This graph shows a direct squared relationship between radius and velocity. This can be seen through our graph because the line is curved meaning that there a square relationship and because the slope of the line is positive, there is a direct relationship. We chose a polynomial fit due to the fact that he thought it would be squared and we right with an R squared value of .9994.
 * Graph**:

Velocity vs Radius (banked angle)
 * Results taken from Emily, Elena, Amanda, Emily.

Radius vs Banked Angle


 * Results taken from Nicole, Jillian, Spencer, Dylan

Theoretical Mu (µ) calculation done by Ani's group by taking the tangent of the angle before the penny slid down the piece of wood. In this case it was tan of 12.13º.
 * Calculations:**

Experimental Percent Error








 * Conclusion**

Our hypothesis for the relationship between radius and velocity for an unbanked turn was correct. Radius and velocity of an unbanked turn have a direct square relationship. This is shown in our graph because the equation of the graph is a quadratic equation with a positive slope. We also know our data illustrated this relationship because the R^2 value of the graph was .9994. Also, for example if you take one of the first trials for example. The square root of the radius .09 is 0.3, which is relatively close to the experimentally determined value for velocity of 0.4. Granted, based on the equation the square root of the radius could not be equal to the velocity because there is also g and the coefficient of friction involved.

Our hypothesis for the relationship between radius and velocity of a banked turn was not correct. This is primarily because we based our hypothesis on the wrong equation for banked turns. The equation we used only applies when there is no friction. However, in the case of this experiment there was friction. The equation derived from the combination of the net force equations for the x and y axes should be: When the equation is rearranged to solve for velocity it shows that radius and velocity of a banked turn should have an inverse square relationship. This shown in the graph taken from Emily and Amanda's group. The shape of the graph indicates it is an inverse square, which we can trust due to the R^2 value of .9963.

Our hypothesis for the relationship between radius and angle of a banked turn was correct. Once, again we based our rationale on the wrong equation for banked turns without friction. However, the correct equation shown above also indicates radius and angle should be inversely proportional. This relationship is shown in Nicole and Jillian's graph. The line on the graph has a negative slope and there is no exponent on the x in the equation (representing the angle). Once again we can trust their results because their R^2 value was .9643.

The greatest amount of percent error experienced in our experiment was 13.23% and all of the other percent errors for the velocities were no greater than 4%. This is a reasonable amount of error because it is not too much so we know our experimental results were not too far from the theoretical values. One source of error in this lab could have been we did not initially calculate the coefficient of friction for our penny on the board, which was most likely different from the coefficient of friction we obtained from the other class. Second, as we tested radii sometimes the penny experienced gradual acceleration whereas other times it experienced immediate acceleration. In this way we were not consistent in our methods because sometimes we moved the penny gradually as the circle spun, giving it time to adjust to the new acceleration. Other times we turned on the record player with the penny already placed at what we believed to be the radius, therefore, it experienced a sudden acceleration and spun out when normally if gradually placed there it would not. This error obviously affected the radii we determined in our experiment resulting in some error. Also, we experienced the greatest amount of error for the largest velocity because for this velocity we had to use a piece of balsa wood to extend the turn table in order to obtain the necessary larger radius. Even at its longest extent, this setup did not seem to allow for the correct radius; it was too short. It was essentially very difficult to test. This different setup for this velocity resulted in a different coefficient of friction because the penny was on balsa wood instead of composite board as in the other trials. Also, we most likely did not determine the correct radius because we could extend the board only so far. These factors obviously caused our experimental velocity to deviate so much from the theoretical. In order to fix the error experienced in this lab we would be more consistent in our methods. This meaning we would consistently gradually move the penny while testing the radius in order to ensure that every time the penny was adjusting to the change in acceleration. This would give us a more accurate experimental radius and therefore less error. Also, for the largest velocity it would be ideal to have a very large turntable so that we could obtain infinite lengths for the radius. This would also prevent us from having a different coefficient of friction for this velocity because it would be all on the same surface.

**Maximum and Minimum Velocity of a Vertical Circle.**

 * Objective**: To determine the minimum velocity at the top of a vertical circle and the max velocity at the bottom of a vertical circle.


 * Hypothesis**: The larger the mass that your group chooses, the smaller your minimum and maximum velocities will be. The larger the radius is that your group uses, the larger the minimum and maximum velocities will be.

Minimum Velocity: >
 * Procedure:**
 * 1) Obtain a mass and attach it to your chosen string
 * 2) Measure the length of the string, this is the radius of your circle
 * 1) Spin the string in a vertical circle and slow down until there is no tension in the string at the top of the circle
 * 2) Do several trials calculating the velocity for each

Maximum Velocity:
 * 1) Attach mass to a piece of your chosen string until it breaks to find the maximum tension[[image:Photo_6.jpg width="384" height="288"]]
 * 2) Take a mass significantly smaller than your maximum tension and attach it to a string.
 * 3) Measure the radius of your string.
 * 4) Swing the string as fast as you can until it breaks. Make sure to count the number of rotations it completed before breaking.
 * 5) Record the data and then repeat.
 * 6) Change the length of the radius and repeat steps 2-5.

Finding velocity from the # of rotations/ second and vice versa: divide one by the number of rotations in one second and that is the time it takes for the mass to travel the circumference of the circle once. then plug that time into the equation for velocity: v = 2pi r/ t if you have velocity, use the same equation but solve for time then do one divide by that and that is the number of rotations you must do in one second

Minimum Velocity:
 * Data**:

Maximum Velocity

Theoretical Min Velocity Experimental Velocity Percent Error Theoretical Max Velocity Experimental Velocity Percent Error Minimum Velocity The data table shows all of the elements that were used in order to calculated our experimental and theoretical minimum velocities. In order to get the theoretical velocity, we used the equation. This equation gave us the 2.04 that we were trying to get to with our experimental value. However, there is error in this lab so we got results that were close to 2.04, but not exactly that. Because it is impossible to swing the mass at a constant velocity, our value will be off by a little bit. Also, there is no way to experimentally calculate velocity when tension is equal to zero because we are spinning the mass at a speed that would make tension a little bit more than zero. In order to find our min velocity, we multiplied the number of rotations it took by the circumference and then divided that by the time. This explains why our experimental values are a little bit larger than our theoretical value.
 * Calculations**:
 * Analysis:**

Maximum Velocity The data table shows all of the components in order to calculate theoretical and experimental maximum velocity. In order to get the experimental velocity, we used to equation. This equation gave us a number much larger than our experimental velocity. This is because when getting the theoretical maximum velocity, we found found our max tension by gradually adding on masses. When finding the experimental velocity, we did not gradually spin the mass so the time for both were very different. In order to find our max velocity, we multiplied the number of rotations it took by the circumference and then divided that by the time.

Although some of our data was not good in terms of how it compared to our theoretical values for maximum and minimum velocities, it generally did show our hypothesis was correct. In our experiment for minimum velocity we changed mass while keeping the radius constant. We hypothesized that a larger mass would generate a smaller velocity. The velocity of our largest mass (4.005 kg and 2.08 m/s) when compared to the velocity of our smallest mass (.005 kg and 2.35 m/s) shows this relationship, however, the data for masses in between those values does not exactly correspond. As shown in the data table the velocities for 2.005 kg and 3.005 kg were larger than the velocity for .005 kg, which should not be the case according to what we hypothesized. This could, however, be due to various errors in our experimental process (see below). For maximum velocity, our data was more consistent with our hypothesis. We hypothesized that as radius increases, maximum velocity increases as well. In our experiment for maximum velocity we changed the radius while keeping mass constant, and our data consistently showed the relationship we hypothesized. Our radii ranged from .194 meters to .631 meters and our corresponding maximum velocities ranged from 1.22 m/s to 4.46 m/s.
 * Conclusion**

There were many sources of error in this lab. First, there was the issue of keeping certain variables constant. We cannot be sure, considering we used no measuring instruments other than a meter stick that we kept certain variables constant. This is especially true of the tension force for minimum velocity. We assumed as we spun the mass that the tension was zero at the top of the circle, however we have no way of knowing this was actually so because we never measured it with a force meter for example. Second, we did not measure our radii in between trials for minimum force to ensure it was kept constant, we merely assumed that if we were holding the string in the same place it was constant. Subsequently, we had a great amount of error in our experiment for maximum velocity. Our experimental values (as shown in the data table) were extremely far off from the theoretical values for maximum velocity. This is most likely due to the fact that as we did these trials we did not gradually get up to maximum velocity. By essentially immediately starting at maximum velocity as we spun the string, the string had no time to adjust to the sudden increase in tension and therefore broke very quickly. This therefore led to us counting an incorrect amount of rotations per second for an also lower tension, which resulted in our lower experimental velocities.

Important methods of improving this lab would be to first use a machine to spin the mass on the string rather than rely on humans to do so. This would ensure all variables that we want constant are kept constant, that tension is zero at the top of the circle, and that we can move the mass up to speed gradually. This type of machine would also help us to calculate more accurate experimental velocities, especially if it could measure the velocity for us.

**Circular Motion: Radius**

 * Objective**: To determine the relationship between the radius of an object's circular path and the center force pulling the object toward the middle by keeping speed and mass of an object constant.

In this equation, when radius gets larger, the force should get smaller and when the radius gets smaller, the force should get larger.
 * Hypothesis**: The radius and force are inversely proportional. This is because of the equation:

1. Determine a standard speed for each experiment by using the equation (rotations)(πr^2)/t where t is time and rotations is the whole distance around the circle. In our case the standard speed was 9.42 m/s/ 2. Tie a mass to the end of a string. The amount of mass is not important as long as the mass stays constant throughout all trials. 3. Determine how long your string needs to be in order keep a constant speed with your first radius. 4. Attach the Data Studios Force meter to your string at the correct length. Select force meter and make sure the setting is set to pull force, NOT push. 5. Swing your mass in a circle at constant speed and record the y-intercept of the line you recieve from the force meter. 6. Repeat step 5, 3 more times. 7. Repeat steps 5 and 6 with different radii until you feel you have a sufficient amount of data. 8. Graph the data is excel and determine relationship.
 * Procedure:**

Sting Mass Data Studios Force Meter Meter Stick
 * Materials:**

** Data **** : ** 

** Graphs: **  *Our graph
 * Graph by Chris Hallowell, Scott Siegel, Ross Dember and Bret Pontillo


 * Graph by Tom McCullough, Tyler Samani, Richie Johnson, and Rory Vanderberg

**Calculations:** Calculating Velocity: Trial 1: r = 10 cm 15 rotations in 10 seconds = 1.5 rotations in 1 second C = 62.832 cm v = 9.42 m/s

Another way of Calculating Velocity: 1.5 rotations in 1 second t = time for one rotation t = .67 s

v = 9.40 m/s

Equation for Centripetal Force ( Tx force of string): Trial 1 R = .10 m m = .012 kg v = 9.40 m/s

Tx = 10.6 N

**Analysis:** Our graph portrays an indirect relationship between force and radius. We came to this conclusion because the line in our graph has a negative slope meaning that they are inversely related. Our R^2 value is a little bit low because of the many sources of error that will be addressed in our conclusion. We are certain that the data collected from data studio for the force was off, for data studio consistently gave us higher values for the force than expected with each change in radius. However, we therefore then know all of our trials' data was affected in this manner, which is why despite this difference in theoretical and experimental force we still got a nice linear inverse relationship between force and radius on our graph.

**Conclusion:** In this lab, our hypothesis was correct. We had stated that radius and force were going to have an inverse relationship and based on the results we produced, this was proven to be true. When our radius was 10cm our tension was 0.144N however, when our radius was 15cm the tension became 0.130N. This shows that as the radius increased, the tension decreased. Our graph has a negative slope which also shows that radius and force are inversely related. There was definitely a lot of error when performing this lab. The most error probably came from us guesstimating the radius of our circle. We decided that it would be sufficient to swing the string around the ruler and just try to get it as close to the number we wanted as we could. This would then result in error because there is no way that we could have done it exactly the same way each time. This was also hard to do due to the fact that the person's who was swinging the mass hand was moving and not staying centered. Also, when performing this experiment velocity needed to remain the same. It is almost impossible for velocity to stay the same for the length of the entire experiment because our velocity was calculated using our judgment to try and get the specific number of swings per 10 seconds. Another source of error comes from the fact that the force censors were only detecting the y component of the tension in the string. Without knowing the angle of each swing it would be impossible to know that actual tension in the rope. Even if we did know the amount of error caused in one of these trials it would be impossible to fix that easily as the error would be different for each different trial, with different angles being used. There are multiple ways to try and fix the error in this lab, just as there are multiple sources of error. To fix the guesstimating problem you would need to have at least one more variable so that it would be possible to calculate an exact radius. Another idea would be to have a centripetal force machine that would be able to swing the object from a stationary position. This machine would also help with keeping speed the same for each trial. There would be no guessing how many rotations needed to happen in a certain amount of seconds. It would be set on the machine. To fix the problem of the force censor only detecting the y component could be dealt with in 2 ways. You could move the hand holding the censor so that is always pointing in the way the string is being pulled, however this might cause some extra error in other places. A more efficient way, which would need to be tested to see if it works, would be to tie a string to the other end of the force censor and swing the whole thing as one system. This way the circle would still be made and the force censor would be pulled in the same path as the mass. Although it is unlikely that if you are turning a circle you will have to take note of the force being applied, however this concept does come up in many everyday activities. When you are turning a car, when a washing machine does a spin cycle, and so on, centripetal force is at work. If you know how small or large your circle is then you will know how much force you will need to make that circle possible.