Bloom,+Listro,+Fihma

=toc= =Moving in a Horizontal Circle= Group Members: Sam Fihma, Evan Bloom, Ryan Listro Class: Period 2 Due Date: January 24, 2011

//Credited Sources:// For A): Chloe, Justin, Steve, and Andrew For C): Anthony, Aaron, Navin, Jimmy


 * Overall Objectives:** How is the maximum velocity keeping a car moving in a horizontal circle dependent on: A) the radius of an unbanked turn, B) the radius of a banked turn, and C) the banking angle?

A) To find the relationship between radius and maximum speed that allows an object to remain in its uniform circular motion. B) To find the relationship between radius and speed on a banked turn. C) To find the relationship between a banking angle and the value of the radius at which maximum speed is reached.
 * Purpose:**

A) As the radius increases, the maximum speed for the turn increases at a rate directly proportional to the square root of the radius. This relationship is modeled by Newton's Second Law F = ma, where a = v^2/r. When we solve for velocity using a frictional force to the right, weight down, and normal up, we see that the v = (Rµg)^1/2. Because of this derivation (shown above), we can infer the aforementioned relationship. B) We predict that the faster the velocity of the penny, the larger the radius will be. This is based off of the pre-lab equation that is shown below. In this equation of maximum velocity for a banked turn, the velocity increases when the radius is increased and vice versa. Also, the radius increases when the velocity does too. The relationship between the velocity and the radius is most likely a direct square due to the fact that in centripetal force equations, velocity is always squared. C) We hypothesize that as the angle increases, the radius will also increase. This can be proven by the equation for maximum velocity of a banked turn.
 * Hypothesis with Rationale:**


 * Materials:** The key material used during this lab was a record player, which supplied the velocity of the horizontal circle. We also utilized a timer, a platform that could sit on top of the spinning record player, and an inclined block, which represented a banked turn. We also used a penny to simulate a car moving around a banked turn.




 * Procedure:**

1. Set a platform on top of the record player. Using Velcro, place an inclined block on the platform. Measure this angle using a protractor.

2. Set the record player at a constant velocity.

3. Time ten revolutions of the record player and find the velocity using an equation that is done below.

4. Place a sliver of tape on the incline, starting low. This is where the penny will rest as the record player starts to spin. If the penny falls off place the tape even lower. If it does not, place the tape a bit higher and try again. Keep using this method until you find the point at which the penny barely stays on the incline.

5. Measure the distance to the center from inner edge of the penny once it stops sliding. This is the radius.

6. Perform steps three through five multiple times to determine an average. Make sure you also perform these trials for multiple speeds, which can be changed using the knobs of the record player.

A) 
 * Data:**

B)

C)


 * Calculations:**

Centripetal Force

These calculations were completed on the pre-lab sheet. Set "N"s equal to each other:



Angular Velocity (ω: radians/ second)

Theoretical Radii

We used the above equation to convert RPM to Angular velocity in order to calculate the theoretical radii for each speed of the record player. To solve for one variable, we plugged in ωR for V, 15˚ for theta, 9.8 for g, and .214 for µ in the following equation:



57.416 RPM= 6.01 rad/s



45.593 RPM= 4.77 rad/ s



34.247 RPM= 3.59 rad/s



17.386 RPM= 1.82 rad/s



57.416 RPM:
 * Error Calculations:**

45.593 RPM:

34.247 RPM:

As seen in the graph, as the velocity increases, the radius will decrease. This is represented by the following equation: R=c*v^2.
 * Graphs:**

//Other Objectives://

A) Radius of Unbanked Angle

C) Banking Angle

= = =Vertical Circle: Maximum and Minimum Tension= Group Members: Sam Fihma, Evan Bloom, Ryan Listro Class: Period 2 Due Date: January 14th, 2011
 * [[image:Colors_are_jimmys_friends.png height="398" caption="Colors_are_jimmys_friends.png"]] ||
 * **Conclusion:** The first part of our hypothesis was proved to be wrong. After experimenting, we found that velocity is actually inversely related to the square root of the radius. As shown in our graph, as velocity increases, the radius decreases**.** The second part, saying that the velocity of the penny and the radius are directly related, was also shown to be false. As we increased velocity, the distance from the penny to the center of the circle actually decreased. The third part of our hypothesis was the only one that was correct. From our lab, we determined that the angle and the radius are directly proportional. As the angle is increased, the radius, too, increased. From the graph, we can see that the two have a linear relationship. One major source of error in this lab is the fact that we were not able to use one of the speeds on the record player. When we tried to put the penny on the block of wood at 78 rpm, we could not get the penny to stay on block of wood. Experimentally, 78 rpm was not a real value for the velocity, showing that it is above max velocity. However, experimentally, 78 rpm is a usable velocity. However, since we could not get it to stay, we could not use it as part of our experiment, therefore losing an experimental value. Also, we calculated that the was some error in the record player itself. While it was said to spin at either 44 rpm, 35 rpm, or 17 rpm, we calculated error in these. We found the error to be 13.64%, 9.76%, and 7.91%, respectively. Also, as with every other lab, there is human error in terms of the timing.


 * Purpose:** The purpose of this lab is to find the maximum and minimum tension/velocity for our string and mass. We will then compare the numbers we received from experimentation to the theoretical and numbers and see how well our method worked.


 * Hypothesis with Rationale:** Our group believes that when the maximum speed will be reached, the string will break due to the increased tension. The minimum speed will occur when the mass spins so slowly it will barely be able to complete a revolution without falling (not in a circular path).


 * Materials:** The materials that we used include white thread, mass (rubber stopper), a mass hanger that was used for spinning the string in a vertical circle, a stop watch, a meter stick that was used for measuring the radius, and scissors.

Maximum Tension: 1. Cut a desirable amount of thread (Note: The length does not matter). 2. Tie one end of the thread to a mass hanger and tape the other end to the top of a table. 3. Allow the mass the hang freely in the air. 4. Carefully add weight to the mass hanger. 5. While adding weight, make sure you hold down the taped side of the string in order to prevent it from sliding underneath the tape. 6. Add weight until the string breaks. Record this value in an excel spreadsheet. (Note: You may need to perform the experiment multiple times to determine a more precise result.) 7. In the spreadsheet, multiply your recorded mass by gravity (9.8). The product will be the maximum tension of the thread.
 * Procedure:**

Minimum Velocity: 1. Cut a desirable amount of thread. 2. Tie one end of the string to a mass and the other to a mass hanger (Note: The hanger is optional as its sole purpose is a grip). 3. Once tied on each end, measure the length of the string. This is the radius of the circular motion. Also, weigh the mass being spun. 4. Once the mass and hanger are secure on each end of the string, spin the mass in a vertical circle. Try to spin the mass as slow as possible while still maintaining a decent amount of tension on the string. 5. As the mass in being spun, use a timer to measure the time of ten revolutions. Divide this time by ten to determine the time of one revolution. 6. Repeat step five multiple times to determine the average time of each revolution. 7. After each trial, record the time, mass, and radius in an excel spreadsheet. 8. To determine the experimental value of minimum velocity, divide circumference by time. To do this, you first need to find the circumference which can be found by multiplying your radius by 2π. Divide this number by average time of one revolution.

media type="file" key="azaesrxdctufyvigubohinpjok.mov" width="300" height="300" Maximum Velocity: 1. Measure a desirable amount of thread. 2. Tie one end of the string to a mass and the other to a mass hanger (Note: The hanger is optional as its sole purpose is a grip). 3. Once tied on each end, measure the length of the string. This is the radius of the circular motion. Also, weigh the mass being spun. 4. Spin the mass in a vertical circle as fast as possible. Do this until the string eventually breaks. 5. At the same time, use a timer to record the time it takes for the string to break. Divide this time by the number of revolutions it took for the string to break. The result will be the time of one revolution. 6. Record the time, mass, and radius in an excel spreadsheet. 7. Repeat steps five and six until you reach a desirable result.

The theoretical and experimental values found in these charts used formulas that can be seen below.
 * Data:**

Maximum Tension: This occurs when the ball is at the bottom of the circle.
 * Free Body Diagrams:**



Minimum Tension: This occurs when the ball is at the top of the circle.




 * Calculations:**

Maximum velocity was found using the following calculations:



This is the percent error between the two numbers we found.



Minimum velocity was found using the following calculations: This is the percent error between the two numbers we found.


 * Conclusion:** For our maximum velocity, our hypothesis was correct. Once we reached maximum velocity when spinning the mass on the string, our string broke. Using the formula velocity = displacement over time, we were able to come up with our experimental velocity for this experiment. Displacement is equal to 2-pi times the radius (circumference), and time was found to be the period, or time over number of revolutions. After doing our calculations, we found the experimental velocity to be 12.26 m/s. We compared this to the theoretical value, or what the ideal velocity should be. We calculated this velocity to be 13.10 m/s. We found the percent error between these two to be 6.41%. Error could come from multiple sources. For one, the period could be off because human reaction cannot measure the exact second from when we started spinning the mass until the exact moment when the string breaks. Also, the string could have already been worn down or not tied tightly. This may cause the string to break at a slower velocity and a lower tension than the true maximum. For our hypothesis for minimum velocity, our hypothesis was also true. We found the lowest possible velocity that we could while still keeping circular motion. We calculated minimum velocity the same way we found maximum velocity: we used the equation v = d/t, with d = circumference and t = period. We did this multiple times and found several different minimum velocities in order to get a more accurate and precise experimental answer. We determined that the experimental minimum velocity is 2.24 m/s. Our theoretical minimum velocity was 1.97 m/s, creating an error of 13.71%. This is not terrible, but it is not as low as our percent error for maximum velocity. While human timing error is obviously a source for this part of the lab as well, another place where error could be found moving the mass in a circular motion as possible. As humans, we are not capable of getting the lowest velocity absolutely perfect. Therefore, the period we get may actually be smaller than what it theoretically should be, creating a larger velocity. Our lab is comparable to a roller coaster with peaks or a car going over a hill. A maximum velocity must be found so that the car (whether on a roller coaster or a road) can still get over the hill, but is not going so fast as to travel over the hill in a linear path and going off path.

= = =Circular Motion=

Group Members: Sam Fihma, Evan Bloom, Ryan Listro Class: Period 2 Date Due: January 7, 2011


 * Purpose:** Find the relationship between speed and centripetal force.


 * Hypothesis:** The speed or velocity will be directly related to the force or tension of the system in circular motion. As the speed increases, so will the tension, increasing the chances that the string will snap.


 * Materials:** The materials used during this lab include a force sensor with USB Link, string, rubber stopper and additional masses, data studio, and timer.




 * Procedure:**

1. Set up the experiment. Set up data studio by plugging the sensor into the computer. Tie a mass to a string and the string to the force sensor.

2. Measure radius and mass by measuring the length of the string and weighing the mass on a scale. To find the circumference or distance that the mass will travel, multiply the radius by 2π.

3. Spin mass and string in a circular motion. While this is occurring, use the timer to measure the time it takes for the mass to travel 10 revolutions. Divide this time by 10 to find the time it takes for one revolution. Also, record the trial in data studio.

4. After each trial, take the mean of the data in Data Studio. You can also find the y-intercept of the data as an optional data point. Both of these numbers will represent the force.

5. In excel, record the force, mass, radius, circumference, y-intercept, time of one revolution, and velocity, which is found by dividing circumference by time.


 * Data:**

This is all the data we compiled for this lab. The mass that hung on the end of the string was .4197kg, while the string was .348m. Since we spun the mass holding the end of the string, the string length was our radius. The circumference was found using the equation Circumference=2(Pi)(Radius). We did multiple trials, each one spinning the mass at a different velocity. This is why our times are different each time. The way velocity was found is shown below. The Y-Int and Centripetal Force were found using Data Studio.


 * Graphs:**

Our Graph: This is a velocity v. force graph (using our own data) that proves that velocity and force have a direct square relationship.

Force v. Mass Graph: (Hallowell, Dember, Seigel, Pontillo) This is a graph of force v. mass and it proves that centripetal force and mass are directly related.

Force v. Radius Graph: (Ani, Ariel, Rachel, Sammy) This is a graph of force v. mass and it proves that centripetal force and mass are directly related.

For speed:
 * Sample Calculation:**

Sample Calculation for Radius of 0.348m and Period of 0.744s:

Originally, we stated that the velocity of the mass will be directly related to the tension force of the string. In other words, as the velocity increases, the force will increase as well. As evident by our data table and our graph, this hypothesis was proven true to some degree. The equation of our Velocity vs. Force graph showed that force increased as velocity did. However, the equation was also quadratic. This means that velocity is a direct square to centripetal force. While our results were accurate enough to come to this conclusion, there were many places where possible errors could occur. The first one has to do with the velocity of the mass at the end of the string. It may have been accelerating not only in terms of direction, but also in terms of magnitude. Velocity may not have been constant throughout, causing the force to fluctuate based on the velocity. Another source of error may come from the fact that the circle being created may not be completely horizontal. The radius would extend out at an angle, creating angular forces which would differ from if the theoretical force if the circle was completely flat. Another human problem that may be present is the reaction time of the person controlling the stopwatch. It is very hard to start the stopwatch at the precise moment that the circle starts making its revolutions and stopping at the precise moment that the circle has finished making the desired amount of revolutions. Since the time is not completely correct, the calculated velocity will not be 100% accurate, either. In addition to human miscalculations, there were also some problems with the equipment being used too. The tape being used to keep the string attached to the sensor could have distorted the results of the force. Also, the string may not have been completely stabilized. Therefore, the string may have shifted, changing the radius, even if only slightly. However, it would still change our results. The final source of error would be with the force sensor. While it is a high tech piece of equipment that is more precise than the human eye and older technology, it is still not completely accurate as shown by our y-intercept. Solutions to these sources of error include becoming more meticulous in our experimental process to reduce human error, buying an even more advanced, more precise force sensor, and spinning the force sensor while holding the mass instead of the other way around. Overall, we made many mistakes in the beginning of this lab. There were many trials in which we got data that did not give us the results we wanted. We had to go back and redo trials multiple times to get what we wanted. ||
 * Conclusion:**