Chloe,+Justin,+Steve,+Andrew

= = toc = = =Lab: Moving in a Horizontal Circle (unbanked)=


 * Pre Lab Derivations**:

For An Unbanked Turn:





For a Banked Turn:






 * Purpose:**
 * To find the relationship between radius and maximum speed that allows an object to remain in its uniform circular motion on an unbanked turn.
 * To find the relationship between radius and maximum speed on a banked turn as an object moves in uniform circular motion.
 * To find the relationship between the radius and the banking angle around a curve as an object moves in uniform circular motion.


 * Hypothesis:** 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.


 * Procedure:**
 * 1) Using a stopwatch, test the actual RPM of the record playing turn table for its slowest speed (16 RPM)
 * 2) With the record player set to its slowest speed, Test a set radius to see if the penny will stay in a uniform circular motion.
 * 3) Repeat step two and increase the radius until you are able to find the maximum radius of the circular path for this velocity.
 * 4) Repeat steps 1-4 using the next fastest speed for the record player and so on for the rest of the speeds.
 * 5) At some of the slower speeds, we needed a greater radius for the object to escape, so we used a meter stick, which we assumed had the same coefficient of static friction as did the record player, for their surfaces are very similar.


 * Data:**





Other laboratory investigations were conducted to determine the relationships between maximum velocity and radius, and banking angle and radius. Their graphs are shown below, and will be discussed more in detail in the conclusion.



Thanks to Emily, Elena, Emily, and Amanda for the above graph.

Another group investigated the relationship between banking angle and radius. Their graph (thanks to Nicole's group) is shown below:







Our objective was to find the relationship between the radius and the escape velocity. We hypothesized, based on the equation force = mass*velocity squared)/radius, that as the radius increased, the velocity would increase as well, proportionally squared. Our hypothesis is supported by our evidence, as our 4 points fit the v squared versus radius model very well, illustrated by our r value of 0.9976.
 * Conclusion**:

There are some places in this lab where error could have occurred. First, we assumed that the record-player was moving at absolutely constant speed, which, if it was not, could have changed our results. Since we didn't have a spinning disk with a large enough radius to find the radius whose escape velocity were 18 rpm and 37 rpm, we had to attach a plank of wood to add radius. This extra weight could have slightly decreased the velocity we had previous calculated for each of the settings. Also, since this was a different surface (though we tried to find the most similar one we could) than the disk we originally tried, the coefficient of friction could have been slightly different than we had used for the other trial, and for our theoretical calculations. We marked the radii on the disk and moved the coin as the table was turning to find the greatest radius at which the coin would not leave orbit, but since we moved the coin while it was in orbit, there is a chance that the measurements for radius were not completely precise. Lastly, since we measured the velocities for each of the settings by hand with a stopwatch, the measurements could have been more precise.

The most important change that could be made in this lab to make the results more accurate would be to find a much larger disk to spin on the turntable. With a radius that accounted for all the radii we needed, there would be a constant weight and surface friction between all of our runs. To make the radius more precise, we could have stopped the turntable between setting the coin at each of the radii, but this would have been significantly more time consuming. Lastly, we could have figured out the velocity of the turntable at the different settings mechanically, with a device like a photogate, so our readings would be more accurate.

Other groups conducted similar investigations, investigating the relationship between maximum velocity and radius on a banked turn with a constant angle. This group found that as the velocity increased, the radius decreased. This does make sense. Starting with the equation F = ma in a circle, where a is v^2/r, we are left with F = mv^2/r. The centripetal forces in this case are the x components of friction and normal of the incline forces, which then becomes Nsin x + f cos x = mv^2/r, as modeled earlier in the pre-lab derivations. As the radius increases, the net centripetal force decreases (for they can be on the same side of the equation), which then causes the velocity to decrease proportionate to the square root of the net force. Because the radius increases, there is less of a centripetal force pulling the object to the center, meaning that its escape velocity will be lower because there is less of a force that the the velocity must combat.

Another group investigated the relationship between the banking angle and the radius of a circle. They discovered that this relationship is an indirect one. As theta increased, the radius decreased, making it harder for the object to escape (for it is closer to the center), making the escape velocity greater.

These calculations are extremely relevant in the "real world". The same process we used for this investigation can be used to set the speed limits at certain turns. If physicists/engineers find the max speed that a car can stay in orbit around a turn with a certain radius, without friction, this speed is the fastest a car could possibly make it without skidding off the road. Since there will most likely always be friction between car tires and the road, the turn could be make at a slightly faster speed, but shouldn't be!

=Lab: Minimum and Maximum Velocities in a Vertical Circle= Group Members: Justin, Chloe, Steve, Andrew Due Date: January 11, 2011 Honors Physics period 2 = =
 * Purpose:**
 * To find the maximum and minimum speeds at which the tension is at a minimum and maximum.
 * To compare our calculated minimum and maximum velocities we actually receive from the experiment.
 * To change the radius while keeping mass and tension constant to see how this affects maximum and minimum velocities.


 * Hypothesis:** We expect that the maximum speed at which the tension is at a maximum will be the speed that causes the string to break from the tension force. We also expect that the minimum speed at which the tension is at a minimum will be the slowest speed possible for the mass to complete a circular revolution. Also, when finding minimum and maximum tension, the minimum velocity of the mass will increase as the radius increases, but the maximum velocity of the mass will decrease as the radius increases.

Materials:
 * Thin thread
 * A mass (50 g)
 * Force Sensor
 * Computer
 * Microsoft Excel
 * Data Studio
 * Timer


 * Procedure for Minimum Tension/Velocity and Changing Radius:**


 * 1) Tie a mass to a thin thread
 * 2) Tie the other end of the thread to a force sensor, with the thread at a set length.
 * 3) Plug the force sensor into a computer, and be sure that someone is prepared to record the force caused by the moving mass.
 * 4) Have one person hold the force sensor so that the mass is hanging below. If necessary, hold string to remove tension from the force sensor and set the zero for your graph.
 * 5) Holding the force sensor, a person moves the hanging mass in a vertical, circular motion at a constant speed.
 * 6) Using a stopwatch, record the time it takes for the mass to revolve a certain amount of times, as well as using the force sensor to record the tension caused by the string.
 * 7) Repeat steps 4-6 with the mass revolving at a slower speed. Continue doing this until the person moving the mass is unable to have the mass complete a revolution, and gravity pulls the mass before it can complete it's circular path.
 * 8) Use the data recorded from the slowest trial in which the mass completed a revolution along the circular path.

media type="file" key="Min Tension J.mov" width="300" height="300"


 * Procedure for Maximum Tension/Velocity and Changing Radius:**


 * 1) Tie a mass to a thin thread
 * 2) Tie the other end of the thread to a force sensor, with the thread at a set length.
 * 3) Plug the force sensor into a computer, and be sure that someone is prepared to record the force caused by the moving mass.
 * 4) Have one person hold the force sensor so that the mass is hanging below. If necessary, hold string to remove tension from the force sensor and set the zero for your graph.
 * 5) Holding the force sensor, a person moves the hanging mass in a vertical, circular motion at a constant speed.
 * 6) Using a stopwatch, record the time it takes for the mass to revolve a certain amount of times, as well as using the force sensor to record the tension caused by the string.
 * 7) Repeat steps 4-6, but have the person moving the mass to increase the speed. Repeat this process until the string snaps and the mass goes flying. (Make sure that nobody will be hit by the mass when the thread snaps!)

media type="file" key="Max Tension J.mov" width="300" height="300"


 * Data:**

Minimum Velocity Data Table:



Maximum Velocity Data Table:



Max Tension Data Table:



Max Tension Data Studio Graph:



There are two lines here because we accidentally hit stop at about 130 seconds. We started again with the blue line, and the rope broke when the tension was 5.6 N.

Minimum Velocity vs. Radius Graph:



Maximum Velocity vs. Radius Graph:






 * Sample Calculations (with free bodies):**

//Minimum Velocit////y://

//Maximum Velocity://

Between theoretical power value (1/2) and actual power value (.439)
 * Minimum Average Velocity Part Error Calculations:**

Between theoretical power value (1/2) and actual power value (.9204)
 * Maximum Average Velocity Part Error Calculations:**

Essentially, our group set out to find the minimum velocity a weight at the end of a string could be swung in a vertical circle, with out the object falling out of orbit. We hypothesized that as the radius increased, so would the minimum velocity to keep the object in orbit, and our graph and upward trend-line suggest that this idea is correct. Our other objective was to find the maximum velocity the object could be swung at before the string broke - based on the greatest amount of tension the string could take without breaking. We hypothesized that this max speed would increase as the radius did. This hypothesis, too, was supported by our table and upward trend-line.
 * Conclusion:**

For our first and second runs, we got 12.2% and 84% error, respectively. This lab was especially prone to error because many of our calculations were based on actions of speculation. For the lab to find the minimum velocity, we simply spun the weight as slow as it seemed we could without it falling out of orbit. This gage was not at all precise, and led us to, most like, spin faster than 0 tension at the top of the orbit. We recorded the speed by timing with a stopwatch, controlled by human hands. Because our reaction time is nowhere near perfect, that too is a factor in our error. For the second part of our lab, we measured the maximum tension the thread could bear with a force meter. These meters only measure to one significant figure, so they aren't too precise - an area of error. Another possible reason for our error is the fact that both times, but more likely in the second part, we may have been spinning the weight at greater than 0 acceleration, which would confuse our calculations. Again, we timed the speed with a stopwatch, the reaction to the exact moment the string broke, and the amount of rotations we counted and used for our calculations, may not be precise. Lastly, in regard to the second part of the lab, our large percent error could be attributed to the very small variation in radii. With only a few points, one stray point could throw off the entire graph and could add to our percent error.

We could improve this lab by using more precise technology. If we has some kind of machine that could spin the weight on the string at a set constant speed, we could arrange for it to go as slow as possible without falling out of orbit. With timers that somehow started and stopped with a mechanical recognition system, like a photogate, the timing would be more precise, and therefore, so would our velocity measurement. If we had a video camera, properly aligned to our spinning weight so the entire scene was proportional, we could slow down the footage (for the second part of the experiment) and measure more precisely the amount of rotation the weight completed before the string broke. With a better number for this, our velocity for this part of the experiment would be more precise.

This information could be valuable to companies manufacturing orbiting rides. They must make sure that with the given radius and weight of the passengers and passenger-receptacles, the ride isn't going to slow that, once they reach the top, the passengers will fall out. They must also make sure that the ride isn't going too fast that the passengers and their compartments will actually fly off.

=Lab: Circular Motion=

Group Members: Justin, Chloe, Steve, Andrew Due Date: January 7, 2011 Honors Physics period 2


 * Purpose:** To test the effects of changing an object’s speed on the force causing circular motion.


 * Hypothesis:** We believe that speed should relate directly to the force causing circular motion (tension), for as on object is spun faster, it has a greater tendency to break off from the string, thereby increasing the tension needed to keep object attached to the hook.


 * Materials:** Timer, Force Sensor, Mass (50g), String, Computer, Data Studio, Excel


 * Procedure:**
 * 1) Attach string to force sensor at a set length.
 * 2) Measure the length of the string (this is the radius of the circle).
 * 3) Plug force sensor into computer.
 * 4) Hold force sensor in air (so that string does not hit cord).
 * 5) Rotate in air (try to keep the circle as flat as possible).
 * 6) Record the time it takes for 10 revolutions, as well as the Tension in the rope (using data studio).
 * 7) Make a best-fit line in data studio and use the y-intercept for the tension in the rope.
 * 8) Make sure that the radius (string length) and mass are the same throughout all the trials.
 * 9) Divide the time for 10 revolutions by 10 to find the time for one revolution.
 * 10) In order to find speed, divide circumference by time.
 * 11) Compare speed trend to tension trend.


 * Data:**

//Trial Runs Graphs://

Trial 1:



Trial 2:



Trial 3:



Trial 4:



Trial 5:



media type="file" key="Investigation 1 Lab Video.mov" width="300" height="300"

Data Table:



The Graph:





The graphs of the other variables tested by others:



Courtesy of Nicole, Jillian, Spencer, Dylan



Courtesy of Hallowell, Dember, Seigel, Pontillo


 * Sample Calculations:**

To find time for one revolution:

We used a stopwatch to record how long it took for the mass to revolve ten times. Then we took that number and divided it by ten to find the average time it took for one revolution.

T= time for ten revolutions t= time for one revolution. T/10= t

To find circumference:

The circumference of the path for the mass is equal to 2 x Pi x r. The radius of the path for the mass was equal to the length of the string we used, because the mass traveled along the hook of our sensor.

r= radius C= circumference C = 2 x Pi x r

To find velocity:

The velocity of an object is found as the distance traveled over a certain time. Because we know how long it takes to complete a revolution and how far the circumference of the path the mass is, we are able to divide the distance by the time for a revolution and find the velocity of the mass.

C= circumference/ distance traveled for 1 revolution t= time for one revolution V= velocity V= C/t

Error Analysis/Conclusion:

In our hypothesis, we postulated that the speed of an object moving in a circular orbit would be directly related to the tension of the string spinning the object. This is supported theoretically by the accepted physical equation:. Our experiment illustrated that our hypothesis was correct also. As we increased our speed, according to the graph, the tension changed proportionally to velocity squared.

The error is most likely a result of a number of factors. First, since we were spinning the weight on the string by hand, it is possible that the speed was not perfectly constant. Since we are doing our theoretical calculations assuming the acceleration is 0, our results will yield error. Next, we measured the velocity by timing 10 rotations, finding the circumference, and dividing distance by time. Since we measured time by hand-stopping the stopwatch, our delayed reaction time definitely skewed the timing. This would also produce a result that is different than our theoretical. We spin the weight on the string from a fixed metal hook on the force sensor. Since it did not spin with the string, the string jumped from on side of the hook to the other during every revolution. During this jump, the tension was dramatically decreased, which made the mean of our data skewed. Another factor that caused error in our experiment was the fact that we measured the string length and used that as our radius, when, in actuality, we were not spinning the weight completely horizontal. Since we swung it slightly lower towards the ground, the radius was less than we measured and used to calculate. The force sensors that we used were useful, but could only measure one significant figure. To improve our data, a more precision force sensor should be used. Lastly, because we were spinning the weight in a horizontal circle, but the force sensor hook was “facing downwards”, the sensor most likely measured only the y-component of the tensions – which would have also altered our data and added to our error.

There a number of things that we could do to reduce out error in the lab. First, we could a machine to spin the string and stopper instead of spinning it by hand. This would ensure that speed was constant. Secondly, we could set up a timing device that could count the number of revolutions and stop after it reached a certain number. This would be better than doing it by hand and would cut down on the reaction time needed to preform this step by hand, thereby reducing the error. Third of all, we could use a force sensor with a rotating hook so that string and stopper never had to jump from one side of the hook to the other. This would keep all the values of our graph positive and would reduce our error. Also, if we had more money, we could buy a more precise sensor that would not have as much error.

Carnival ride engineers go through the same process that we did in this lab. For example in a carnival swing ride, It is imperative that they choose the right material to connect the swing to the ride and establish the maximum speed they can travel without the material breaking.