Group2_2_ch5

toc =Centripetal Force Lab=

Task A- Dani Rubenstein Task B- Danielle Bonnett Task C- Andrew Chung Task D- Michael Solimano


 * Objective:**

To determine the relationships between velocity, mass, radius, and centripetal force.


 * Hypothesis:**

The graph of velocity v force will be positive and polynomial. The graph of mass v force will be positive and linear.


 * Methods and Materials:**

For this lab, there were numerous materials that were needed to be used to complete it. First, we had to attach the rubber stopper to a string, which went through a tube, that was attached to a force meter that we held down on the ground. We used the USB link to record our data into Data Studio and then took that information and put it into Excel. We used the stop watch to record the time that the stopper went in 10 circles, and then divided it by 10 to get the time for one turn. Finally, the meter stick was used to measure the radius of the circle that the stopper was making.


 * Procedure:**

1. Attach the analyzer to one end of the string. 2. Thread the string through the tube. 3. Attach the other end of the tube to the rubber stopper. 4. Hold the analyzer down on the ground and "zero" it. 5. Begin whirling the stopper. Do this in such a way that the radius doesn't drastically change. 6. Using a stopwatch, measure the time it takes for the rubber stopper to revolve 10 times. 7. Record all the data in an excel file. 8. Repeat steps 4-7 three times. 9. Create a graph for analysis using the Excel sheet.


 * Video:**

media type="file" key="Movie on 2011-12-14 at 08.57


 * Graphs and Data:**


 * Our Data: Radius**
 * Our Data: Velocity**
 * Our Data: Mass**
 * Sample Data: Radius**
 * Sample Data: Mass**


 * Sample Data: Velocity**




 * Our Graph: Mass**



Your Graph: Radius

Your Graph: Velocity

Graphs: These graphs show the various relationships between centripetal force and mass, radius, and velocity. The fits of these graphs help in showing us the relationships between these variables.


 * Analysis:**

Our Net Force vs. Mass Graph resulted in a graph with a linear line of best fit. The r^2 value for this graph is .80101 which indicates a decent pattern. Our slope was calculated as being 1.7262. When you stop to think about it, this makes complete sense. The heavier the weight is, the more centripetal force there is. This is because if you look at the equations, as mass increases force does as well. Thus, a bigger weight yields a bigger net centripetal force. Our graph for centripetal force vs. radius is a negative power graph. Though it doesn't look it on the graph, you must take into account that centripetal force is equal to one divided by the radius. Thus, if you do these calculations and graph the subsequent results, you will get the graph. We also discovered that the graph for centripetal force vs. velocity is a power function, which makes sense because if you look at the Circular Motion equations, centripetal force is proportional to the value of velocity squared. Our group was able to complete a lot of this experiment during the several class periods we were given. However, certain data in our lab needed to be supplemented by the lab results of last year's class.


 * Conclusion**:

Was the purpose satisfied? The purpose of our lab was satisfied, as we observed the relationships between mass, velocity, and radius, and centripetal force. We hypothesized that the graphs of our mass v centripetal force, and radius v centripetal force, would be positive and the graph of velocity v centripetal force would be positive and polynomial. Only one of these graphs, mass v centripetal force, was confirmed in our experiment. As the system mass was increased, the centripetal force also increased in a linear fashion. The graph should be linear, as centripetal force is proportional to mass. The equation of this line was y = 1.7262x and the R^2 value was .80102, which although is not perfect, does show that it was a good fit to our data. Our graph of radius v centripetal force was a positive polynomial fit, where it should have been a negative power graph. The sample data used, however, does fit a power graph, with equation y = 2.5143x^2 - 1.5537x + .2848, with an R squared value of .99925. This graph should be a power fit, because centripetal force is proportional to 1/radius. Our final graph was not conclusive, as we did not have time to complete the velocity v centripetal force test. However, this data should have fit a positive power fit, as is shown in your data, because centripetal force is proportional to velocity squared. The R squared value for the sample data was .89758, showing a weaker representation of data points that fit the equation y = .0141x^2 - .0516x + .4252.

There was obvious error in our radius v centripetal force data, as well as our mass v centripetal force graph, as the R^2 value was low. Most of the error probably arose from the set up of our lab. For example, when testing the changing mass v centripetal force, it was difficult to keep a constant velocity and constant radius. Had these changed, which was inherent in the lab, it would have led to error in our lab. Also, when taking the mean of our centripetal force in data studio, it was often difficult to locate the data which was representative of our test. Had we used extraneous points it could have led to further error.

Had we done this lab again, we could have used a tool to keep desired variables as constants, for example the radius size, instead of trying to keep them constant by hand. By doing this, we could have reduced error in our lab, and our graphs may have fit the proper trendlines. This lab has many real life implications, including in analyzing the centripetal force of a helicopter's rotors, and the force they put on the helicopter.

=Finding Minimum Speed=


 * Objective:**Determine the minimum speed of an object that is attached to a 1 meter string.


 * Hypothesis:** After solving for the velocity in the equation below, we believe our velocity should be 5.61 m/s.


 * Methods and Materials:** First, we used a meter stick to measure a piece of string that is 1 m long. We then used a weight and tied it to the bottom of the string. Then we slowly swung the string around in a horizontal circle, trying not to let the string pick up any slack. In order to time how long it took to complete ten revolutions, we used a stopwatch. We then repeated this process two more times.


 * Our Data:**


 * Class Data:**


 * Calculations:**
 * Theoretical Value:**
 * Percent Error:**


 * Percent Difference:**

After calculating the percent error to be 80.095% we were curious to find out why is was so large. Some of the main factors that could have caused this could have been not being able to move the string at the exact minimum velocity. It was also very difficult for us to get the least possible amount of tension due to us being so inconsistent. It was also hard to get exactly a 1 meter radius the whole time. The slowest velocity possible also created a problem for us due to the lack of tension and inability to maintain a circular motion. To fix this main source of error, one can constantly remeasure the length of the string before and after each trial. Someone can also try to use a device that would allow the system to spin that is slow enough to be close to the minimum velocity, but also maintain enough tension to have a complete circular motion.
 * Analysis/Conclusion:**

=Class Lab: Conical Pendulum=

The purpose of this lab was to calculate the velocity and period of the pendulum as it moved in a conical circular motion at different radii. We were looking to find the relationship between the radius of a conical pendulum and its period and our group hypothesized that the larger the radius becomes the longer the period will be.
 * Purpose, Hypothesis and Rationale:**

A swinging pendulum was attached to a string (2.48 m long found using a meter stick) that hung from the ceiling. As Ms. Burns pushed the ball in a circular motion, the class observed and each used a stop watch to mark down a certain period. Each person's data was recorded and a class data table was created using Excel.
 * Materials and Methods:**

First, we discussed the lab as a class and each recorded certain information in our notebooks such as the length of the string (2.48 m) and the mass of the object (1.744 kg). Then, we observed as Ms. Burns pushed the pendulum in a circular motion. First, the radius of the situation was .2 meters. Then it was moved to .5 meters. Then it was moved to .75 meters. Finally, it was moved to 1 meter. Each student observed and recorded what they observed to be the period of the pendulum. We then went around and each read our result, and a class data table was recorded.
 * Procedure:**

This data was taken from the results of the entire class. At first, it was difficult for everybody to obtain close results. However, as we got further and further into the lab, a much closer range of results was observed. One can see that as the radius increased, the time (period) of the pendulum decreased.
 * Class Data:**

Below is a sample calculation of our computations when the pendulum at a .5 radius.
 * Sample Calculation:**
 * Percent Error:**


 * Using the above calculations, we got:**

//Period//:

.2m radius = 3.16s .75m radius = 3.086s 1m radius = 3.024s

//% Error://

.5 radius -> .96% error .2 radius -> 2.84% error .75 radius -> 1.43% error 1 radius -> 1.19% error


 * Analysis Questions: **

1. View calculations above

2. View calculations above

3. The accuracy of our data is certainly at question. For the radius of .5 meters, we calculated a percent error of .96%. This value is certainly huge and I do not know what could have caused this. It could definitely be due to a simple calculation error. The value of the period at .5m radius is suspect. The rest of the results, however, were quite accurate. The percent errors were on par with the rest of the class'. We Our results were also fairly precise. They were close to the values that every other group got and they were also close together, with the exception of the .5 radius.

4. The tangential axis was not usable in this lab. This is because there were no forces acting on it, and thus it would not have been usable when solving our data.

5. The bigger the mass of the object swinging, the slower the velocity will be, and thus, the slower the period will be. Because of our equation Tx = m * (v squared)/radius, the more mass, the more that the root of the velocity will be divided by. This thus means that the velocity will be slower.

6. The period was longer as the radius decreased. Because our radius was used in three distinct areas of our lab, it had an effect on many variables including our angle, the velocity, and the tension. This will make the graph a non-linear fit.

7. There were many sources of error inherent in the lab. One source of error was the path of the object in its circle. It would have been difficult to have it move in a perfect circle. If it moved, forming an oval, its radius would be shorter at some points, and longer at others. Thus, a perfect circle would have been necessary to have accurate results. Also, there was error associated with the timing aspect, where different students were timing the rotations from different perspectives, which could have led to error in timing, and thus in our calculations.

=Car Around a Turn Lab=

Analysis: Andrew Chung Task B: Michael Solimano Task C: Daniele Bonnett Task D: Dani Rubenstein

The purpose of this lab is to examine the relationship between the radius of a turn and the maximum speed at which a car can make the turn. Also, we will analyze the effect that banking, and the steepness of banking, has on the radius at which a maximum velocity can successfully be navigated.
 * Objective:**

1. The larger the radius, the larger the maximum velocity with which a car can make a turn will be. This is because the turn will be less sharp, and less force will be needed to keep the car in line with the circle. 2. Banking creates a force pointing the car towards the center, or around the circle. Thus, banking will allow a car to round the turn at a larger speed, and with a smaller radius. 3. The steeper the angle, the smaller the radius will be that can hold the car in its maximum speed around the circle. This is because banking creates a normal force towards the center, so the turn can be more sharp and facilitate the car moving at a max speed.
 * Hypotheses:**

The materials that were given to conduct this lab included a rotational turntable, a power supply, a photogate, a mass, and a metric ruler. The mass is placed on the rotational turntable, and a photogate is also attached to this. The photogate is used to measure the period of the mass traveling around the spinning turntable. The power supply is used to power the spinning turntable, which will allow us to conduct the experiment.
 * Methods and Materials:**

A mass was placed on a turntable, attached to a powersource, at a designated radius. The voltage of the power source was slowly increased, turning the turn table with increasing velocities, which were measured from data taken from our photogate and our given radius. The velocity was slowly increased until the mass slipped off of the turntable, thus the maximum velocity that it could travel while still turning about a uniform radius. With the data collected we were able to find the maximum velocity that the mass could travel around our radius.
 * Procedure:**


 * VIdeo of Procedure:**

media type="file" key="Movie on 2012-01-04 at 08.22.mov" width="300" height="300"


 * Data and Graphs:**

Our Data for .2 m Radius The above graph shows our trials attempting to find the maximum speed at which the mass can travel around our designated radius. It will be used to analyze the maximum velocity of a mass around changing radii when joined on a graph with the class data.

Class Data Tables Radius vs. Max Speed Radius vs. Mu The above graphs are of the classes data. This data will be graphed to find a trend associating max velocity and radius length.




 * Analysis**

1. Discuss the shape of the graph and its agreement with the theoretical relationship between R and V.

The shape of the graph is very unique. In the past, we have dealt with linear and polynomial functions. However, both of these graphs do not fit the theoretical relationship between R and V. Instead, the shape of the graph would be a power graph. This becomes clear when we look at the equation for velocity. We found velocity to be equal to the square root of Mu times gravity times the radius. If we do a little bit of shifting around, we can re-write the equation as velocity equals the square root of mu times gravity times the square root of R. If we instead substitute R raised to the 1/2 instead of the square root of R, our graph fits the equation y = Ax^B. In this case, B would be .5 or 1/2.

2. Derive the coefficient of friction between the mass and the surface.







3. Compare your coefficient of friction with that of all the other groups. (Be sure to post a data table with the class values).


 * Radius || Mu ||
 * 10 || .68 ||
 * 20 || .47 ||
 * 25 || .64 ||
 * 30 || .55 ||
 * 40 || .56 ||
 * Average= || .58 ||





% Difference of graph µ and our µ:

4. A car goes around a banked turn. Find an expression for its maximum velocity in variables only. How do you think the graph would change if you performed the same procedure but with an angled surface, instead of the level surface we used?

a)

for Flat:



b) The equation for a banked circle would be different than the equation above. We would get:

which is different in that it includes various trigonometric symbols. This equation indicates that with a smaller radius on a banked turn, you can achieve faster speeds.


 * Conclusion:**

We hypothesized that the greater the radius was, the greater the maximum velocity would be. Also, we predicted that banking would be what allowed a car to round a turn with a smaller radius at a higher velocity. Finally, we hypothesized that the steeper the angle was, the smaller the radius would be. After completing the lab and observing our data, we learned that our first hypothesis was definitely correct. Although we did not perform an experiment to test the next two predictions, by observing our data and using outside knowledge we can conclude that the rest of our predictions were correct as well. We saw that at a higher velocity the object flew off the side so that helped us to conclude that banking would allow a car to make a turn at a higher velocity. Also, by thinking hypothetically about the situation in comparison to the lab, we concluded that the steeper angle would definitely mean that the radius would be smaller. At first, our results were slightly off from what they should have been and we received results that did not match the correct ones. However, after completing a new set of trials, our results were much more accurate and that can be seen through our percent error and percent difference. The percent error of our exponent (B) was 6% and our percent difference from the class average was 18.96%. Both of these percents fall clearly in the 20% range, evidently showing that our results matched the expected ones. However, as expected, these percents show that our results were not perfect and that there definitely were some sources of error that contributed to it. One could possible be basic human error throughout the lab. From reaction time to adjusting the dial that increased/decreased the velocity, it was hard to get constant results throughout the entire lab. One could have easily had delayed reaction time, thus resulting in slightly different results. Another possible source of error, also coming from basic human errors, could be the rate in which the dial was turned up. It was difficult to remain constant throughout the entire procedure, as it was impossible to know the exact point to which it was turned each time. Although our small percent error shows that we were pretty accurate, the dial could have been turned to slightly different levels during each trial. In the future, one could fix this errors by having more people working on data studio, so there would be less chance of a delayed reaction time and an average could have been taken for each trial. Also, if the machine was able to be set to turn to a certain speed each trial it would have got rid of the human error source of turning the dial. Although we did not apply banking to this lab, it still easily can be related to driving in real life. While making a turn, there always is a certain maximum velocity that one could go at before an accident could occur. Although banking helps to decrease the risk in this situation, we still were able to relate actual driving to the lab where the mass flew off of the spinner.