Deanna,+Nikki+D,+Maddy,+Sam

toc =Lab: Elastic Potential Energy= Deanna, Sam, Maddy, Nikki Period 4 Lab date: 3/2 and 3/3 Due: 3/7/11

Part 1:Finding the spring force constant (k).

Purpose: The purpose of this part of the lab was to determine the spring force constant of our spring. This was important because we need k to solve the conservation of energy equation in part 2.

Hypothesis:

Materials:

Procedure:

Graph:

Data:

Sample Calculations:

Error Analysis:

Percent Difference Between Our Result and the Class Result:

Part 2:Determining the relationship between the distance the string stretched and the final velocity. Purpose: The purpose of this part of the lab was to determine the relationship between the horizontal distance the spring stretched and the final velocity of the cart after the spring was released.

Hypothesis:

Materials:

Procedure:

Graph:

Data:

Sample Calculations:

Error Analysis:

Percent Difference Between Our Result and the Class Result:

**Lab: Moving in a Horizontal Circle**
Deanna, Sam, Maddy, Nikki Period 4 Due 1/19/11

The goal of this lab is to investigate the relationships between radius, max velocity, and banking angle. We want to know how the presence of banking affects the relationship between radius and max velocity, and also how different banking angles have an effect on the radius.
 * Purpose**:

A. Relationship between radius and max velocity (unbanked):The radius and max velocity are indirectly related. As the velocity increases, the radius decreases. We believe this based on the equation .
 * Hypotheses**:

__B. Relationship between radius and max velocity (banked):__ __The relationship between radius and max velocity is indirectly proportional (exponentially). We believe this because in solving theoretical calculations, we found that the radii decreased as the velocity increased.__ (This is the one we tested)

C. Relationship between radius and banking angle: The relationship between radius and banking angle is an indirect relationship. We hypothesized an indirect relationship because we believe this will be the same relationship as the other banked experiment.

Turntable with at least 4 speed settings, long wooden plank, tape, duct tape (to raise up the turntable), a nickel, ruler, protractor, wooden wedge to use as an incline, stapler to use an a counter weight
 * Materials**:

1. First, gather the necessary materials. Use a nickel for a mass, since it is easier to move than a block and allows for more possible radii. 2. Experimentally determine the coefficient of friction between the nickel the incline (wood). Use the protractor is measure the coefficient of static friction by observing at which angle the nickel begins to slide down the incline. Use the formula µ=tan(theta) to determine the coefficient of friction. 3. Put the top on the turntable and measure the velocities by timing the number of rotations in 10 seconds. Do this for all speed settings. Do not assume the labeled rotations are correct. 4. Attach the chosen incline to the turntable via velcrow. 5. Assemble the turntable by first putting down the duct tape in order to raise the platform of the turntable. Place the round platform on the duct tape roll, and place the platform on top. Turn on the turntable, and align the platform so that it is directly in the center and rotating evenly. Tape down the platform to ensure that it does not move. 6. Place a counter weight on the side opposite the wedge (a stapler will work) in order to allow the turntable to spin evenly. 7. Secure the long wooden plank to the incline to allow for longer radii. Use tape. 8. Put the nickel on the incline and set the velocity to the first setting. 9. Start the nickel close to the center of the turntable, and slowly move it back until it begin to slide.Do this until the minimum radius at which the nickel does not slide is determined. 10. Record the radius, and repeat. Do at least three trials for each speed. Compare the results with the theoretically determined radii.
 * Procedure:**

Video Procedure: (the second block acts as a counterweight since the turntable board was wobbly)
 * media type="file" key="dm video 4.mov" width="300" height="300" ||

This is our first data chart. The banking angle remained constant for each trial. We changed the velocity on the turntable and measured the radius at which the nickel just began to move. To find the relationship between velocity and radius on a bank, we graphed the velocities and the average radii. However, the velocities we used were incorrect. We had solved for velocity using the radius of our block. Because the velocity is different at each radii, we needed to use the radii we measured to solve for velocity instead. We redid the experiment a second time. Using the formula for angular velocity, we solved for the four velocities on the turntable and their theoretical radii (see sample calculations). This was the correct way to solve for velocity. The theoretical radius for the largest velocity was around 2.2m - much too big for us to test - so we have no data for that velocity. As the following velocities increased, the radii needed for the nickel to move on the board decreased. This is an indirect relationship, so our hypothesis was correct.
 * Data**

This is the graph that was created using the incorrect data from the first time we performed the experiment. We decided that a graph was not an effective way to display the relationship between velocity and the radius on a banked angle. The r2 value was 1, which most likely means that we used the velocities to solve the radii, and so the points were perfect. We did not create a graph for our new data because we did not believe it would better help us understand the relationship.
 * Graph**

Equation for Banked Turn:
 * Sample Calculations**



Theoretical Max Radius at 33RPM (angular velocity of 3.46m/s) mass of nickel: 0.00496kg


 * Analysis of Other Conclusions:**
 * [[image:deanna_other_4.png width="864" height="317"]] ||
 * [[image:deanna_other_5.png width="864" height="369"]] ||

= = In this lab, we found most hypotheses to be correct. Results showed radius and banking angle, and radius and max velocity on a banked angle, to be indirectly related. We were incorrect in hypothesizing that radius and velocity on an unbanked angle would have an indirect relationship. Based on the data of other groups, we found that we were wrong to assume that the indirect relationship between radius and velocity on a banked angle would be the same for radius and velocity on an unbanked angle. Both theoretically and experimentally, we found our relationship to be true. Although the exact quantitative values of the relationship differed a bit between the theoretical and experimental, the overall qualitative relationship was the same. The factors involved in this investigation (incline angle, radius, max velocity) are all involved in any car turning on a road. Although we often do not realize, turns are banked, and the incline angles are determined by engineers that try to make the roads least dependent on friction for centripetal force. Understanding the relationships helps us understand how we decide to turn our cars on the road, and why we are able to succeed in turning a partial circular pathway.
 * Conclusion**

Our hypothesis for part A of the investigation was incorrect. We thought that the max velocity and radius were indirectly related because of the equation f=(mv^2)/r. According to the findings of Jae, Tucker, and Danielle the relationship is actually a direct squared pattern. This makes more sense than our original interpretation of the equation. When the equation is moved around to rf=mv^2 it becomes more clear that it is a direct squared relationship.

According to the data collected by Tom, Rory, Tyler, and Richie our hypothesis for part C of the investigation was correct. We thought that the relationship between the radius and banked angle would be indirect and it was. Radius and the angle are inversely related meaning that as the radius decreases the angle increases.

For this investigation lab we had many sources of error. We performed the investigation twice. Both times there were many things wrong with the lab, however the second time we were able to address the fundamental mistakes made the first time. Our hypothesis ended up correct. We thought that, the radius and maximum velocity were indirectly proportional a d in fact as the velocity increased the radius decreased.

The first time we attempted this investigation we measured the radius of the disk and also measured the velocity at this radius for each speed setting on the turntable. We measured the velocity at each setting by timing how many seconds it took for the table to make ten rotations. We used the seconds per ten rotations to find the period and then the velocity using the radius of the entire disk. Our first major mistake was using the velocity for the radius of the entire disk as the velocity for all of our experimental radii. This was incorrect because as the radius changes, so does the velocity (the period stays the same though). From our data table you can see how for our velocities changed drastically. For example, with our fastest speed setting the velocity changed from 0.868 m/s in our original investigation to 8.030 m/s in our second investigation when our calculation of velocity was corrected.

There were also many sources of error in finding our experimental radii. To start there was the error from the stopwatch when we were timing the rotations per second to find the velocities and period. When we were finding where the coin was in order to move we had to put the coin while the turntable was in motion or else the coin would fly off as soon as the table was turned on. Putting the coin on while in motion made it difficult to measure the exact location of the coin on the block when it moved. Also the block was not long enough for most of the radii so we had to add a piece of wood to the block. This made the disk on the turntable unbalanced and so we had to try and balance the other side of the disk with some sort of mass. In the future we could weigh the block and extra piece of wood and putt exactly that mass on the other side however during the investigation we simply put a mass on until it looked even. Also at the highest speed option on the turntable when the wood was attached to the block the entire disk with block and wood and everything flew off despite the tape we used to keep it attached to the turntable. For the highest velocity we had to use only the block so the coefficient of friction may have been slightly different, however our results seemed to fit with the other two points and because both surfaces were wood we assumed the coefficient of friction would be very similar. The data seemed to fit because as the velocity increased the radius decreased. With the wood and block there was a velocity of 4.5m/s and radius of .277m and without the wood the next setting was a higher velocity at 8.03m/s and the radius decreased to .079m. This assumption definitely could have contributed to our error. To make the lab better we could be more precise when balancing the disk on the turntable (as mentioned in previous paragraph). We also could find more experimental radii. We only did three trial for each speed. To get better results we could do more trials for each velocity. Also we only had four different speed settings on the turntable, one of which we could not test because according to the theoretical radius would have been too long of a radius for us to find with the materials that we had. At a velocity of 1.690m/s the theoretical radius was about 2.2 meters. It would have been more beneficial to see more than three different velocities to compare. Also when finding the period for each velocity we could time for more rotations so that the effect that the stopwatch had on our error could be minimized. Lastly we also could not assume that the coefficient of friction was the same between the piece of wood and the block of wood.

(these percent error calculations were based on the average experimental radius and the theoretical radius for a velocity of 3.32m/s)

=** Velocity at the Top and Bottom of a Circular Path **= Deanna, Sam, Maddy, Nikki Period 4 Due 1/13/11

Our purpose to find out how certain factors affect the minimum and maximum velocity of an object in centripetal motion. We want to know the effect of mass on an object's minimum velocity (at the top of the circular path) and maximum velocity (at the bottom of the circular path).
 * Purpose **

The minimum velocity is not affected by mass. The maximum velocity at the bottom of the circular path decreases as mass increases.
 * Hypothesis **

Part I: min. at top of circle First, cut a piece of string and tie one end to a mass. The length of the string will be your diameter. Record the mass of the object at the end of the string. Leave enough room at the other end of the string for your hand to get a good grip. Second, swing the mass in a circle, going as slow as possible without their being slack in the rope. Then, another person with a stopwatch timer will start timing, and someone else will count the amount of rotations the mass completes in ten seconds. Record this data, and repeat three times. Then, change the mass, and repeat.
 * Procedure **

Part 2: max at bottom of circle First, experimentally determine the maximum tension of your selected string (thread in our case). Measure your radius (the length of the string). We tied the string to a force meter just to have something to hang the string off of. Then, slowly add masses until the thread breaks. Make sure the string is twisted or moving before placing the next weight; this will allow the string to readjust to the new tension in the string. Once the string breaks, record the mass, and repeat at least three more times.Then, pick a mass smaller than the mass used to find the max tension. Measure out another piece of thread using the same radius and tie the mass to the end of the thread. Swing the mass in a circle. Have one person count the number of rotations and have another person timing you with the stopwatch. Continue until the threat breaks. Record the number of rotations over the time recorded from the stopwatch. Repeat at least three times for each mass. Change the mass and repeat.

Part 1: Minimum Velocity of an Object media type="file" key="ChengProcedure1.mov" width="300" height="300"

= = Part 2: Finding Max Tension: media type="file" key="ChengProcedure2.mov" width="300" height="300"

Part 2: Maximum Tension of an Object media type="file" key="ChengDeanna.mov" width="300" height="300"

Part 1 Minimum Velocity: The above chart displays our results for the finding the minimum velocity at the top of a circular path. When the radius of the circle was kept constant and the mass was changed, the velocity we calculated was always the same. From this we drew the conclusion that mass does not have an effect on the minimum velocity. (see analysis for reason that theoretical velocity is less than experimental velocity)
 * Data **

Part 2 Maximum Velocity: This chart displays the data collected during Part 2. We first conducted four trials to find the maximum tension that the thread could handle and averaged the masses together. We multiplied that mass by 9.8 to solve for the max tension in newtons which is 4.9 N. We kept the radius of the circle constant in this part of the experiment and changed the mass each time. The velocities did change when the mass changed this time, and from this we can draw the conclusion that mass does effect maximum velocity.

Theoretical Minimum Velocity for .010 kg Experimental Minimum Velocity for .010 kg Theoretical Maximum Velocity for .050 kg Experimental Maximum Velocity for .050 kg
 * Sample Calculations **

Part 1: Our data in part one agrees with our hypothesis that mass is not relevant to the minimum velocity at the top of a circle. The reason the theoretical velocity was consistently smaller than experimental velocity because when doing calculations, tension was 0. However, when we were actually swinging the mass in a circle, we tried to keep the tension as low as possible, but it was not zero. Therefore, our experimental results make sense.
 * Analysis**

Part 2: We proved hypothesis correct, about an object's mass being indirectly related to its maximum velocity. By maintaining a constant radius, we solved for the theoretical maximum velocities for three different masses. Our theoretical values showed that increasing the mass, decreased the velocity. Upon performing the trials, we achieved good results for the last two masses. The experimental velocities of the last two masses (0.04 and 0.05 kg) were fairly close to the theoretical velocities, and the indirect relationship proved true in these two values. The first mass's experimental maximum velocity was a definite error, because it did not follow the pattern of the other two, and the experimental value was very different from the theoretical value. This is probably due to the many sources of error further discussed below. Also, all our theoretical values depended on the maximum tension being consistently the centripetal force. However, the error this assumption causes becomes more prominent with smaller values, when the string might not be at maximum tension. This is perhaps whey the experimental and theoretical values are closer in value at our largest mass used. The masses we used yielded weight that was far below what the string could really support.


 * Conclusion **

Part 1: Our first hypothesis was that the minimum velocity of a vertical circle is not affected by mass. Our hypothesis was proved correct during the investigation. Through out the experiment we kept the radius of the circle a constant 0.45 meters and changed only the mass. Our experimental velocities were very similar for each mass. For example for 0.01 kg the experimental velocity was 3.4 m/s and for 0.03 kg the velocity was 3.1 m/s. As you can see these values are fairly similar. For the theoretical velocity we calculated 2.1 m/s at the minimum. Using this information and the experimental values we can find our percent error.

Our percent error came out to be 52 percent and there are a lot of possible sources of error. First we had to find velocity by finding how many rotations occurred in a second. We decided to find how many rotations occurred in 10 seconds and then calculate from this information how many rotations occurred in one second in order to minimize the error. The timing of the stopwatch still contained error due to imperfect reflexes. Also if the mass was slightly shy of or past the exact starting or ending point it was not accounted for in the calculations, which would have added to the percent error. Also we did not find a way to measure that the tension was actually zero at the top of the circle and had try and make the rotations as slow as possible without having slack in the string. Because we could not measure this it we had to assume that the tension was zero when it may not have been. In the future for this lab to further minimize error we could find how many rotation in a minute and then calculate from that value the seconds. Making the time longer would minimize the impact of our stopwatch reflexes on the error.

Part 2:

Our second hypothesis is that the maximum velocity at the bottom on the circle decreases as the mass increases. Our hypothesis was proved correct through our experiment. For example when the mass was 0.02 kg the experimental velocity was 9.5 m/s and when the mass was 0.04 kg the velocity was 6.6. This makes sense because to break the string you have to reach maximum tension. When there is more weight on the string the maximum tension is reached with a smaller velocity. (This error calculation is using the theoretical maximum velocity and experimental maximum velocity form the sample calculations) Our possible sources of error include stopwatch timing, which due to our reaction time was not able to capture the exact moment that the weight broke the string. Also not being able to discern the exact number of rotations played a part in the experimental error. Also when finding the maximum tension by putting weight on the still string until it broke we had to be sure to put the masses on gently. If the masses were put on too quickly then the string would break prematurely. Part of the error was that when we put on the masses the string would move and we had to wait until the string was still to put more masses on. If the string was not perfectly still it would contribute to our error. To minimize this we could wait longer to put more masses on and increase the masses in smaller increments. Also to minimize error we could use smaller masses so that it would take more time for the string to break, which would minimize the effect the stopwatch had on the error. Real life applications of this investigation include using a pulley in construction. It would be necessary to know the maximum tension the string could hold to keep the job safe and the materials unbroken. If a truck with a crane and pulley needed to turn the crane it would be helpful to know how slow the turn had to be in order for the velocity to not cause the pulley to reach maximum tension.

** Factors of Centripetal Force **

 * Deanna, Nikki, Maddy, Sam **
 * Period 4 **
 * Due 1/7/11 **

To determine the effect of a circle's radius on the force towards the center.
 * Purpose**

The relationship between force and radius is directly proportional. We hypothesize this because it requires more force for someone to spin the stopper around a larger circle since they have to spin faster to keep a constant speed.
 * Hypothesis**

Materials: circular motion kit, force meter (data studios), timer, ruler, computer
 * Procedure (prior to error correction in conclusion, improvements in procedure discussed further in conclusion)**

1. Zero the force meter. 2. Attach the string from the circular motion kit to the force meter, and plug the force meter into the computer. 3. Determine a radius (11cm), and mark the diameter of the circle on a surface which the circular motion kit will swing above. 4. Using a timer, time how many full rotations the object makes in ten seconds when speed is constant. Set a ratio of rotations to seconds. 5. Using the force meter, collect data of the tension force of the object's motion at that radius. Perform three trials of each radius. 6. Next, choose a different radius. Using a proportion with the ratio of the first radius, determine what the number of rotations should be in ten seconds (of the next circle). Example: 7. Check that the object is moving at constant speed by counting the number of rotations per ten seconds. The number of rotations that should occur should coincide with that of the ratio derived from the proportion. 7. Perform three trials using the second radii. 8. Collect data from at least three different radii, performing three trials for each radii, and adjusting the frequency (proportion) before each new radii. 9. Using excel, graph the relation between the radius and the Force (tension).


 * Force Meter || How we swung the ball || Apparatus that does the same thing we did in this lab, instead it was a person's hand and a force meter ||
 * [[image:force_meter.png width="187" height="131"]] || [[image:dm_motion.png width="189" height="225"]] || [[image:apparatus.png width="238" height="210"]] ||

Data We took our data by putting a linear fit line in on Data Studio and using the y-intercept as our force results. The above chart displays the radii we solved for using the proportions for how many rotations could be made in 10 seconds. We completed three trials for each radii and averaged them together. The first graph we created was using the radius as the x-coordinate and the average force as the y-coordinate. The mass remained the same for each trial, .13 kg. The velocity was arrived at by multiplying the circumference of each circle by the number of rotations for each circle and dividing it by time (10). We then converted the velocities to meters. In theory, the velocities should have all be the same as we were trying to keep velocity constant. However, when calculated, the velocities varied. New calculations were completed in attempt to fix this error. When the first graph did not predict the relationship, we selected the three points above to graph because they were the most closely related on the original graph.

__*Recalibrated Data Table* (See error analysis in conclusion for explanation on the recalibration of data*__


 * Sample Calculations**

Calculating speed:

This is the original graph generated by all of our trials. With a terrible r2 value, the graph predicts nearly no relationship. Error in the experiment caused our graph to show no relationship. This second graph was generated by eliminating 5 of the points on the graph. The relationship here shows that the radius of a circle and the force are inversely related. However, in order to find this relationship, we had to selectively eliminate over half of our data. The r2 value is much better, but because we had to eliminate most of our data to create this graph, it is essential to address the sources of error that affected our data such an extreme amount.
 * Graph**

This graph from Sam, Ryan, and Evan's group shows the relationship between velocity and centripetal force. As velocity increases, the force increases exponentially. This is a direct exponential relationship. This graph from Alyssa, Rebecca, and Niki shows a directly proportional relationship between mass and centripetal force (the tension is the centripetal force). As the mass increases, so does the tension.


 * Conclusion:****

Our hypothesis was incorrect. We thought the relationship between radius and centripetal force was direct however according to the investigation we were incorrect. According to both of our force vs. radius graphs the trend-line showed an indirect relationship. For example when the radius was 11.5 cm. the average tension was 0.1827 newtons and when the radius was 18.07 cm. the average tension was 0.1767 newtons. This clearly shows that as the radius became larger the force decreased. Because the force decreased as the radius increased that means they are indirectly related. This makes the relationship Fc=1/R. When this equation is combined with the relationships of Centripetal force vs. mass and Centripetal force vs. speed the equation becomes Fc=(mV2)/R. This equation that we found during the investigation matches the theoretical equation discussed in class. It is important to understand the relationship between radii and force, because cars commonly experience centripetal motion on the road. Better understanding the relationship between the car's path and the force will help the driver make good decisions on how to go about the turn.

This lab is wrought with potential sources of error. Firstly, we are attempting to find relationships between variables in uniform circular motion. The speed should ideally be constant, but it probably was not. It would be best if we had a motor device that could keep speed constant. We had only ourselves to judge if our object was moving constantly or not. Also, we had to judge if the circle was following the radius properly. We tried to draw a circle on a surface for the object to circle on top of, but doing so will not guarantee that our object is in fact moving around the proper radius. Another source of error is the stopwatch timing. If we had a device to count the number of rotations when the time reached 10 seconds, then this source of error would have been avoided. Although we tried to be as exact as possible, our reaction time is not perfect, and could not time exactly to when we wanted to. In terms of technology, the force meter sensor was not the ideal instrument for this lab. The sensor would mainly measure the y-component of the centripetal force (tension), and not the overall tension force. This was because we swung the object at different heights depending on how we changed the radius. Therefore, this source of error is not systematic either, because the magnitude of the y-value compared to the overall value changes with each radius. Also, another problem with having a person swinging the mass instead of a machine was that the hand was moving during the trials, as it was attempting to keep the speed constant. It was difficult to know if the hand was exactly at the center, so it probably was not. And since the hand was moving, the center did not remain constant either. Like the height, which changed for each trial and resulted in non-systematic error, the amount of movement of the person's hand varied from speed to speed, therefore making that not a systematic source of error either.


 * Most challenging to us, was keeping speed consistent. We must readjust how many rotations we have per time, each time we change the radius. When doing the lab, we thought that as radius increased, the frequency must increase too. Therefore, we calculated the speed incorrectly, changing the speed each time we changed the radius. In actuality, the radius and frequency are indirectly related. As the radius increases, we should expect less rotations in the timed ten seconds. To attempt correcting this error in our data, I used an indirect proportion equation (k=xy), where k is a constant, and x=radius, and y=rotations. Using the number of rotations we used in the lab, I calculated the radii we should have used with each number of rotations. Then, using the calculation above, I solved for velocity. The recalibrated data table shows the new radii I solved for and the velocities they yielded. The velocities are now fairly constant. However, we did not post the graph, because the force values do not match the new radii. The force values still correspond to the changing speeds. However, we have realized that the cause of our inconstant velocities was the calculation method for maintaining constant speed. To maintain constant speed, we should have used an equation showing an indirect relationship between the number of rotations in a given time and the radius.

Centripetal force is very common in our daily lives from planets to the spin cycle in our washer. These things use centripetal force however they do not involve tension as the centripetal force. While not very common here in New Jersey a cowboy would be very familiar with the centripetal force involving tension. A cowboy with his lasso would have to know how large of a radius and how much force he would need in order to have good aim. He would also have to be aware of tangential velocity. While a cowboy most likely would not actually take measurements, just like we do not actually measure the force of friction on a turning car in order to make the turn, the cowboy would get to know how much force and what radius he needs by getting to know the feeling of that force. While not many of us in this class are likely to be using a lasso anytime soon depending on the area of the country someone is from this could be a very practical use.

In addition to the car example mentioned above, there are practical uses for this lab. The washing machine, for example, is a situation where normal force caused the clothes to move in a centripetal motion. The engineers who design these labs surely have to make sure the clothes reach a certain speed in order to properly dry, and this can vary with the size of the barrel. Anyone who has been to a carnival has seen the spin ride where the riders are fixed to the sides of a giant moving circle, and this is accomplished through centripetal motion. The size of the ride affects how fast the ride has to go in order to function. This lab has increased how knowledge of centripetal motion, and increased our awareness of it's presence in our everyday lives.