Anthony,+Aaron,+Navin,+Jimmy

=**Angle vs. Radius of a Horizontal Circletoc** =

(Investigation 3) Anthony Iannetta, James Ferrara, Navin Raj, and Aaron Chang ﻿ Period 2 Due Date: January 21, 2011

How does changing the banking angle change the value of the radius at which max velocity is reached?
 * Purpose: **

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/Rationale: **


 * [[image:THIS_IS_GETTING_OLD_JIMMY.png width="221" height="256"]][[image:THE_EQUATION_OBEYS_JIMMY.png width="219" height="211"]][[image:SHORT_JIMMYSON.png width="122" height="51"]]

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By putting in two different angles, we found that the higher angle gave us the higher radius, thus supporting our hypothesis.


 * Materials :**

Four wood blocks of varying angles, ruler, protractor, tape, penny, cardboard "record" disk, velcro, Cold War era record player that looks like portable missile launcher, laptop with Excel


 * Procedure: **
 * 1) Perform a pre-lab to gain information on the subject of banked turns.
 * 2) From the pre-lab, form a purpose and hypothesis.
 * 3) Use any given lab devices to create a lab to test the hypothesis.
 * 4) Choose a record player and determine the rpm at different speeds.
 * 5) Next, find the angle for the four different wood blocks that you will be using as the banked turns.
 * 6) Attach the block to a cardboard disk on the record player suing velcro.
 * 7) Place a penny on the block and begin the rotation.
 * 8) Keep moving the penny up the block until you reach the minimum height that it falls off.
 * 9) Record your data and graph.

** Data ** :

Data Table from Excel:


 * [[image:The_table_of_james_the_third.png width="206" height="79"]] ||

Graph showing the relationship of Angle vs. Radius (banked):


 * [[image:Colors_are_jimmys_friends.png height="398"]] ||

[|WHY DO JIMMY DANCE SO WALRUSLIKE.xls]

Radius vs. Velocity (banked) by Denna, Nikki D, Maddy, Sam:


 * [[image:THIS_GROUP_IS_COOL_NOTTTTT.png height="491"]] ||

Velocity vs. Radius (Not Banked) by Rebecca, Niki, Alyssa


 * [[image:JIMMY_U_NO_SEE_THE_COUNTRY.png height="447"]] ||

** Calculations ** :

This calculation is using a rps of 1.317, the coefficient of static friction of 0.214 and an angle of 20.59 degrees


 * [[image:WATCH_ME_PRANK_JIMMY.png]] [[image:Dance_like_the_moon_likes_jimmy.png]]

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Percent Error:


 * [[image:PERCENT_JIMMY_IS_JIMMY.png]] ||

**Conclusion:**

Our hypothesis turns out to be correct; as the angle increases so does the radius. This is a direct relationship in which the radius depends on the angle. This means that the farther a car at max speed goes around a banked turn, the safer it is. A low bank angle and big radius allow the driver to go very fast around the turn and not spin out. By discussing the construction of roads this relationship is clearer. As the road becomes steeper horizontally (the bank angle increases), the turn must become wider (radius increase) or cars will not be able to make the turn at maximum, or even normal, speeds. The Banking Angle vs. Radius graph clarifies that the relationship is direct. The graph is linear with a positive slope and an R2 value of 0.9193. Also the data that generated the graph shows that as the angle increases from 9.46 to 32.2 degrees, the radius increases from 0.0245 to 0.1932 meters. Other factors that effect a car’s turn are also displayed with the graphs of Radius vs. Velocity (unbanked) and Radius vs. Velocity (banked). The unbanked Radius vs. Velocity graph has polynomial fit demonstrating that radius decreases as velocity increases at a tremendous rate, making this relationship an inverse square. The banked Radius vs. Velocity graph also has a polynomial fit with an R2 value of 0.9996. The graph confirms that the relationship between velocity and radius with a banked turn is a direct square because as velocity increases, radius increases exponentially. Overall the graphs and data collected prove the relationships between these factors are varied.

It turns out that for this lab, we have 19.62% error. The error could have occurred both on the theoretical calculation side, and the spinning side. First, many of the angles that were calculated were close, but not exactly, the same as the angle read on the protractor. To bridge the gap, some numbers were approximated. Similarly, the radii length of the penny was often approximated by .01 or .02 of a decimal place. This approximation was done in an effort to make data work a little smoother, as opposed to having crazy numbers. However, the errors in angle measuring could have been due to improper measurement of sides, whereas the error in radii length could have been due to the error in angle size. If we calculate the angle to be something it might not exactly be, then the theoretical radii would be different from the actual. Error also occurred when keeping the penny in place on the wood block. Our first idea was the put a piece of tape under the penny to keep it from sliding down. However, we quickly realized that this would affect our results. So, we took the tape off and continued. What we forgot to do was wipe the excess sticky, tape residue off of the block. So, much of the reason that the radii did not change sooner was because the penny could have been stuck to the tape residue and trying to break free of that before continuing on its merry way. While on the subject of the penny, more error could have occurred when the penny was just starting on the wood block. Since the penny kept sliding down, it was necessary to hold it in place initially. However, we observed that the movement of the holders' hands away from the penny caused it to move left, right, up, or down slightly. Last, we never took into account the effect added weight from wood blocks, disks, and pennies would have on the velocity of the record player. It was very likely that this slowed down the record player, and thus would have changed the results. If the record player was going at constant speed before, which we were not sure of either, it would definitely have changed with more weight.

To address the error present in this lab it would have to be done on a bigger scale. By either using bigger, better technology or actual cars with varied turns, this lab would produce better results. The turntables limit the velocity and radius as well as the bank angles (depends on wooden blocks supplied). It would be more ideal to use a car of course because that is what the classroom lab is based off of. With a car you could test many different turns by changing the variables involved. Velocity could be altered with the tap of a foot or manually by a machine controlling the speed. By simply driving along different curves radius can be changed (wide vs. narrow). Finally a system of tilting roads could change the bank angle. All these factors would make a perfect situation to test the capabilities of a turning car. A real-life application of this, besides a car making a turn, is a slide. As a person goes down the “chute” it banks on turns to keep the person going in the right direction. With water slides this is very prevalent and different water slides have different radii. The various radii allow the person to slide down the ride faster of slower. This is a basic demonstration of the relationship between velocity and radius with banked turns. Another application like a slide is the luge. The difference here is that the luge has less of a coefficient of friction than the slide so the person sledding down a luge course can go faster than a slide would allow. In addition to this a luge course has very high banks on the turns allowing the person to increase the radius of the turn. This makes them go faster illustrating the relationship between bank angle and radius. Although the lab overall was reduced in scale from car to penny it still encompasses the major factors that effect a car when turning. These relationships are everywhere, as seen in the applications, and constitute our daily lives.

= =

=Maximum and Minimum Velocities of a Vertical Circle Lab =

 (Investigation 2)  Anthony Iannetta, James Ferrara, Navin Raj, and Aaron Chang  Period 2 Due Date: January 11, 2011

To figure out how mass affects the maximum and minimum velocities of a centripetal force experiment.
 * Purpose: **

1. We hypothesize that the mass will effect the max velocity directly. It seems like adding more mass would require less velocity to break the string and vice versa. This can be proved by the force equation:
 * Hypothesis/Rationale: **

Since radius and tension are constant, by increasing the mass, the lower the numerator and the greater the denominator. This will make the max velocity less.

2. We hypothesize if the mass is increased the minimum velocity will stay constant when the string is swung in a vertical circle at minimum tension T + w = mv2 / r 0 + w = mv2 / r mgr = mv2 sq(gr) = v
 * There is no mass in the reduced equation, thus there is no reason it should change

Force sensor, string, rubber stopper, metal rod, clamp, masses, mass hanger, stopwatch, masking tape, laptop
 * Materials :**


 * Procedure: **

1. Place and clamp a metal rod on a table and allow a small part of it to hang over the edge of the table. 2. Tie a piece of string on the force sensor and tie the other end to a mass hanger. 3. Place the force sensor through the metal rod and allow the mass hanger to hang down. 4. Add weight to the mass hanger until the string breaks. 5. Again, add weight to the mass hanger but less weight than in step 4 to prevent the string from breaking. 6. Begin recording on DataStudio and add a little weight onto the mass hanger until the string breaks. 7. Press stop on DataStudio after the string breaks, and the tension force at which the string broke will be the y-intercept.
 * (Figuring out Max Tension): **

1. Tie string onto the force sensor and tie the other end onto a rubber stopper. 2. Spin the rubber stopper around in circles until the string breaks. 3. While spinning, use the stopwatch record the time in which the string takes to break.
 * (Figuring out Max Velocity): **

1. Calculate the theoretical minimum velocity on paper for comparison later. 2. Tie a string to a rubber stopper; the radius should be an extra .03m so that the string can be held and spun. 3. Spin the stopper vertically making sure to slow the string down to its slowest velocity while still making a full circle each time you spin. 4. Perform this for 5 revolutions each period and record the time after 5 revolutions. 5. Do each trial 3 times for each mass before changing. 6. Add a single washer weight to the stopper after 3 trials to find how an increases in mass effects the velocity. 7. Tape new weights down. 8. Perform steps 3-5 for each new weight added. 9. Record all data.
 * (Figuring out Min Velocity): **

** Data ** :

Finding Max Tension (Expanded View):



As shown by this data, as we added more mass to the string, the tension in the string increased. When the string broke it shows the tension decreasing rapidly, but right before it broke lies the data we want.

Finding Max Tension (Close View):



This is a closer view of the tension right before the string broke. We used two tools to determine the max tension of the rope. First we used the cross-hairs to single out the y-value (or in this case, tension) right before the string broke. We did not use this value because it only contained two significant figures. Next we took the mean of the relevant data shown by the highlighted component of the data. By knowing that the cross-hairs revealed the max tension was 4.7, we knew that 4.71 (the y-intercept) was the number that was rounded, and because of its greater quantity of significant figures, we used 4.71 as our max tension value.


 * Determining the Max Velocity:**



For this experiment the radius was kept constant at 0.45 meters.


 * Determining the Min Velocity:**


 * Trial || Time for 5 Revolutions (s) || Period (s) || Mass (kg) ||
 * 1 || 4.21 || 0.842 || 0.01249 ||
 * 2 || 4.35 || 0.870 || 0.01249 ||
 * 3 || 4.56 || 0.912 || 0.01249 ||
 * 4 || 4.33 || 0.866 || 0.01849 ||
 * 5 || 4.64 || 0.928 || 0.01849 ||
 * 6 || 4.01 || 0.802 || 0.01849 ||
 * 7 || 4.15 || 0.830 || 0.02449 ||
 * 8 || 4.41 || 0.882 || 0.02449 ||
 * 9 || 4.33 || 0.866 || 0.02449 ||

Average Period 1 – 0.875 Average Period 2 – 0.865 Average Period 3 – 0.859

Resulting Velocities 1 – 3.2m/s 2 – 3.3m/s 3 – 3.3m/s

** Calculations ** :

This sample calculation is the theoretical value for maximum velocity when using 12.15 grams and 0.45 meter radius and a max Tension of 4.71.






 * Percent Error:**

Maximum

Minimum


 * Calculation for the Experimental Minimum Velocity**

Circumference/Period = Velocity 2πr/T = V

2π(0.45)/0.875 = V 3.2m/s = V

2π(0.45)/0.865 = V 3.3m/s = V

2π(0.45)/0.859 = V 3.3m/s = V


 * Calculation for the Theoretical Minimum Velocity**



T + w = mv2 / r 0 + 9.8m = mv2 / 0.45 4.41m = mv2 2.1m/s = v


 * Assuming tension is 0 at the minimum velocity

**Conclusion:**

We hypothesized that mass would directly impact the maximum velocity and that it would not effect the minimum velocity. Both hypothesizes held up. When finding out the actual values for maximum velocity, we increased the mass of the object being spun, and it led to having to use less velocity to break the string. As shown in the data table, we increased the mass from 12.15 grams to 84.70 grams the max velocity decreased from 9.02 m/s to 6.19 m/s respectively. In the minimum velocity experiment we increased the mass but the minimum velocity stayed close enough to assume that it had no affect. All of the velocities were 3.2 m/s, 3.3 m/s, and 3.3 m/s, which are all very close.

For the maximum velocity the error was 30.83% and for the minimum velocity the error was 57.14%. The sources of error vary but are similar to that of the previous lab. We kept the time and period for each revolution using a stopwatch operated by a human which has a tendency to include human error to correctly starting and stopping the stopwatch. To fix this, an automated stopwatch could be used so that when the spinner starts to spin the cork, the stopwatch will start and keep track of the revolutions, and when the string broke, the stopwatch will stop automatically. Even though we practiced keeping the velocity constant, it is very hard to tell if that is actually true. Another source of error could come with the circle that we created while spinning the cork. It is assumed that the circle is perfectly vertical but when the cork is spun fast, it has a tendency to wobble. We did not calculate the wobble angle of the cork into our actual value. The last two problems could be fixed if there was an automated rotator that made sure that the velocity was kept the same and that the circle was perfectly vertical. An additional source of error could come from inaccuracies in the radius length. The radius was taken as .45m,

A real life application of this experiment can be used in the weapon called the flail or ball and chain. During the medieval times, the knights would spin the ball that was connected to a handle through iron links. The centripetal motion that was caused by the spinning motion was used to strike their enemies. This experiment plays into the production of these weapons or anything similar. If the ball at the end weighs too much it will prevent the knight carrying it from spinning it fast enough to do damage to their enemies. It could also break the iron links that connect it to the handle, making it ineffective as a weapon. But if the ball has too little mass, the knight will be able to spin it with great velocity but the weight force of the ball with prevent it from doing any damage. The manufacturers might want to think of fortifying the iron links to prevent the ball from breaking off also. They have to find the perfect damage to combine mass with velocity to cause the most damage.

== ==

=  Circular Motion Lab =

 (Investigation 1)  Anthony Iannetta, James Ferrara, Navin Raj, and Aaron Chang

 Period 2 Due Date: January 5, 2011

To ascertain the effects of mass on the force that causes circular motion.
 * Purpose: **


 * Hypothesis/Rationale: **
 * If the mass of the object in circular motion be increased, then we hypothesize that the tension in the rope will increase because more mass spinning means that the rope is being pulled much harder on both sides to keep the motion circular. **


 * We can hypothesize this while we look at the centripetal force equation. We know that it is: F=m(v^2/r). The FBD of the cork with weights has the weight force pointing down, and the tension force pointing towards the radius. Since this circle is a horizontal one, the only force is the tension mass making the equation ** ** T=m(v^2/r) ** **.** This equation supports our hypothesis because as mass increases, it forces the tension force to increase, creating a direct proportion.

Force sensor, string, rubber stopper, masses, masking tape, laptop with Excel
 * Materials: **

1. Tie a piece of string to the hook on the force sensor and tie the other end to a rubber stopper. 2. Add 2 masses onto the rubber stopper by putting it through the string. 3. Let the mass and rubber stopper hang down and spin them around in circles holding onto the force sensor. 4. Record data on laptop with DataStudio. 5. Record 3 trials for each different mass. 6. Repeat steps 2 to 5 four more times for a total of 15 trials.
 * Procedure: **

It was important to keep the speed and radius constant in order to decrease the chance of error.

** Data ** :

Data Table of Our Experimental Results:



Graphs are like JIMMY.xls

These are some examples of how we collected our data using a Force Sensor on DataStudio. We took 3 trials for each mass, found the mean of each trial and averaged them all together. We also witnessed the y-intercept of each trial and made sure that the mean was close.

Trials of 35.52 g:



Trials of 47.06 g:



Trials of 58.61 g:



** Calculations ** :

This is the actual data for 47.06 gram (0.04706 kg) mass with a 0.45 meter radius. 4 revolutions took 1.1 seconds.

Setting up the equation: Finding the amount of time it takes for one revolution

Find the velocity of the circular motion:

Using the previous data to find tension force:

__Percent Error:__



** Graphs: **

Graph of Our Results (Tension Force vs. Mass):



Graph of Sam, Ryan, and Evan's Results (Tension Force vs. Velocity):



Graph of Nikki, Deana, Maddy, and Sam's Results (Tension Force vs. Radius Length):



**Conclusion:**
 * It was hypothesized that mass and tension are directly proportional (if mass is increased, tension will increase). This was found to be correct and the evidence of this is demonstrated in the data collected. As the masses increase from 0.02358Kg to 0.07032Kg, the tensions also increase from 1.94n to 2.65n. The tension force vs. mass graph is a linear fit with an R2 of 0.9886 clearly demonstrating that mass and tension are directly proportional. The two other graphs show different relationships with tension, including velocity and radius. The velocity vs. force graph is a direct square relationship and the radius vs. force graph is an inverse proportion relationship. The tension force vs. mass graph is different than these other graphs because mass is directly related to tension while velocity and radius are related to tension in different ways. **


 * There are many sources of error in this lab because of all the human interaction and lack of precision with the equipment. First of all it is unreasonable to assume that the stopper was spinning at constant speed. Also it is not really known if it spun with the full radius of the strong extended each revolution. The string can also be inconsistent because of knots or interferences. Other human error generated was from not being able to stop the stop watch exactly at the right time. Besides all these factors, the force sensor only has one significant figure making the results somewhat imprecise. Also the wire attached to the force sensor made it more difficult to spin the stopper. To fix this error there would have to be less human interaction and more technology. The experiment could have been done with a more expensive, precise force sensor. In addition, the entire force sensor could have been spun almost as if it is the mass if there was no wire attached. **


 * A real life application of measuring the relationship between tension and mass is taken into account when building certain rides. Rides with wires or ropes must maintain a certain tension depending on the mass of what the support wires are holding up or moving around. The experiment that was carried out is similar to the ride that has a cart at the bottom where thrill seekers sit and is spun around in a circular path by a strong support wire. The tension in the support wire must be able to withstand the maximum mass of the cart with all passengers on board. If the ride is not engineered with the tension mass relationship in mind, the ride will break down often because wires supporting the cart will snap under the overriding mass. **