BKLS

__//The Lunchables Brigade//__ toc
 * Eric,** **Sean, Chris,** **Phil**

=Lab: Moving in a Horizontal Circle= Period 4 Due Date: 1/21/2011


 * Objective:** To discover the relationship between radius and velocity with banking. (Hypothesis #2)

__//Radius vs. Vmax (unbanked)://__ When controlling the maximum velocity, the higher the maximum velocity, the larger the radius will have to be. gggggggg //Rationale:// When rounding a turn in a car, the sharper the turn (short radius), the slower the car must travel (small velocity) to avoid skidding out or rolling over. Conversely, if the turn is very gradual (large radius), the car may travel faster (large velocity). Velocity and radius are directly related.
 * Hypotheses:**

__//Radius vs. Vmax (banked)://__ When using a constant banking angle, and controlling the velocity, the higher the maximum velocity, the smaller the radius will have to be, however it will not be as small as in an unbanked situation. The presence of banking will reduce the required length of the radius. gggggggg //Rationale:// When rounding a turn in a car, the faster the car is travelling (large velocity), the smaller the turn (short radius) should be when a banking angle is present. The high speed ensures that the car will not slide down the bank. Conversely, if the turn is very gradual (large radius), the car may travel slower (small velocity) as a skid is less likely to occur. Skidding and sliding is dependent on friction. If the force on the car is greater than the friction on the car's tires, a rollover or skid will occur. One thing that can increase the value of friction is banking. Banking allows for a smaller radius.

__//Radius vs. banking angle://__ When using a constant velocity, the larger the banking angle, the smaller the radius will have to be. gggggggg //Rationale:// Larger banking angles will require more revolutions per second, thus the radius will need to be smaller.

//Finding the coefficient of friction of the wooden block:// 1. Attach a Pasco scientific Angle Indicator to the wooden block. 2. Place the penny (we consistently placed it face down to avoid error) about halfway up on the wooden block. 3. Slowly lift the wooden block (less shaking = less error) until the penny begins to slide. 4. Record the angle from the Angle Indicator. 5. Do this 5 times and then take the average of the angles. 6. Use µ=tan(theta) to find mu:
 * Procedure:**

//Procedure for Calibrating the Turntable:// media type="file" key="lunchablesrecordspin.m4v" width="300" height="300" The turntable has four preset speeds (78rpm, 45rpm 33rpm and 16rpm). These turntables are old however, so their settings may be slightly off. Instead of trusting the presets, we decided to measure the rpm for each speed ourselves: 1. Acquire a stopwatch 2. Make some sort of marking on the turning disk. In our case, we used the already present strip of Velcro. This marking will provide a point to watch which will help when measuring the revolutions. 3. Choose a setting. 4. DO NOT begin timing immediately. The disk must accelerate to its full speed before being timed. Allow the disk to make a few revolutions. 5. Choose a point to be the beginning and ending of one full revolution. Every time the marked section of the disk comes to this point is one revolution. 6. Start the timer when the disk hits the selected point. Count 10-20 revolutions in order to ensure accuracy. 7. When the marked section has reached the selected starting/ending point the desired amount of times, stop the timer and record this data. (ex. At setting, the disk made revolutions in _ seconds). 8. Convert this data into rpm (revolutions per minute). This information will be used later. Revolutions per Minute Calculation:



//Procedure for Finding Experimental Radius//

1. If in the theoretical calculations for each speed, the max radius is more than the radius of the turntable, attach a piece of balsa wood to the wedge to extend the amount of testable radii//.// Like so:



2. Using your calculations for the theoretical radius, place the penny roughly at that point. Make sure to measure horizontally outward from the center, not along the diagonal piece of balsa wood. 3. Turn on the turntable to the required speed. Keep adjusting the penny, moving it either further or closer from the center, until it is at the point where it is //just about// to fall off. To this several times. 4. Repeat steps 2-3 several times for each speed.


 * Data:**
 * Setting || Time || Rotations || RPS's || Radius (m) || Velocity || RPM || Period ||
 * 45 || 13.33 || 10 || 0.750187547 || 0.322 || 1.51699925 || 45.01125281 || 1.333 ||
 * 78 || 10.6 || 10 || 0.943396226 || 0.206 || 1.22045283 || 56.60377358 || 1.06 ||
 * 33 || 15.98 || 10 || 0.625782228 || 0.475 || 1.866708385 || 37.54693367 || 1.598 ||
 * 16 || 30.11 || 10 || 0.332115576 || Too large to test || #VALUE! || 19.92693457 || 3.011 ||
 * Angle in º that makes penny move ||  ||   ||   ||   ||   ||
 * || Mew ||  ||   ||   ||   ||   ||   ||
 * Avg = 22 || 0.4 ||  ||   ||   ||   ||   ||   ||
 * || Mew ||  ||   ||   ||   ||   ||   ||
 * Avg = 22 || 0.4 ||  ||   ||   ||   ||   ||   ||


 * Calculations**

//Centripetal Force of a Banked Turn//





//Revolutions per Minute to Angular Velocity (//**ω**//) (radians/second)//



//Theoretical Radii for Varying Speeds//

We converted the various settings on the record player from RPM to Angular Velocity by the above formula and used this value in rad/s to calculate the theoretical radius. We then plugged in **ω****R** for v in order to isolate one variable, radius, in our equation. We also plugged in the banking angle for theta, 9.8 for g, and .4 for mu into this equation to solve for the radius of the varying speeds:

Plug in **ω****R** for v



The three different speeds we used are below. The calculation for radius is below each speed.

56.60 RPM = 5.93 rad/s



45.00 RPM = 4.71 rad/s



37.55 RPM = 3.93 rad/s



//Percent Error//

Radius for speed of 56.60 RPM



Radius for speed of 45.00 RPM



Radius for speed of 37.55 RPM




 * Graph:**

Using the relationship, F=v^2/r, we hypothesized that the values of v vs r should be a binomial value, which it was. Also, we had figured out during the lab that as the velocity increases, the radius should decrease. This is all shown by our lab information.


 * Other Objectives**

//Max Velocity vs. Radius// (Graph courtesy of Danielle, Jae, Jessica) This graph shows the direct square relationship between velocity and radius. The relationship clearly shows this (derivation shown below). If the maximum velocity were say tripled, the radius for the penny would be 9 times greater. This conclusion holds true to our hypothesis that as the max velocity increased, so would the radius. We did however learn that it is not a directly proportional relationship, but a direct square relationship. Derivation:

//Banking Angle vs. Radius// (Graph courtesy of Tom, Tyler, Richie and Rory) This graph depicts the relationship between a turn's banking angle and the radius needed to keep the penny from sliding when at a constant velocity. By keeping velocity constant and changing the angles, one will find that as they increase the size of the angle, the size of the radius decreases. The equation we found proves our hypothesis was correct in asserting that a larger angle will yield a shorter radius for the aforementioned reasons found in our rationale. Equations Proving Hypothesis:



gggggggg Our results supported our hypothesis. Our objective was that by using a constant banking angle, and controlling the velocity, when we increased the velocity, we would also have to increase the radius. We also expected that in the presence of banking would reduce the size of the radius in comparison to an turn without banking. In short, our hypothesis was that the presence of banking will reduce the minimum length of the radius**,** however a high speed will require a large radius. Our results supported this hypothesis. As you can see from our graph, as we increased velocity, the length of the radius also increased. gggggggg As you can see from the results of Jae, Jess and Danielle included above, their conclusion holds true to our hypothesis that as the max velocity increased, so would the radius. We did however learn that it is not a directly proportional relationship, but a direct square relationship. Also, in Tom, Tyler, Richie and Rory's experiment on radius and banking angle, the results of their trials support that our hypothesis was correct in asserting that a larger angle will yield a shorter radius. gggggggg There was a lot of error within this lab, not only human error but also mechanical errors with our equipment. First, when finding the coefficient of friction of the strip of balsa wood, we tilted the wood by hand and used a visual angle indicator to determine the angle when the penny started sliding. Also, when calculating the time that it took for 10 revolutions of the turntable we timed it with a stopwatch. In both of these examples, we relied on human perception and reaction time. In terms of mechanical error, the turntable was very difficult to use efficiently. It was hard to attach the wooden block to the turntable and keep it straight and prevent rocking. Also, the added weight on one side of the turntable caused a warping effect. To counter this, we attempted to add a weight on the other side. But the weight was crude and we did not know how similar it was to the weight of the block and balsa wood. gggggggg Em pty swimming pools are frequented by urban skaters because of their steep sides. The steepness of the sides of the pool (banking angle) allow for greater speeds, which provide more of a thrill to inline skaters and skate boarders. Similarly, when exiting off a highway the exit ramps will frequently be banked. This way, cars traveling at high speeds on the highway will need to decelerate less in order to complete the turn safely. The same idea can also be applied to roller coasters, but in reverse. If you have ever been to Six Flags you know that on the rides not all the turns are level. Many are banked, from slightly to dramatically. Generally, the rides with steep banks go faster. This is because the steep angles allow for a much higher speed than a level turn, and thus more of a thrill for the quests in an amusement park.
 * Conclusion:**

=Max and Min Velocity of a Vertical Circle=

1.To find the maximum tension a string can handle. 2. To apply this data and find the maximum velocity that the string can handle at both the top and bottom of its circle.
 * Objective:**

Rationale: A string can only handle so much tension. When we thought about it, we came to the conclusion that a heavier mass is exerting more of a preliminary tension force than a lighter mass. Because of this greater tension to start with, the heavier mass does not have to be swung at as high of a velocity as a lighter mass.
 * Hypothesis:** As the mass increases, the max. velocity for a mass at the bottom of a circle will decrease.


 * Procedure:**
 * Minimum Velocity**

1. Cut a piece of string of any length. 2. Tie a mass to one end of the string. 3. Make sure to record both pieces of information - string length (radius) and mass. 4. Swing the mass at the lowest possible velocity in order to maintain circular motion. This effectively makes tension zero at the highest point. 5. Swing the mass for 20 rotations while timing how long it takes. 6. Calculate the velocity and repeat several times and take the average of your results.


 * Maximum Velocity**

1. Cut a piece of string of any length and record that length. This will be the radius of your circle. 2. Attach one end to a ring stand and keep adding weight to the other end until the string breaks. This will be the maximum tension of the string. 3. Repeat step 2 several times to be as accurate as possible. 4. Use another string of the same length and attach a small mass to one end of the string. 5. Swing the mass as fast as possible until it breaks, counting the number of rotations and timing it to see how long it took. 6. Record the number of rotations and time. Repeat several times to be as accurate as possible. 7. Repeat steps 1-6 varying mass and record the results.


 * Data:**
 * Maximum Tension ||  ||   ||   ||   ||   ||   ||
 * Mass (kg) || Tension (N) ||  ||   ||   ||   ||   ||
 * 3.5 || 34.3 ||  ||   ||   ||   ||   ||
 * 3.664 || 35.9072 ||  ||   ||   ||   ||   ||
 * 4.06 || 39.788 ||  ||   ||   ||   ||   ||
 * 3.741333333 || 36.66506667 ||  ||   ||   ||   ||   ||
 * Minimum Speed ||  ||   ||   ||   ||   ||   ||
 * Mass (kg) || Time per Rotation || Velocity (m/s) ||  ||   ||   ||   ||
 * 0.05 || 1.006 || 3.12285552 ||  ||   ||   ||   ||
 * 0.05 || 1.132 || 2.775258528 ||  ||   ||   ||   ||
 * 0.05 || 1.018 || 3.086043864 ||  ||   ||   ||   ||
 * 0.05 || 1.043 || 3.012073493 ||  ||   ||   ||   ||
 * 0.05 || 1.056 || 2.974993043 ||  ||   ||   ||   ||
 * 0.05 || 1.051 || 2.99424489 ||  ||   ||   ||   ||
 * Maximum Speed ||  ||   ||   ||   ||   ||   ||
 * Mass (kg) || Circumference (m) || Rotations || Time (s) || Rotations per s || Time per rotation || Velocity (m/s) ||
 * 0.05 || 0.628 || 36 || 3.15 || 11.42857143 || 0.0875 || 7.177142857 ||
 * 0.05 || 0.628 || 32 || 2.85 || 11.22807018 || 0.0890625 || 7.05122807 ||
 * 0.05 || 0.628 || 42 || 3.04 || 13.81578947 || 0.072380952 || 8.676315789 ||
 * 0.1 || 0.628 || 40 || 4.43 || 9.029345372 || 0.11075 || 5.670428894 ||
 * 0.1 || 0.628 || 31 || 3.35 || 9.253731343 || 0.108064516 || 5.811343284 ||
 * 0.1 || 0.628 || 36 || 4.38 || 8.219178082 || 0.121666667 || 5.161643836 ||
 * 0.05 ||  ||   ||   ||   ||   || 7.634895572 ||
 * 0.1 ||  ||   ||   ||   ||   || 5.547805338 ||
 * 0.05 ||  ||   ||   ||   ||   || 7.634895572 ||
 * 0.1 ||  ||   ||   ||   ||   || 5.547805338 ||

Graph:

As you can see from our data, our hypothesis was proven correct. This graph shows the relationship between Max. Velocity and Mass. In the experiment, we would increase the mass in order to find out how fast we would need to spin the string in order for it to break. The trend that we had noticed that as our masses increased, we would need to spin the string at a slower pace in order for it to break.

Minimum Velocity Free Body Diagram and Calculation:
 * Calculations:**



Maximum Tension Free Body Diagram and Calculation:





Maximum Velocity

Percent Error (Minimum Velocity)



Percent Error (Maximum Velocity)



gggggggg Our hypothesis was correct. As the mass of an object increases, the max velocity it needs in order to break the string is slower than that of a lighter mass. The relationship between these factors as shown by our graph is indirectly proportional, so as the mass increases the max velocity decreases and vice versa. For example, if a mass of .06kg was tied to the string we used with the same radius, the speed at which it would break would be 7.22 m/s. If a mass of .08kg was tied to the same string with the same radius, the string would break at 6.38m/s, a slower speed. Part II: When performing the minimum tension part of the experiment, it was very difficult to spin the hanging mass at exactly the correct speed to just barely cause a tension of 0N. The mass' acceleration was also very difficult to keep uniform. These factors played heavily in giving us our percent error of 35.3% in the minimum tension portion. gggggggg The maximum tension portion on the lab was easier to do more accurately. Due to our string's strength, we chose a very short radius at which to spin the mass. The spinner would basically spin it as quickly as they could, and notify the stopwatch holder when the mass' acceleration was uniform. The error here occurred in the stopwatch holder. The mass was moving at such a great speed, that it was difficult to tell exactly when it had made a full rotation, therefore the measured data for the vertical rpm was off: about 7.105% off. gggggggg In order to make all the performing easier, we would choose a weaker string. A weaker string will break with less tension, therefore it would be easier to more accurately measure the speed at which it breaks (a weaker string will require less speed of the same mass to break). Ancient South Americans and North Americans were familiar with the concepts of max tension. Bolas were devices used for hunting comprised of two heavy rocks tied to a rope. They would swing the bolas over their heads and throw them at their prey. The spinning balls would wrap the animal's legs together and immobilize him. The natives were sure to create the strongest ropes that they could, for they knew that the stronger the rope, the faster they could swing and hurl the bolas. If the rope was not strong enough, they could swing it too fast, and watch their prey escape as their bolas broke and flew off in the wrong direction.
 * Conclusion:**

=Circular Force vs. Speed Lab= Period 4 Due Date: 1/07/2011


 * Objective:** To determine the relationship between centripetal force and velocity.

Rationale: Just using our prior knowledge and experiences, our group concluded that when an object spins at a greater rate, there is more tension upon the string to which it is attached. For instance, when spinning keys that are attached to a lanyard, one can feel the keys trying to pull away more and more as they increase the speed. Another, more scientific instance of proof comes from the equation "a= V^2/R". This proves that if the radius of a centripetal system were to remain constant, and if the velocity increases over time, than the acceleration will have to increase in order to compensate. This information will be compensated in the total force equation, Sigma F(circle)= ma(circle). Therefore, with a constant mass and increasing acceleration, it is necessary for the Tension force to increase in response. We therefore concluded that this tendency to move away from the center with greater force when at higher speeds means that the tension force between the center and the object is greater at higher speeds.
 * Hypothesis****:** As speed increases, force will also increase. Speed and force will be directly proportional.


 * Procedure:**

Materials Required: Force Meter, Timer, Data Studio, Excel, String of any length

1. Cut a piece of string of any length and record that length. 2. Tie one end of the string to the hook on the force meter and plug in the meter to the computer and open up Data Studio. 3. Set the graph in Data Studio to record a pulling force, not a pushing force. 4. Pick up the loose end of the string and begin swinging the force meter in a circle, making sure the cord attached to the meter does not get in the way. 5. When at constant speed, start timer, start recording on data studio, and swing the meter for 20 rotations. 6. Do this for varying speeds. 7. Highlight the data on the graph in Data Studio and select a fit line. Record the y-intercept; this is the tension force. 8. Calculate the circumference of the path of the meter based on the length of the string (radius). 9. Divide this distance by the time - 20 seconds, to get the velocity of the force meter. 10. Put the velocities and tension forces into an excel spreadsheet and make a graph.

As can clearly be seen, the radius and mass remain constant. The only changing variable is the velocity, which, also as shown, produces an increase in tension force. This, experimentally at least, proves our hypothesis.
 * Data:**
 * Data for Circular Motion Lab ||  ||   ||   ||   ||   ||   ||
 * Radius(cm) || Circumference || Mass (kg) || Time for 20 Rotations || Time Between Rotation || Velocity (m/s) || Force (Tension) ||
 * 31 || 0.19468 || 0.0917 || 16.68 || 0.834 || 0.233429257 || 1.56 ||
 * 31 || 0.19468 || 0.0917 || 17.35 || 0.8675 || 0.224414986 || 1.54 ||
 * 31 || 0.19468 || 0.0917 || 14.58 || 0.729 || 0.267050754 || 2.09 ||
 * 31 || 0.19468 || 0.0917 || 18.69 || 0.9345 || 0.208325308 || 1.31 ||
 * 31 || 0.19468 || 0.0917 || 19.63 || 0.9815 || 0.198349465 || 1.22 ||
 * 31 || 0.19468 || 0.0917 || 11.2 || 0.56 || 0.347642857 || 2.87 ||


 * Sample Calculations:**

The above two graphs three of our experimental trials.
 * Graphs:**

This graph shows (albeit sloppily because of our many sources of error) that with a greater velocity, the tension will also increase.

(Taken from Nikki, Deanna, Maddy, and Sam) This graph shows, through the use of the radius as compared to the force, that as the radius increases, the force will decrease. One likely reasoning for this is because the speed will need to stay constant in order for this experiment to be correct. That means that the longer the radius, the more the mass attached to the string will fall. The angle from the horizontal will increase, therefore producing a lower x value of tension and a higher y value. Also, using the accepted equations when calculating tension force, (shown above in hypothesis rationale), that the radius and the acceleration are indirectly related. As the radius increases, with a constant speed throughout multiple trials, the acceleration will decrease. Since the x value of tension is the force that causes a system to pull away from its environment, a lower x value will effectively show a lower tension force.

(Taken from Hallowell, Dember, Siegel, Pontillo) This graph shows, in an experiment dealing with the mass - force relationship involved in circular motion, that as the mass increases, so will the force. First off, this can be proved using the equations that we have learned dealing with the derivation of the total force in a centripetal motion-based system. (Again, as shown above in the hypothesis rationale) The mass is a factor of the equation that, when multiplied with another factor (acceleration), it will produce the total centripetal force. Therefore, when the mass is increased, that side of the equation is multiplied by a larger number, therefore producing a larger tension on the other side of the equation. Theoretically, the same result can be produced. This explanation is fairly simple. When the mass is increased, regardless of how much the system is on an incline (it won't matter because the radius should be constant in order to show proper results), the two components of tension will be increased. Therefore, the entire centripetal force will increase.

Our assignment was to find how differences in speed affect the tension on a rope connecting a moving mass to the center. Speed was supposed to be the only variable while the mass and radius were meant to stay constant. This however, was not the case. While our mass did stay the same, the radius of the circle changed from speed to speed. At slower speeds the mass swung in a conical-pendulum type revolution, where the tension force acted on an angle rather than directly on the x or y axis. Because of this, the measured force was actually a sum of the tensions on the x and y axes. When the hanging mass was swung with a greater speed, the string came closer to creating a horizontal circle, in which the length of the string would be a true radius. One obvious application of this lab is in centrifuges. These are spinning apparatuses used for chemistry experiments. They use the centripetal force obtained in circular motion in order to separate liquids of different densities. Centripetal force is also used in the steering of cars and motorcycles, where the centripetal force pushes the vehicles inward, thus making a more efficient and effective turn. Another application of this is possibly the most prevalent in the entire universe. This is the revolution of heavenly bodies around each other. For example, this can be applied to the motion of the moon around the earth, the earth around the sun, comets around the sun and planets, and many others.
 * Conclusion:**
 * Part I: (Shown Above)**
 * Part II:** One erroneous method in our procedure can be found in the way we used the force sensor. By holding a string attached to the sensor and swinging the sensor around in a circle, the wires attached to the sensor became troublesome. At first we held the stray wires tightly, but found that this method added in unwanted forces to our DataStudio graphs. The wires we were holding jerked the force meter outside of its circular motion slightly, and therefore changed the tension force being measured. We then decided to let the wires flail about as the force sensor was swung. Though this provided better results, it is still possible that they affected the sensor's circular motion by pulling it in undesired directions, for they too were moving circularly.
 * Part III:** The simplest way to address the aforementioned error pertaining to the wires would be to use a wireless sensor. This would eliminate all the interfering factors that the cables played within our trials. As for the problems with the radii, having a motorized apparatus that could swing a mass in a horizontal circle would eliminate the inconsistencies between our radii lengths, for the circle's radius would always be the length of the string (provided it was swinging at a great enough velocity for the machine. This can also be solved if we could have devised a way to measure the angle of the incline that the mass follows as it spins in a circle.