Group+1

__Lab: Rotational Motion

Objective__: What is the relationship between tangential velocity and the radius; the Angular velocity and the radius?

__Hypothesis__: Tangential velocity and the radius are directly proportional relationship, angular velocity and radius are inversely proportional.

__Materials__: Turntable, disk with Velcro tab, photogate with ports, Data Studio, ruler.

__Procedure__: Staring at 16RPM, measure a large radius, using time the tab passes through the gate, find velocity. Record three or more times. decrease radius by roughly 5 cm, repeat recording of times, until you have four different radii. then repeat the whole process for all RPM's.

__Data__:





__Sample Calculations__: Average time

Tangential velocity Angular velocity Theoretical Tangential Velocity (with given rpm of device and radius): V = rev * 2 * pi * r / 60 V = 16 * 2 * pi * .194 / 60 V = .3249 m/s

% error: = ((actual - theoretical)/(actual + theoretical) / 2) * 100 = ((.3389 - .3249) / (.3389 + .3349) / 2) * 100 = 1.0547%

__Discussion Questions:__ 1. As the radius increases, so does tangential velocity. 2. As the radius increases, angular velocity decreases. 3. The slope of the graph represents the angular velocity, because v=wr. 4. If we had measured velocity by using the period and the circumference, our result would have been an average velocity, and it would not have been as precise as our method. 5. Even though we can convert all quantities to linear values, angular quantities are still useful because they measure a value that linear velocity does not account for.

__Conclusion__: Based on our data, we certainly proved our hypothesis correct. There was a direct relationship between tangential velocity and the radius, because as we decreased the radius, the velocity decreased as well. In addition, we showed there was an inverse relationship between the angular velocity and the radius, because as we increased the angular velocity (rpm), it was a lower radius that resulted in an equal tangential velocity, showing the indirect relationship.

Overall, we were extremely successful with this lab. Our highest error on a trial was still less than 5% (4.9649%), and almost all were under 2%, with some under .5%. The only real flaw in this lab that may have accounted for some of the error was that our "actual" values were attained with an average of three trials, creating some inaccuracies due to it being an average. In the future, for better accuracy, an error calculation could be done with all of the values, not just the average.

Lab: Ballistic Pendulum Group: Matt Levy, Sarah Caspari, Jon Brizzolara Date Begun: 3/16/10 Date Submitted: 3/23/10

__Objective:__ What is the relationship between initial and final momenta of a ball fired into a ballistic pendulum?

__Hypothesis:__ Because of the law of conservation of momentum, we know that the initial and final momenta of the ball should be equal.

__Materials:__
 * Steel ball
 * Meter stick
 * Carbon paper and white paper
 * Launching apparatus (Projectile launcher + pendulum + stand)
 * Clamp

__Procedure:__
 * 1) Set up launcher (attach it to the stand and clamp it to the table).
 * 2) Measure and record vertical distance with meter stick.
 * 3) Load the launcher to medium range.
 * 4) Launch ball and tape down carbon paper and white paper (carbon paper face-down on the white paper) approximately where the ball lands
 * 5) Launch the ball various times and measure the range from the mouth of the launcher to the carbon marks on the paper.
 * 6) Take the average of these measurements: this is the range to use in calculations.
 * 7) Measure the vertical distance from the mouth of the launcher to the floor.
 * 8) Use the dynamics equation D y =Vi t +(1/2)a y t 2 to solve for time.
 * 9) Use the dynamics equation V x =D x /t to solve for V.
 * 10) Add pendulum to launching apparatus.
 * 11) Launch the ball various times, recording the angle measure that the needle on the stand reads (we ended up adding a degree to account for friction, and subtracting two degrees because we saw that the needle started at about two, instead of zero. These approximations account for some error).
 * 12) Take the average angle //a// (for us, it was the same every trial, so we didn't need to average them).
 * 13) Measure the length of the pendulum and use trigonometry to find the final height above zero of the pendulum (the hypotenuse is the length //l// of the pendulum, so the vertical side of the triangle is //l//cos(//a//)//.// The height //h// above zero is //l-l//cos(//a//)//.//
 * 14) Use the Work-Energy principle to solve for the velocity of the ball at the collision.
 * 15) Weigh the ball (m 1 ) and the pendulum (m 2 ) and record their masses.
 * 16) Use the momentum principle to solve for the initial velocity of the ball (at launch).
 * 17) Calculate percent difference between the velocity achieved in step 9 and the one from step 16.

__Data:__ range (m) height (m) Angle Moved(degrees) initial velocity (m/s) vertical displacement final velocity


 * range (m) || height (m) || Angle moved (degrees) || initial velocity (m/s) || final velocity (m/s) || [[image:http://www.wikispaces.com/i/editor/insert_table.gif]] ||
 * 1.45 || .85 || 27 || 3.49 || .80 ||
 * 1.44 || .85 || 27 || 3.49 || .80 ||
 * 1.44 || .85 || 27 || 3.49 || .80 ||
 * 1.45 || .85 || 27 || 3.49 || .80 ||

__Sample Calculations:__ Given info: m1 = .066 kg (steel ball) m2 = .246 kg (pendulum) Vertical distance = .845 m Avg. x-distance = 1.45 m Avg. angle of pendulum = 27 o Length of Pendulum (radius) = .301 m

Initial velocity using kinematics: D = vit + 1/2at2 -.845 = ½(-9.8)t2 t2 = .172 t = .4153 s → Vi = d/t Vi = 1.45/.4153
 * Vi = 3.492 m/s**

Initial Velocity Using Pendulum: cos(27) = x/.301 x = .268 m → .301 - .268 = .033 m (vertical displacement) → Kei = PEgf 1/2mv2 = mgh 1/2v2 = 9.8(.033) v = .80 m/s (vf) (perfect inelastic collision) → m1v1 + m2v2 = m1v1f + m2v2f .066v1 = .80(.066 + .268)
 * v1 = 3.79 m/s**

Derivation: r = radius; theta = displaced angle; g = force of gravity

% Difference: (a-b)/[(a+b)/2] * 100 (3.492 – 3.79)/[(3.492+3.79)/2] *100
 * 8.18%**

__Discussion Questions:__ 1. In general, elastic collisions conserve Kinetic energy, and inelastic collisions do not. A perfect inelastic collision with friction would result in the greatest loss in Kinetic Energy. 2. The ball and pendulum collision is inelastic (because they stick together after the collision). Because of this, kinetic energy is not conserved, but momentum is. 3. At the risk of sounding redundant, we say once again that in this collision kinetic energy is not conserved, but momentum is. 4. KEo = 1/2mv^2 KEo = ½(.066)(3.79)^2 KEo = .474 N KEf = ½(.312)(.8)^2 KEf = .099 N .474-.099 = **.374 N lost** ---percentage: .099/.474 * 100 = 79% The percent loss comes out to be about 79%, making it wrong to assume that energy was conserved in the collision. The ratio of M/(m+M) is about .8, which is close to the .79 from part b. 5. In the simulation, increasing the mass of the ball causes a greater velocity at collision (v'), while increasing the mass of the pendulum leads to a lesser v'. 6. The two velocities were not very different (one was around 3.5 and the other was around 3.8). This was partly because of the friction of the needle on the pendulum, so the angle we used was not perfectly accurate. If we built a ballistic pendulum, we could use a pencil instead of a needle so that it would not need friction to hold the spot of the angle; it would just make a mark.

__Conclusion:__ Our procedure was largely successful. By first solving for the initial velocity of the ball using kinematics and then comparing it to the value determined through the law of conservation of momentum, we were to able to complete our objective and prove our hypothesis, thus confirming with only 8% difference that the initial and final momenta of a ball fired into a ballistic pendulum are equal. Our error came from friction from the needle on the pendulum that measured the angle. While we tried to compensate for this problem (we added one degree to make up for friction and later subtracted two because we realized that the needle did not start exactly at zero), we still ended up with some error. Another problem could have been our measurements from the kinematics section of the lab: in measuring the range, we took an average of the ranges of various trials, so the number we used may have caused some error. Also, the launcher itself is not perfectly consistent, which would have affected our range and thus our velocity. There isn't much we could do to address the error in this lab; unless we could find a more consistent launcher, that problem cannot be resolved.

Lab: Energy of a Projectile Launcher Group: Matt Levy, Sarah Caspari, Jon Brizzolara Date Began: 2/9/10 Date submitted: 2/23/10

__Objective__: Determine the relationship between elastic potential energy of the compressed spring, the kinetic energy at initial projection, and potential energy at max height for a ball shot vertically.

__Hypothesis__: All values of energy mentioned earlier will be equal, since at their points of existence (prior to launch, right at launch, and max height), they are the only forms of energy present. This is reinforced by the Law of Conservation of Energy, which states that energy cannot be created or destroyed in a system.

__Materials:__
 * Projectile launcher and ball
 * Photogate timer
 * Caliper
 * Various masses
 * Plastic cup
 * Tape
 * Tape measure
 * Balance
 * Ramrod
 * Computer

__Procedure:__
 * 1) Attach photogate timer to launcher, and clamp launcher to a desk or other surface
 * 2) Link photogate to computer with cord, and open up DataStudio on computer
 * 3) Launch ball vertically and record its maximum height (determine by measuring the height with tape measure)
 * 4) Record times from photogate
 * 5) Weigh the ball and calculate Gravitational Potential energy with the equation (mass)(g)(height)=Energy
 * 6) Use the caliper to measure the diameter of the ball
 * 7) Use the kinematics equation distance=(initial velocity)(time)+(1/2)(acceleration)(time) 2 to determine the initial velocity. Use the diameter of the ball for distance.
 * 8) Calculate initial Kinetic energy with the equation (1/2)(mass)(initial velocity) 2 =Energy
 * 9) Tape plastic cup to the ramrod and put it in the launcher
 * 10) Load the cup with different masses
 * 11) Use tape measure to record the distance the spring inside the launcher compresses with each addition of mass
 * 12) Use Excel to graph Spring Force (in this case, weight) against Distance
 * 13) Using the slope from the graph as the k value, calculate Spring Force Potential energy: (1/2)(k)(distance) 2

__Data:

Potential Energy constant__





mass of ball - .01 Kg diameter ball - .025 m x = .03m k = 168.24 N/m average time through photogate = .0051 s avergage max height = 1.01 m
 * __Calculations__**:
 * givens/taken from graphs:**

PE = mgH PE = .01(9.8)(1.01) PE = .09898 J
 * Gravitational Potential Energy:**

d = Vit + 1/2 at^2 .025 = Vi(.0051) + 1/2 (9.8)(.0051)^2 Vf^2 = Vi^2 + 2ad 0 = 4.877^2 + 2(9.8)(d) d = 1.21 m PEg = mgH Peg = .01(9.8)(1.21) Peg = .1189 J --> most accurate theoretical value (used in later error calculations)
 * Theoretical Gravitational Potential Energy:**
 * Vi = 4.877 m/s

PEs = 1/2 kx^2 PEs = 1/2 (168.24)(.03)^2 PEs = .0757 J
 * Spring Potential Energy:**

d = Vi t + 1/2 at^2 .025 = .0051Vi + 1/2 (9.8)(.0051)^2 .02487 = .0051Vi --Vi = 4.877 m/s Ke = 1/2mv^2 Ke = 1/2(.01)(4.877)^2 Ke = .1189 J
 * Initial Kinetic Energy:**

(theoretical - actual) / theoretical (.1189 - .09898) / .1189 * 100 16.75 %
 * % error for Peg:**

(.1189 - .0757) / .1189 * 100 36.16%
 * % error for Pes:**

(.1189 - .1189) / .1189 * 100 0%
 * % Error for Kei:**

__Discussion Questions__
 * 1) **Why didn't we calculate Work due to spring or due to gravity?** We did not calculate either of those values because work done by the spring is just spring force potential energy, and work due to gravity is simply gravitational potential energy.
 * 2) **How do you explain the relationship between PEs, PEg, and KE?** They are all equal because at the start of the launch, the only thing acting on the ball is spring energy; once it leaves the launcher, only kinetic energy is present; and at the top of the trajectory, the only energy present is gravitational. These have to be equal, beause you cannot add/take away energy to/from a system.
 * 3) **What do you think would happen if you used a ball with more mass?** Using a more massive ball would only mean using more energy. The values would still be equal, just greater.

__Conclusion__ Overall, we satisfied our objective, which was to determine the relationship between Kinetic, Gravitational Potential, and Spring Potential Energies in a closed vertical launch system. Our hypothesis was proven to be fairly accurate, in that we found then all to be close to equal, and we predicted they would be due to the Law of Conservation of Energy. There was a considerable amount of error, however, due in part to so much human error, and inconsistencies with the spring. The percentages were particularly high for two of them (Peg and Pes), mainly because the numbers were so small, so small discrepancies threw off the error considerably. However, one of our measurements (Initial Kinetic Energy), had no error at all, further supporting our hypothesis that the energies are the same.

A lot of our error came from the fact that we were unable to determine the exact maximum height of the ball when we launched it, because we simply had to watch it and estimate its height. A way to address this would be to attach the end of a ball of string to the plastic ball, and hold the ball of string by the launcher. Then, we would launch the ball, and measure the amount of string that was unraveled during the launch. A way to verify our height values would be to use the kinematics equation d=v i t+1/2(a)t 2 to find the initial velocity (d=diameter, t=time between gates, a=-9.8). Then, use the equation V f 2 =V i 2 +2ad to find the maximum height (d), when V f =0. This would give a more accurate maximum height, thus minimizing error due to our height measurements. For knowing the Pes, this knowledge is used when designing kitchen garbage bins with a top that closes on it s own, if its done with a spring. Knowing how strong you need the spring to be at a certain distance to close the garbage can is essential becasue if the spring its not strong enough then it won't work, if its too strong then it will not open properly, or it will break the system itself.

Sarah Caspari, Matt Levy, Jon Brizzolara (Period 2) Due: 2/9/10
 * Lab: Work Done by Friction**

__Data__:
 * mass(kg) || T1 || T2 || T3 || avg tension || Normalforce(mg) ||
 * 0.17627 || 0.5 || 0.6 || 0.5 || 0.533333333 || 1.727446 ||
 * 0.27627 || 0.8 || 0.8 || 0.8 || 0.8 || 2.707446 ||
 * 0.37627 || 1.1 || 1.1 || 1.2 || 1.133333333 || 3.687446 ||
 * 0.47627 || 1.5 || 1.4 || 1.4 || 1.433333333 || 4.667446 ||
 * 0.47627 || 1.5 || 1.4 || 1.4 || 1.433333333 || 4.667446 ||



block of wood; string; Force sensor; motion sensor, data studio; tape measure __For Force of Friction/Work:__ __For Initial Kinetic Energy:__ V initial
 * Objective:** Determine the relationship between initial kinetic energy and the amount of work done by friction.
 * Hypothesis/Rationale:** Initial kinetic energy will be equal to the work done by friction. This is because the Law of Conservation of Energy states that final energy is equal to the sum of initial energy and work. In the case of our lab, the only energy present is initial kinetic energy, because the block starts and ends on the ground (so there is no potential gravitational energy, initial or final), there is no spring present (so there is no Potential elastic energy, initial or final), and the object ends at rest (so there is no final kinetic energy). This leaves us with K Ei +W=0, or W=-K Ei, but since you cannot have negative energy, W=K Ei.
 * Materials:**
 * Procedure:**
 * attach string to block of wood and attach force meter to string
 * using data studio, find the coefficient of friction by pulling the block on the ground at constant speed, and using the slope of the graph created
 * repeat 3-5 times
 * Next, throw the block (pointed at the motion sensor), and measure the distance it travels
 * repeat 3-5 times
 * to find work, use formula W = FxDxcosø where F is the force of friction, D is the distance traveled, and ø is the angle between the direction of the force and direction of motion.
 * Using the motion sensor and data collected from the other trial, find the initial velocity of the block by finding the average acceleration
 * Repeat 3-5 times
 * plug this and data collected previously into the equation V^2 final = V^2 initial +2ad, where v is velocity, a is acceleration, and d is distance traveled
 * to find Kinetic energy, use the Formula Ke = 1/2mv^2, where m is the mass of the block, and v is the initial velocity of the block
 * Compare this result to that calculated of Work
 * Sample Calculations:**

K Ei

F //f// Work (KE - W) / [(KE+W) / 2] *100 (.981-1.059) / [.981 + 1.059) / 2] * 100 7.708%
 * % difference:**

There was a small difference between Work and Kinetic Energy. even though the two are the same they came out to be different in the experiment due to possible error. Work is the amount of energy used to move an object, kinetic energy is the energy in the object when it is moving. The work done by friction would increase because of the greater friction force due to greater mass, the distance might not change, but work done will increase. The distance traveled would be lessened, the work done by friction would not change unless the mass was changed. This is because the coefficient of friction is greater in creasing the force of friction. The block would travel much further because friction is reduced so much that the block may continue until it hits another object.
 * Discussion Questions**
 * How does the magnitude of Work compare to the Kinetic energy?**
 * How do you explain the relationship between the Work Done and the Kinetic Energy?**
 * What do you think would happen if you used a block with more mass?**
 * What do you think would happen if you used a rubber block instead of a wooden block?**
 * What do you think would happen if you did this experiment on ice instead of on tile floor?**

Our purpose was satisfied: through our procedure, we were successfully able to determine the relationship between initial kinetic energy and the amount of work done by friction. We predicted that the two would be equal, because of the Law of Conservation of Energy, and we were correct. Our values of work and initial kinetic energy were always pretty close to each other, thus supporting our hypothesis.1
 * Conclusion**

Overall, while percentages of error are higher than we would like (specifically the 16.2%), this is only due to the fact that the numbers being used are so small, that a small fraction of a change leads to a high percentage of error. A good example of this is the first run in comparing Kinetic energy and work. While there was only a difference of .078 J, there was still 7.7% error, so in reality, the lab was quite successful. One possible sources of error for this lab could have been the distance the block was thrown. Although it was measured with a tape measure, we still eyed the exact number that it was. In the future, using a ruler to accurately measure this would be best. In addition, when calculating the force of friction, the coefficient of kinetic friction was an average, and so was not precise. In the future, to prevent blips in the data reading, a constant motion vehicle could be used to pull the block on the ground to accurately measure the coefficient of kinetic friction.