Group+5

Lab 19: Rotational Kinematics By: William Fassuliotis, Noah Feit, Danny Schneider, Tony Xu Period: 6 Due: 4/8/10

Tasks A- Noah B- Danny C- Will D- TXU

Objective: For an object in circular motion, what is the relationship between tangential velocity and the radius? Between the angular velocity and the radius?

Hypothesis: The tangential velocity and the radius will have a directly proportional relationship where as one increases, so will the other. However, as the radius increases the angular velocity will remain constant.

Materials: - Turntable - Pasco photogates and Data Studio software - Cardboard disc and marker - Ruler

Procedure: 1) Measure out a radius of 0.04m on the cardboard disc and attach the marker to the velcro strip with a radius of 0.04m. 2) Set the Revolutions per Minute(RPM) to 16 on the turntable. 3) Hold the Pasco photogate censor over the marker so that time between gates can be marked with every revolution of the turntable. 4) Start the photogate and the turntable and record 3 complete revolutions of the turntable so that a total of 3 times between gates can be recorded. 5) Take the average of times between gates. 6) Set the RPM to 16, 33, 45, and 78 and repeat steps 1 and 3-5. 7) Measure out four different radii (0.04m, 0.08m, 0.12m, 0.16m) and do step 6 for each different radius.

Data:
 * Trial || Radius (m) || Velocity (RPM) || Time

1 || Between

2 || Gates (s)

3 || Avg. Time (s) || Experimental Tangential Velocity (m/s) || Theoretical Angular Velocity (rad/s) || Experimental Angular Velocity* (rad/s) || Percent error (%) || Graph:
 * 1 || 0.04 || 16 || 3.52 || 3.50 || 3.48 || 3.50 || 0.072 || 1.67 || 1.82 || 8.87 ||
 * 5 || 0.08 || 16 || 3.45 || 3.42 || 3.40 || 3.42 || 0.147 || 1.67 || 1.82 || 8.87 ||
 * 9 || 0.12 || 16 || 3.49 || 3.48 || 3.47 || 3.48 || 0.217 || 1.67 || 1.82 || 8.87 ||
 * 13 || 0.16 || 16 || 3.42 || 3.43 || 3.43 || 3.43 || 0.293 || 1.67 || 1.82 || 8.87 ||
 * 2 || 0.04 || 33 || 1.81 || 1.80 || 1.79 || 1.80 || 0.140 || 3.45 || 3.52 || 1.88 ||
 * 6 || 0.08 || 33 || 1.80 || 1.77 || 1.79 || 1.79 || 0.281 || 3.45 || 3.52 || 1.88 ||
 * 10 || 0.12 || 33 || 1.80 || 1.79 || 1.78 || 1.79 || 0.421 || 3.45 || 3.52 || 1.88 ||
 * 14 || 0.16 || 33 || 1.78 || 1.78 || 1.78 || 1.78 || 0.564 || 3.45 || 3.52 || 1.88 ||
 * 3 || 0.04 || 45 || 1.32 || 1.32 || 1.33 || 1.32 || 0.190 || 4.71 || 4.75 || 0.85 ||
 * 7 || 0.08 || 45 || 1.32 || 1.32 || 1.32 || 1.32 || 0.381 || 4.71 || 4.75 || 0.85 ||
 * 11 || 0.12 || 45 || 1.34 || 1.32 || 1.32 || 1.33 || 0.568 || 4.71 || 4.75 || 0.85 ||
 * 15 || 0.16 || 35 || 1.32 || 1.32 || 1.32 || 1.32 || 0.761 || 4.71 || 4.75 || 0.85 ||
 * 4 || 0.04 || 78 || 0.76 || 0.77 || 0.76 || 0.76 || 0.329 || 8.16 || 8.26 || 1.20 ||
 * 8 || 0.08 || 78 || 0.76 || 0.76 || 0.76 || 0.76 || 0.661 || 8.16 || 8.26 || 1.20 ||
 * 12 || 0.12 || 78 || 0.76 || 0.76 || 0.76 || 0.76 || 0.992 || 8.16 || 8.26 || 1.20 ||
 * 16 || 0.16 || 78 || 0.76 || 0.76 || 0.76 || 0.76 || 1.322 || 8.16 || 8.26 || 1.20 ||
 * Experimental Angular Velocity was found using the slopes of the graphs below
 * Experimental Angular Velocity was found using the slopes of the graphs below



Experimental Tangential Velocity = 2πr / avg time = (2π(.04))/3.5 = 0.072 m/s
 * All Calculations**:

Theoretical Angular Velocity = 2*RPM*π/60 = 2*16*π/60 = 1.67 rad/sec

% Error = |theo - act|/theo*100 = |1.67-1.82|/1.67*100 = 8.87%

ANALYSIS: __Discussion Questions:__ 1) What happens to tangential velocity as the radius increases? The tangential velocity increases as the radius increases because they are directly proportional. 2) What happens to angular velocity as the radius increases? As the radius increases, the angular velocity stayed the same. 3) What does the slope of each line indicate? The slope of each line indicates the angular velocity in radians per second or (2*pi)/T. 4) Why didn't we measure the velocity by measuring the period and circumference? For our group, we did measure the velocity by measuring the period and circumference. This is why in our data table, we had to use the equation (2*pi*r)/T in order to find the tangential velocity. This is also why our R^2 value for two of the lines are both 1. 5) SInce we can convert everything to linear anyway, what do you suppose is the point in using angular quantities? The usefulness of using angular quantities is that if we are given angular displacement, velocity, or acceleration, then we can work with rotational quantities instead of having to convert everything back and forth into linear, thus saving time and effort. Angular quantities also provide us with a different perspective to the movement of objects.

Evaluation/Conclusion (1, 2, and 3): In this lab, the purpose was satisfied and the hypothesis was supported by the data. Our objective was to determine the relationship between tangential velocity and the radius, as well as the relationship between the angular velocity and the radius. Through our experiment, we were able to determine these correlations. First, for tangential velocity and radius, as the radius increased so did the tangential velocity. This is supported by the data. For instance, in Trial 1 the radius was 0.04 meters, and the resulting tangential velocity was 0.072 m/s. When the radius increased to 0.08 meters, the tangential velocity increased to 0.147, as did it when the radius increased to 0.12 and 0.16. This supported our hypothesis in which tangential velocity and the radius will have a directly proportional relationship where as one increases, so will the other, thus determining the relationship and satisfying the purpose. As for the second part, the relationship between the angular velocity and the radius, our experiment determined the relationship where as the radius increased, the angular velocity remained constant. For example, in Trial 1, as the radius was 0.04 meters and the angular velocity was 1.82 rad/s. As the radius increased to 0.08, 0.12, and 0.16 meters, the angular velocity (experimental) remained at 1.82 rad/s, as did the theoretical angular velocity for all trials. This, too, supported our hypothesis in which as the radius increases the angular velocity will remain constant. Therefore, we determined the relationships between tangential velocity and radius, and angular velocity and the radius, thus satisfying both our purpose and our hypothesis, all supported by the data.

Now there were some errors in our experiment. In this lab, we should have had 0 percent error since we found the average velocity due to the fact that we had to find the velocity by measuring the period and circumference. Also, in our graphs, we got R^2 values of 1 meaning that the slope of the line was perfect. Since the slope gave us our experimental angular velocity in radians per second, it should have been the same as the theoretical angular velocity in radians per second. However, we did obtain some error between our experimental and theoretical angular velocities. For example, for the 45 RPm trials, there was a percent error of 0.85%. As for the sources of error, one of the biggest cause of our error was the method we used to find the time between gates. The photogate we used was held by hand above the marker. Since we used our hands instead of a stand, the photogate was constantly moving about the rotating turntable meaning that the time between gates was not exactly one revolution.

There are several ways to eliminate these sources of error, as well as to use real world applications. One way to avoid this error would be to use a ring stand that would be able to clamp the photogate down so that the turntable would turn exactly one revolution every time it passed the photogate. One real life application of the relationship between linear velocity and radius would be in track. On the track field, the starting position is marked in different places for the 6 different lanes. This is required because the person running in the lane farthest from the center would have a greater distance to cover since the radius is the greatest. If all 6 lanes had the same start and finish line, then the person in the outermost lane would have to run with a greater velocity than the person in the innermost lane due to the extra distance needed to be covered.

Lab 18: Ballistic Pendulum By: Stephanie Cha, Noah Feit, Vicki Shopland Period: 6 Completed on: March 23rd Due: March 24th

Objective: Find the relationship between initial and final momenta of a ball fired into a ballistic pendulum.

Hypothesis: The initial and final momenta of the ball will be equal. According to the Law of Conservation of Momentium, initial and final momenta will be equal to one another. Additionally, the initial velocity of the ball exiting the projectile launcher should be the same using projectile concepts and work-energy concepts.

Materials: Projectile Launcher, Steel Ball, Plumb Bob, Meter Stick, Mass Balance, Carbon Paper, Tape

Procedure:

Data: Projectile velocity


 * Trial # || Mass of Ball (kg) || Range (m) || Height (m) || Time (s) || Velocity initial of Launched Ball (m/s) ||
 * 1 || 0.066 || 1.975 || -0.85 || 0.416 || 4.742 ||
 * 2 || 0.066 || 1.977 || -0.85 || 0.416 || 4.747 ||
 * 3 || 0.066 || 1.987 || -0.85 || 0.416 || 4.771 ||
 * 4 || 0.066 || 1.978 || -0.85 || 0.416 || 4.749 ||
 * 5 || 0.066 || 1.973 || -0.85 || 0.416 || 4.737 ||
 * Average || 0.066 || 1.978 || -0.85 || 0.416 || 4.749 ||

Conservation of energy initial velocity

(kg) || Angle of Pendulum (degrees) || Initial Height of Pendulum (m) || Final Height of Pendulum (m) || Potential Energy Intial (Joules) || Potential Energy Final (Joules) || Initial Velocity of Pendulum (m/s) || Initial Velocity of Ball (m/s) ||
 * Mass of ball (kg) || Mass of Pendulum
 * 0.066 || 0.245 || 40 || -0.27 || -0.207 || -2.646 || -2.028 || 1.112 || 5.241 ||
 * 0.066 || 0.245 || 39.5 || -0.27 || -0.208 || -2.646 || -2.042 || 1.099 || 5.178 ||
 * 0.066 || 0.245 || 41 || -0.27 || -0.204 || -2.646 || -1.998 || 1.139 || 5.366 ||
 * 0.066 || 0.245 || 42 || -0.27 || -0.201 || -2.646 || -1.967 || 1.165 || 5.491 ||
 * 0.066 || 0.245 || 42 || -0.27 || -0.201 || -2.646 || -1.967 || 1.165 || 5.491 ||
 * 0.066 || 0.245 || 40.9 || -0.27 || -0.204 || -2.646 || -2.000 || 1.136 || 5.353 ||

% difference


 * Percent difference between initial velocities || Final velocity (m/s) || Initial Momentum of Ball (kgm/s) || Final Momentum of ball (kgm/s) || Percent Difference of Momentum ||
 * 10.516 || 1.112 || 0.346 || 0.346 || 0.000 ||
 * 9.080 || 1.099 || 0.342 || 0.342 || 0.000 ||
 * 12.478 || 1.139 || 0.354 || 0.354 || 0.000 ||
 * 15.623 || 1.165 || 0.362 || 0.362 || 0.000 ||
 * 15.916 || 1.165 || 0.362 || 0.362 || 0.000 ||
 * 12.722 || 1.136 || 0.353 || 0.353 || 0.000 ||

Excel-

Mass of ball: 0.066 kg Mass of pendulum: 0.245 kg Length of pendulum: 0.30 m Angle of elevation: 40 degrees (assume for calculations; based on Trial #1) __Time:__ dy = voyt + 1/2ayt^2 dy = 1/2ayt^2 2dy = ayt^2 2dy/a = t^2 t = square root[(2)(-0.85)/(-9.8)] t = 0.416 sec __Initial Velocity (only horizontal component) :__ dx = voxt vox = dx/t vox = 1.975/0.416 vox = 4.742 m/s __Momentum:__ p = mv p = (0.066)(4.742) p = 0.313 kg m/s __Final Height of Ball/Pendulum:__ cos(40 degress) = x/-0.27 x = -0.207 m __Initial Velocity of Pendulum (upon ball striking):__ PEg + KEi = PEf mgh + ½mv^2 = mgh (mass cancels out) (9.8)(-0.27) + ½v^2 = (9.8)(-0.207) vi = 1.112 m/s
 * __Calculations:__ **

__Initial Velocity of Ball (upon exiting projectile launcher):__ (m1)(v1) + (m2)(v2) = (m1)(v1final) + (m2)(v2final) (m1)(v1) + 0 = (m1 + m2)(vfinal) vfinal = v1final = v2final (because inelastic collision, final velocities are same) (0.066)(v1) = (0.066 +0.245)(1.112) v1 = 5.2451 m/s __Initial Momentum of Ball:__ p = mv p = (0.066)(5.241) p = 0.073 kg m/s __Final Momentum of Ball:__ p = mv p = (0.066 + 0.245)(5.241) p = 0.073 kg m/s __Percent Difference between Initial Velocities Found with Projectile Concepts and LCM Concepts:__ __Absolute value(Theoretical – Experimental)__ x 100 Theoretical __Absolute value(0.346 - 0.313)__ x 100 = 0.952 = 9.52% 0.346

Discussion Questions: What kind results in maximum loss of kinetic energy?** Elastic collisions conserve kinetic energy and inelastic collision do not conserve kinetic energy. A perfectly or completely stuck together inelastic collision with friction will result in maximum loss of kinetic energy. Also, in a perfectly inelastic collision, the objects stick together and has the same final velocities. But, in elastic collisions, kinetic energy is conserved and the velocities are opposite and equal.
 * 1. In general, what kind of collision conserves kinetic energy? What kind doesn’t?

a. Is it elastic or inelastic?** The collision is inelastic because after the ball was launched into the pendulum, the two objects stuck together after collision and moved as one body.
 * 2. Consider the collision between the ball and pendulum.
 * b. Is energy conserved?** No, energy is not conserved.
 * c. Is momentum conserved?** Yes, momentum is conserved because of the Law of Conservation of Momentum. It was conserved when the ball hit the pendulum until the maximum height.

a. Is energy conserved from the moment just before the ball strikes the pendulum to the moment the pendulum rises to its maximum height?** Kinetic energy is not conserved because it is an inelastic collision. In elastic collisions, energy is conserved.When the pendulum swung with the ball as one body, momentum was conserved. The LCM states that the invial and final momenta are equal. This was proved from our lab. Also, the energy from the initial when the ball hits the pendulum and the energy at its maximum height is not conserved and energy can't be the same.
 * 3. Consider the swing and rise of the pendulum and embedded ball.
 * b. How about momentum?** Yes, momentum is conserved.

and pendulum. a. Calculate the loss in kinetic energy as the difference between the kinetic energy before and immediately after the collision between ball and pendulum.** KEo=1/2mv^2 = 1/2(.066)(5.241)^2 =0.9064 J kEf=1/2(.066)(1.112)^2 =0.048 J 0.9064-0.048= 0.858 J lost original kinetic energy.** =Loss/KEo =100(0.858/0.9064) =94.66% difference in that collision?** It would be wrong to assume that energy was conserved because the difference of loss of kinetic energy was 94.66%, which is a big difference. part (b). Theoretically, these two ratios should be the same. State the level of agreement for these two quantities for your data.** =M/(m+M) = (0.245/(0.066+.245))100 = 95.1% The two ratios are very similar, the ratio in part b was 94.66 and this ratio equals 95.1. Therefore these two ratio are close and has a 4.4% difference.
 * 4. It would greatly simplify the calculations if kinetic energy were conserved in the collision between ball
 * b. What is the percentage loss in kinetic energy? Find by dividing the loss by the
 * c. According to your calculations, would it be valid to assume that energy was conserved
 * d. Calculate the ratio M/(m+M). Compare this ratio with the ratio calculated in

Select “Ballistic Pendulum” from the column on the left. What is the effect of increasing the mass of the ball? What is the effect of increasing the pendulum mass? Try it. (NOTE: You have to read “Student Notes” first before you can run the simulation.)** When the mass of the ball was increased, velocity at collision was increased.This means that the height of the pendulum traveled was higher. Also, the more mass an object has, the more force and momentum. When the pendulum mass was increased, velocity was decreased because gravity was acting against it.
 * 5. Go to** [|**http://www3.interscience.wiley.com:8100/legacy/college/halliday/0471320005/simulations6e/**]**.

What factors would increase the difference between these two results? How would you build a ballistic pendulum so that momentum method gave better results?** The differences between the two velocities weren't very different. The percent difference was about 9%. The differences between the two velocities was caused by the friction of the needle. Because of the friction from the needle on the pendulum, we didn't get exact results. Therefore using a pen or pencil to mark the angle would be a better choice to decrease friction.
 * 6. Is there a significant difference between the two calculated values of velocity?

__Evaluation/conclusion:__ Through this lab, our purpose was satisfied and our hypothesis was correct. We were able to find the relationship between initial and final momenta of a ball in a ballistic pendulum. In trial 1, the initial velocity was 0.346 m/s and the final velocity was 0.313 m/s. This resulted a 9.5% difference which proves that this was an elastic collision because kinetic energy was not conserved. This is why the initial and final velocities were different. Also, our hypothesis was correct; the initial and final momenta of the ball is equal. Our data was very accurate and had 0% difference which definately proved our hypothesis correct. In trial 1, the Initial momentum = 0.346 kg/ms and the final momentum also equalled 0.346 kg/ms. This is true according to the Law of Conservation of Momentum: the initial and final momenta are equal in collisions. Overall, we were able to satisfy our purpose and prove our hypothesis correct.

=
Based on our percent differences between the initial velocities calculated using projectile concepts and the Law of Conservation of Momentum, our results were precise and accurate. Our percent differences ranged from 8.3-13.7%. All in all, the initial velocity of the ball was approximately 5 m/s, representing the average between using the two different approaches to solving for initial velocity (when the ball/projectile leaves the projectile launcher). The error seemed to become greater and the angle increased- the angle referring to the angle created by the pendulum’s circular motion following the projection of the ball. As the angle became bigger, based on the trials, the percent error grew. This can be accounted for. This increasing percent error indicates that the angle should actually be closer to 39.5 degrees (the smallest trial angle) than to 42.0 degrees (the largest trial angle). Had the angle been smaller, the height would have been more negative. Based on the Law of Conservation of Energy, the initial velocity of the pendulum, would thus, be smaller. Then, based on the Law of Conservation of Momentum, we would have calculated a lower final velocity, closed to the numbers we calculated using projectile concepts. This could account in full for our percent error.======

=
This lab had little error so not many changes would have to be made. The only minor errors that could have taken place would have been because of the friction between the pendulum and the angle-measuring device. To make the date reflect this we could have first solved for the friction force that this caused and added this into our calculations. Another source of error could have been fixed if we had used more precise measuring devices. This lab proves that momentum is conserved when objects collide. This concept is used in many life applications such as a car crash where this idea is used to determine the relative speed of the cars based on the distance that the cars traveled when they crashed.======

Lab 17: Energy of a Pendulum By: Stephanie Cha, Noah Feit, Vicki Shopland Period: 6 Completed on: February 25th Due: February 25th

Objective:
Find the relationship between the kinetic energy and potential energy of a pendulum.

Hypothesis:
The Potential Energy is related to mass and height; the mass is going to be constant so the higher the vertical height, the more potential energy (PE= mgh). Then when the pendulum is released, it is converted to Kinetic Energy.( KE= 0.5mv^2). As height becomes higher, the velocity will increase and mass will stay constant. Therefore, PE & KE have a direct relationship. As the vertical heigh increases, both Potential and Kinetic will increase.

Materials:
String, Pendulum bob, 2 photogates, usb port, photogate port, meterstick, tape

=Procedure:= 
 * 1) Tie a string to the cylinder’s hook.
 * 2) Determine the mass of the cylindrical mass being used and record.
 * 3) Tape the top of the string to an elevated surface, such as a table. At this point, the contraption will become a pendulum.
 * 4) Attach two photogates together by using tape in order to determine the time the pendulum travels over a certain distance.
 * 5) Tape the photogates to the ground so as to point their legs upwards.
 * 6) Open up Data Studio with the photogates connected to the computer. Choose the “time between gates” setting.
 * 7) Measure the diameter of the cylindrical mass using a caliper nd record (in meters).
 * 8) Measure the distance between the bottom of the cylinder and the floor.
 * 9) Place the pendulum at a given height and measure that height.
 * 10) Press “start” on Data Studio.
 * 11) Release the pendulum from that height and record the time between the two photogates. Repeat this three times at each height and solve for the average time. You can now solve for velocity.
 * 12) Repeat this process at five different heights. Use an average of the velocities to find the kinetic energy of the pendulum. This will enable you to solve for velocity of the pendulum at its lowest point, when it is perpendicular to the surface to which it is attached.
 * 13) Solve for kinetic energy using the equation 1/2mv^2.
 * 14) Use the mass and the change in height to solve for potential energy (in Joules). Assume that the ground is zero. Use the equation mgh.
 * 15) Compare kinetic and potential energies.

Data:
Theoretical Velocity: Height (m) || Theoretical Velocity (m/s) ||
 * Trial || Initial Height (m) || Final Height (m) || Change in
 * 1 || 0.470 || 0.0455 || 0.4245 || 2.885 ||
 * 2 || 0.324 || 0.0455 || 0.2785 || 2.520 ||
 * 3 || 0.250 || 0.0455 || 0.2045 || 2.214 ||
 * 4 || 0.170 || 0.0455 || 0.1245 || 1.825 ||
 * 5 || 0.100 || 0.0455 || 0.0545 || 1.40 ||

Actual Velocity:
 * Trial || Diameter (m) || Time 1 (s) || Time 2 (s) || Time 3 (s) || Average Time (s) || Actual Velocity (m/s) ||
 * 1 || 0.0136 || 0.0055 || 0.0056 || 0.0056 || 0.0056 || 2.443 ||
 * 2 || 0.0136 || 0.0074 || 0.0069 || 0.0068 || 0.0070 || 1.934 ||
 * 3 || 0.0136 || 0.0081 || 0.0083 || 0.0082 || 0.0082 || 1.652 ||
 * 4 || 0.0136 || 0.0115 || 0.0107 || 0.0105 || 0.0109 || 1.248 ||
 * 5 || 0.0136 || 0.0166 || 0.0180 || 0.0175 || 0.0175 || 0.776 ||

Kinetic Energy:
 * Trial || Mass (kg) || Velocity (m/s) || Kinetic Energy (J) ||
 * 1 || 0.069 || 2.443 || 0.2059 ||
 * 2 || 0.069 || 1.934 || 0.1290 ||
 * 3 || 0.069 || 1.652 || 0.0941 ||
 * 4 || 0.069 || 1.248 || 0.0537 ||
 * 5 || 0.069 || 0.776 || 0.0208 ||

Potential Energy:

(J) || Change in Potential Energy (J) ||
 * Trial || Initial Height (m) || Final Height (m) || Change in Height (m) || Gravity (m/s) || Mass (kg) || Initial Potential Energy (J) || Final Potential Energy
 * 1 || 0.470 || 0.0455 || 0.4245 || 9.8 || 0.069 || 0.3178 || 0.0307 || 0.2871 ||
 * 2 || 0.324 || 0.0455 || 0.2785 || 9.8 || 0.2190 || 0.117 || 0.1883 || 0.1882 ||
 * 3 || 0.250 || 0.0455 || 0.2045 || 9.8 || 0.069 || 0.1691 || 0.0307 || 0.1383 ||
 * 4 || 0.170 || 0.0455 || 0.1245 || 9.8 || 0.069 || 0.1150 || 0.0307 || 0.0842 ||
 * 5 || 0.100 || 0.0455 || 0.0545 || 9.8 || 0.069 || 0.0676 || 0.0307 || 0.0369 ||



Sample Calculations: (Trial 1)
Average Time: Change in Height:

Theoretical Velocity: Actual Velocity: Kinetic Energy: Initial Potential Energy: Final Potential Energy: Change in Potential Energy:

Velocity: Data Table:
 * Theoretical Velocity (m/s) || Actual Velocity (m/s) || Percent Error (%) ||
 * 2.885 || 2.443 || 15.301 ||
 * 2.520 || 1.934 || 23.268 ||
 * 2.214 || 1.652 || 25.378 ||
 * 1.825 || 1.248 || 31.647 ||
 * 1.400 || 0.776 || 44.595 ||

Percent Difference:
Data Table:

**Discussion Questions:**
1) What role did work play in this situation? Why? Work does not play a role in this lab. This is because the two forces acting on the pendulum bob are not parallel to the pendulum. The force of gravity has no external work and the Tension force is perpendicular to the motion of the pendulum. That’s why work is not present in this situation 2) What types of energy are present when a pendulum is swinging? When a pendulum is swinging, Kinetic energy and potential energy are present. 3) How do the changes in PE and KE compare? Why? Potential Energy and Kinetic Energy both increase and they add up to the mechanical value because there is no work added in this experiment. 4) Ideally, as we've viewed it, a pendulum will swing forever. Explain why this is not actually possible, in terms of the law of conservation of energy. According to the Law of Conservation of Energy, energy cannot be created of destroyed. But since there is air resistance against the pendulum bob, the pendulum cant swing forever. Therefore, when air resistance exists, it works against the force and causes it to lose energy through heat and will not have enough energy to keep swinging. 5) What do you think would happen if you used a pendulum bob with more mass? If we used a pendulum bob with more mass, the bob will swing faster. This will cause more air resistance and cause the bob to slow more quickly. Also if the pendulum bob had more mass, it would have more potential energy which also means more kinetic energy, resulting in a higher velocity.

Evaluation:
In this lab, our purpose was satisfied. The purpose of this lab was to find the relationship between kinetic energy and potential energy of a pendulum. Through the experiment, we found both the potential and kinetic energy and proved the hypothesis correct. As we stated in the hypothesis above, kinetic energy and potential energy have a direct relationship. As one increases, the other increases too. This is because, they are both based on the initial height of the pendulum. In the equation PE=mgh, as height increased, there was more potential energy. For example, in trial 2, the potential energy was 0.1882 J; but when it was increased approximately .15 m in height, the potential energy was increased to 0.2871 J. This height also relates to the Kinetic Energy because, as the pendulum’s height is higher, it increases velocity because of acceleration due to gravity. Therefore, it also increases the kinetic energy. According to the equation KE=0.5mv^2, as velocity becomes greater, Kinetic Energy becomes greater. Therefore, as height increases, both kinetic energy and potential increased, proving our hypothesis correct. Also, when the pendulum bob was held at the highest point, it has potential energy but kinetic energy is equal to 0 because there is no velocity. Even though our objected was successfully solved and our hypothesis was correct, there were many errors in this lab.

One of the biggest obstacles to saying that our results supported our hypothesis is the percent errors we acquired, which we never less than 15%. Given that our percentage error was based on our velocity calculations, distance and time measurements could be the only reasons for inaccurate results. The chances of the distance being immensely incorrect is nearly impossible because we used a tool that is made to measure very accurately and precisely. So, the error must have come in timing. The photogates are responsible for measuring a time period so minute that it is likely that the times were measured incorrectly. For our kinetic energy and potential energy comparison, the major percent different likely comes from incorrect measurements of not the pendulum, but the heights. Our percent difference for this statistic was never less than 28%, a shockingly high number. Given that velocity was calculated incorrectly, kinetic energy is naturally calculated incorrectly given that KE = 1/2mv^2. Gravitational potential energy is based on height in addition to mass and gravity (which remain constant). Using a meter-stick, a somewhat inaccurate measurement device, could account for incorrect height measurements. These minor errors could account for our major percent errors.

In this lab there was a lot of error. A major reason for this error was the force of air resistance acting on the pendulum that we did not take into account while calculating the velocity. Also, the initial height along with the final height could have been measured more accurately. There is no real way to fix the problem of air resistance acting on the pendulum other than to calculate for it in the data. This would make the calculations more complicated, but much more accurate. Another way to handle this problem would be to do the whole experiment in a vacuum that way no air resistance is acting on the pendulum. To deal with the measuring, a more accurate way to do it would be to use a measuring machine with a larger amount of significant figures present than on the meter stick. If you do not have a machine like this, it might still help just to use a measuring device with more significant figures. There are many places where pendulums are used in the real world. Not only is a pendulum an actual object people buy, but it is also present in simple things like a swing at the park. The swinging motion is used for many random things, such as religious ceremonies and making clocks.

By: Stephanie Cha, Noah Feit, Vicki Shopland Period: 6 Completed on: February 10th 2010 Due: February 11th, 2010
 * 1) 16 Work Done By Friction

__**Objective:**__
Find the relationship between initial kinetic energy and the amount of work done by friction.

**__Hypothesis:__**
The initial kinetic energy and the amount of work done by work will be directly proportional. The more kinetic energy, there will be more work done by friction. This is because the faster you throw the block, there will be a greater velocity which will increase kinetic energy based on the equation KE = mv^2. Also, the faster you throw the block, the distance will be greater which will increase work by friction based on the equation W= F * d * cosθ.

**__Materials__** :
wooden block, 10 meter tape measure, string, force sensor, data studio, motion sensor, balance, photogate

__Procedure:__
i. Find initial velocity by using kinematics equation Vfinal^2 = Vinitial^2 + 2a D d. ii. Mass remains constant; thus, you can solve for KE. 2. Calculate work due to friction by using equation W = F x d x cos f i. You have distance from having measured it. ii. You have friction, which is substituted in for F because f = m N. iii. You have f, which is 0. The cos(0) = 1. 3. Compare the two in a graph, whose slope should demonstrate a directly proportional relationship based on our hypothesis.
 * 1) Find the coefficient of friction by using the force sensor and Data Studio.
 * 2) Mass the wooden box.
 * 3) Attack a string to wooden box.
 * 4) Use the force sensor to measure the tension at each weight.
 * 5) Tension is equal to friction because the object is moving at constant speed (a = 0; thus in ma = T – f on x-axis, T = f).
 * 6) No acceleration is occurring the y-axis, so N = W and W = mg. Therefore N = mg. Solve for N with the given mass and gravity.
 * 7) Graph Normal Force vs. Friction/Tension Force in Microsoft Excel.
 * 8) Do five trials at various weights with three trials for individual weights.
 * 9) The slope of the graph will represent the coefficient of kinetic friction.
 * 10) Set up a motion sensor on the ground and open Data Studio.
 * 11) Use a v-t graph in order to find acceleration (by determining the slope of a trendline).
 * 12) When opening Data Studio, select “picket-fence.”
 * 13) Lay out a tape measure that extends about eight meters.
 * 14) Tape a photogate to the floor so as to thread the picket fence through the photogate in the direction of the tape measure.
 * 15) Thread the picket fence through and attach it to the wooden box.
 * 16) Place a box at d = 0 meters on the tape measure.
 * 17) Loosen the picket fence, but don’t let it move through the photogate.
 * 18) Press “start” in Data Studio.
 * 19) Wind up and bowl the box, letting it go at d = 0 meters.
 * 20) Measure the distance the box traveled.
 * 21) Calculate kinetic energy and work done by friction
 * 22) Calculate kinetic energy by using the equation KE = (1/2)mv^2.

__**DATA/GRAPHS**__ :
__Data to calculate coefficient of kinetic friction:__ __Graph:__ (Used to determine coefficient of kinetic friction) __ [|Lab 16.xls] Data to calculate kinetic energy:__
 * mass (kg) || Tension 1 (N) || Tension 2 (N) || Tension 3 (N) || Avg. T/f (N) || Normal Force (N) ||
 * 0.695 || 2.0 || 2.2 || 2.1 || 2.1 || 6.8 ||
 * 0.895 || 2.7 || 2.6 || 2.7 || 2.7 || 8.8 ||
 * 1.095 || 3.1 || 3.2 || 3.1 || 3.1 || 10.7 ||
 * 1.295 || 3.8 || 3.7 || 3.5 || 3.7 || 12.7 ||
 * Kinetic Energy ||  ||   ||   ||   ||   ||
 * Trial || Mass (kg) || Acceleration (m/s^2) || Velocity initial (m/s^2) || distance (m) || KE (J) ||
 * 1 || .195 || -2.860 || 5.809 || 5.900 || 3.290 ||
 * 2 || .195 || -2.730 || 5.077 || 4.721 || 2.513 ||
 * 3 || .195 || -2.850 || 5.219 || 4.783 || 2.657 ||
 * 4 || .195 || -2.690 || 4.926 || 4.510 || 2.366 ||
 * 5 || .195 || -2.730 || 4.896 || 4.390 || 2.337 ||

__Graph: (used to determine acceleration)__ Trial 1: __Data to calculate the work done by friction:__
 * Work ||  ||   ||   ||   ||
 * Trial || Force of friction (N) || distance (m) || cos (angle) || Work (J) ||
 * 1 || 0.507 || 5.900 || 1 || 2.991 ||
 * 2 || 0.507 || 4.721 || 1 || 2.393 ||
 * 3 || 0.507 || 4.783 || 1 || 2.423 ||
 * 4 || 0.507 || 4.510 || 1 || 2.287 ||
 * 5 || 0.507 || 4.390 || 1 || 2.226 ||

Link: [|Lab 16.xls]

__SAMPLE CALCULATIONS (trial 1):__
Initial Velocity:

Kinetic Energy: Normal Force:

Friction Force:

Work:

__Percent Difference:__
Data Table:

Discussion Questions:
1. How does the magnitude of work compare to the kinetic energy? The magnitude of work and kinetic energy are the same, but in our experiments they were different by a small percent.

2. How do you explain the relationship between the work done and the kinetic energy? When there is motion, there’s energy that requires the object to move and this is the same as the work or the force done to cause the object to be in motion.

3. What do you think would happen if you used a block with more mass? If we used a block with more mass, this will cause the normal force of the floor to be greater. This is because the equation f = (mu)(N), therefore as Normal force increases, the friction force will increase also. Since there will be a greater friction force, the work will increase too (W = F * d* cos(angle)). Also, in the kinetic energy equation, KE = .5mv^2, the mass and KE are directly related so as more mass is added, kinetic energy will increase too. Therefore, if we used a block with more mass, both Kinetic energy and work will increase because they are the same.

4. What do you think would happen if you used a rubber block instead of wooden block? If we used a rubber block instead of a wooden block, the friction force will be greater because there will be more friction. Therefore the work will be greater because they are directly proportional. And since W = KE, more kinetic energy will be required to move the block.

5. What do you think would happen if you did this experiment on ice instead of on the tile floor? If the experiment was on ice instead of the floor, there would be less friction because ice is slippery. Then, work done will decrease as well as kinetic energy. Since there’s less friction, the block will go further even with a smaller force. But, if there were no friction, the block will keep sliding.

Evaluation/Conclusion Part 1:
In this lab, our purpose was satisfied. The purpose of this lab was to find the relationship between initial kinetic energy and the amount of work done by friction and we successfully determined through this lab. First, by using data studio, we found the acceleration by looking at the v-t graph. Then we used kinematics to find the initial velocity, which then was used to find initial kinetic energy. Lastly, to find the work done by friction, first we used the force sensor to find the coefficient of friction and then used the Work equation in order to find work. But, our hypothesis was incorrect because we thought the two values were only directly proportional, as kinetic energy increases, work will increase too. Our hypothesis was, the faster we throw the block, the velocity increases and there would be more kinetic energy. Also, the faster we throw it, it'll have more distance, which will increase Work also. When we looked at our data, we realized that even though they both increased, their values were very close. Now it makes sense that kinetic energy and work done by friction are equal. This is because when there’s motion, there’s energy that requires the object to move and also work done by friction that causes the object to be in motion. For example in trial 1, the kinetic energy was; KE = 3.29 J and work was W= 2.99. As we did more trials, we found out the values were closer and we knew that Kinetic Energy = Work. Also, according to the Law of Conservation of Energy, it says that final energy is equal to the initial kinetic energy and work. This means that the initial kinetic energy that causes the object to move and the work done by friction that causes the block to stop is equal to each other.

**Evaluation/Conclusion Part 2:**
Our percent difference ranged from about 3.3% to 9.1% in comparing kinetic energy and work. It is, however, clear that work done due to friction and kinetic energy are directly proportional. This is evident based on our graph, whose trendline is a linear line in the positive direction. As work increases, as does kinetic energy, and vice versa. Our percentage difference must be accounted for. A clear pattern is present in our data; kinetic energy is always slightly higher than work. Kinetic energy is based on mass and velocity. Our mass was not likely the cause of error, as we used a standardized scale. Thus, our velocity (initial) could have been slightly higher than it should have been. If this was the case, then it can be accounted for by too large of a distance traveled or too large of an acceleration. Given that acceleration is the derivative of velocity and we used a picket fence and data studio to measure this, our issue likely resulted from a faulty distance measurement. In Data Studio, it should be noted that we found the derivative of the entire set of points, making a major error unlikely. Thus, if our error resulted in determining kinetic energy, it likely came from distance. In our experiment, bowling the wooden box may not have yielded the most accurate results. We may have let go of the box too early or too late in relation to the beginning of the measuring tape. When the wooden box was not in a straight line, we measure it by tightening the tape and bringing it closer to the measuring tape. This could have accounted for a greater initial velocity in the kinematics equation we used. The picket fence tape may not have been tight when the wooden box was bowled. By tightening it, we increased distance, and in turn, increased velocity. Given that velocity has a square proportional relationship with kinetic energy, even a slightly incorrect velocity could dramatically alter kinetic energy. This is a highly likely scenario given our results. Our percent difference was at times high and kinetic energy was always higher than word done by friction. If the work done by friction was a source of error, then the error must have either resulted from the measured distance (likely scenario) or the friction. Friction is based on the equation f = m N. m could have been miscalculated based on imprecise or inaccurate readings of tension on our Normal Force vs. Friction/Tension Force graph ( m is the slope). This could have occurred in either direction, making friction larger or smaller than it was in actuality. Given that work done by friction was always lower than kinetic energy, m or normal force may have been under-calculated.

Evaluation/Conclusion Part 3:
There are many ways that our error could be fixed if the lab was to be done again. One of the main reasons for the error is that when the box was thrown down the hallway it tended to bounce or would swerve off track. A way to fix this would be to have a mechanical way of pushing it down the hallway so the object would not go off track or bounce. Another way would be to use an object with wheels so the track that it went on would be straighter. Also, a way to make the data more precise would be to use a measuring tape with more significant figures. Another reason for the error could have been our calculation of the coefficient of friction was a bit off. This could result from the limited significant figures that we got when measuring tension. Also, it could have resulted from an acceleration when the box was being dragged across the floor. For more precise results we could have used a force sensor that gave us more significant figures, or we could have done more trials giving us a more accurate result. One real world application that this lab would apply to is if it was necessary to determine the tension in a string that a bungee jumper would need to use to jump from a certain height. Also, it would help if you are trying to determine how large of a force you would need to move a car on a specific surface.