Ballistic Pendant and Preservation of Impulse and Potential Measures - Assignment Example

Ballistic Pendulum and Conservation of momentum and energy experiments June 2013 Experiment: 1(Ballistic Pendulum) Introduction This experiment involved the determination of muzzle velocity of a ball that was shot out of a projectile launcher by use of ballistic pendulum. The experiment involve use of laws of conservation of momentum and conversation of energy in the derivation of the equations for muzzle velocity. Aim: The experiment aimed at correction of as many figures of muzzle velocity of a ball shot of a projectile launcher. . Theory The ballistic pendulum is useful in the calculation of launch velocity of a high speed projectile. In the current experient a projectile launcher fired a steel ball with mass designated as mball with a launch velocity of Vo and the steel mass then being caught with a pendulum of mass mpend. after momentum being transferred from the ball to the catcher ball system, the pendulum was set into a swinging motion upwards, thus raising the system centre of gravity by a distance h. The rod of the pendulum was a hollow so as to ensure that its mass was kept low with most of the mass being concentrated at the end such that the entire system could be estimated to a simple pendulum. There is conservation of ttal momentum of the system during the collision of the ball and the catcher. This implies that the ball momentum just before the collision is equal to the momentum of the ball-catcher system immediately after collision has just occurred. mballvo = Mv (1) v is the speed of the catcher-ball system just after the collision, and M = mball + mpend (2) As the process of collision occurs, some of ball’s initial kinetic energy is changed to thermal energy. However, after the collision process, as the pendulum started swinging freely upwards, it can be assumed that there is conservation of energy with all the kinetic energy of the catcher-ball system being transformed into an increase in gravitational potential energy. ½ Mv2=Mgh .(3) g =9.8 m/s2, and h being the vertical rise of the center of mass of the pendulum-ball system. Using equations- V0=√(2gh)… .(4) Setup The ballast mass was attached to the bottom of the catcher. The mini launcher, bracket, table clamp, the mounting rod, and the rotary motion sensor were set up as illustrated in the figure 1. At this point the exact position of Rotary Motion Sensor was not yet important. (In the diagram the side of the Rotary Motion Sensor with no model number on the label is facing the person. I f the Rotary Motion Sensor was to be mounted the other way, then there would be measurement of negative displacement. Figure 1: Launcher and Rotary Motion Sensor mounted on rod The three-step pulley was then slid on the Rotary Motion Sensor shaft with the largest pulley facing out. The pendulum was attached to Rotary Motion Sensor by use of the hole near the end of the pendulum as can be seen in figure 2. Figure 2: Mounting Pendulum Rotary Motion Sensor The next step was the adjustment of the position of Rotary Motion Sensor with the pendulum being aligned with the launcher is seen in figure 3 Figure 3: Aligning Pendulum with Launcher The Rotary Motion Sensor was connected to PASPORT interface and then the DataStudio file called "Ballistic Pend1.ds" was opened Experiment procedure In order to upload the launcher, the pendulum was swung out of the way; the ball was placed in the push the ball down the barrel until the trigger could get into contact with the third position. The pendulum was returned into its normal hanging position with data collection being started. The ball was then launched so that it could be caught in the pendulum. After the pendulum was swung out and back data collection was stopped. (Note: At this point, if statistics was not already selected on the graph, at this points statistics was turned on and Maximum selected. the maximum displaments were recorded in the table. steps 1 through 6 were repeated several times and then the average maximum displacement θmax. was found from table Finding the Mass and Center of Mass The ball was fired one more time 9without recording data). The pendulum was then stopped near the top of its swing so that it could not swing back and hit the launcher (so as to prevent falling out or shifting). The pendulum was then removed from Rotary Motion Sensor and the screw was removed from the pendulum shaft. while the ball still in the catcher, the pendulum was placed at the edge of the table with the pendulum shaft being perpendicular to the edge and the counterweight hanging over the edge. the pendulum was pushed out until it barely balanced on the edge of the table and the balance point being taken as the center of mass (See Figure 4) Figure 4: Balancing the Pendulum The distance r from the center of rotation was measured (where the Pendulum was attached to the Rotary Motion Sensor) to the center of mass. The ball catcher was removed and then the mass of the pendulum without the ball was measured. The mass of the ball was also measured. Analysis The angle found from the data and distance to the center r were used in working out the maximum rise in the system as h = r (1cos(θmax)) (5) The Max h of the system was used in working out the velocity of the ball h = r (1 - cos (θmax)) Figure 5: Finding Height Result: From the experiment Mass of ball mball= 16.8g; Mass of pendulum mpend= 140.99g and The distance to the center = 367mm The velocity of the ball was worked out by photo gates. This result can be taken as an actual velocity of the ball. In terms of angles, there were 3 data sets of result Angle (1) = 15.5 degree Angle (2) =18.9 degree Angle (3) = 17 degree Calculation: g=9.81m/s2  First data  Second data Third data The average velocity, The actual velocity =5.5m/s Discussion: There was a difference of 3.45% in the velocity which was calculated using height and the velocity caught by photogates with the former being lower. This could be attributed to fact that the ball velocity was not high enough to bring about accurate calculation. On the other hand, the photogate was able to catch every moment of the system thus resulting in accurate and precise result. The low accuracy in calculation is also due to the assumption that the system was ideal, with no energy loss and the rotational inertia of the pendulum were ignored Conclusion: The actual rate is 5.5 m/s and the theoretical low is 5.3 m/s. After considering all these factor it can be said that our experiment was a successful attempt Experiment: 2(Conservation of momentum and energy) Introduction With a view to comparing the actual kinetic energy and momentum between theoretical results after a pendulum collision with a ball with velocity V by using the law of conservation of energy and momentum AIM Determining the velocity of pendulum using conservation of energy and momentum. Theory When the system launcher launches a ball with velocity v, due to the conservation of momentum it becomes- m(ball)v=(m+M)V During conversion not all the kinetic energy transfer into the system. The system transfer some of them into thermal energy . However it can be assumed that in an ideal condition there are no energy loss during conversion, all energy transfer into gravitational potential energy. 1/2(m+M) V^2= (m+M)gh V= (m+M)/m (2gh) ^ (1/2) Methods: (From practical manual) The data-taking phase of this experiment has three parts. In Part 1 you will use photogates to measure the launch velocity of the ball. In Part 2 you will measure the maximum rotational velocity and angular displacement of the pendulum after the collision. In Part 3 you will measure the rotational inertia of the pendulum-ball system. Part 1: Measure Launch Velocity Set up the launcher with two photogates and a photogate bracket (see Figure 6). Figure 6: Launcher with photogates Measure the launch velocity (Vlaunch) of the ball on the fastest setting. Do several trials and use the average value. If you will be doing the Advanced Study part of this experiment, also measure the launch velocity for the medium and slow settings. Part 2: Record Ballistic Pendulum Data Set-up 1. Set up the equipment and software as described on pages 4 through 6 with the pendulum rod attached to the RMS at the center hole and the catcher side of the pendulum facing the launcher. Do not attach the ballast mass to the bottom of the catcher. 2. Slide the counterweight onto the pendulum rod above the RMS attachment point. Adjust the position of the counterweight so the pendulum is perfectly balanced without the ball. If you release the pendulum in the horizontal position, it should not move. (See Figure 7) Figure 7: Pendulum balanced horizontally Figure 3.2: Pendulum balanced horizontally 3. Prepare a graphs to display angular velocity versus time and position versus time. Procedure 1. Load the launcher and push the ball in to the third (fastest) position. 2. Move the pendulum into the vertical position. If it does not stay that way by itself, hold it very lightly with one finger. 3. Start data collection. 4. Launch the ball so that it is caught by pendulum. 5. After the pendulum has swung out and back, stop data collection. 6. Note the maximum angular displacement measured by the RMS. Record it in Table 1 7. Record the initial angular velocity (just after the collision) in Table 1. Because this velocity occurs close to the collision, the RMS cannot measure it accurately. Instead, note the maximum negative velocity that occurs when the pendulum swings back toward the launcher (record it as a positive value). Though this might be slightly smaller than the actual initial velocity (due to friction), it is a more reliable measurement. 8. Repeat steps 1 through 7 several times. 9. Calculate the average maximum displacement (θmax) and the average initial velocity (ω0). Table 1: Angular displacement and velocity 10. Measure the mass of the ball. mball = _____________________ 11. Fire the ball into the catcher one more time (without recording data). Stop the pendulum near the top of its swing so it does not swing back and hit the launcher (this will prevent the ball from falling out or shifting). 12. Measure the distance, , from the center of rotation (where the pendulum attaches to the RMS) to the center of ball. = _____________________ Part 3: Determine Rotational Inertia of Pendulum-ball System Set-up 1. Remove the RMS from the mounting rod. (Leave the pendulum attached to the RMS and leave the ball in the catcher.) 2. Clamp the RMS on the mounting rod so that the pendulum can rotate in a horizontal plane (see Figure 8). Figure 8: Setup for determining rotational inertia 3. Clamp a pulley to the RMS and set up a string and hanging mass (approximately 20 g to 30 g) as shown in Figure 8. Wind the string a few times around the middle step of the three-step pulley. Adjust the angle and height of the clamped-on pulley so that the string will unwind and run over the pulley with as little friction as possible. Procedure(Practical Manual) 1. Start data collection. 2. Release the hanging mass. 3. After the string has unwound from the three-step pulley, stop data collection. 4. Determine the angular acceleration of the pendulum (α) from the slope of the angular velocity versus time graph. α = _____________________ 5. Measure the radius of the middle step of the three-step pulley. Rpulley = _____________________ 6. Measure the mass of the hanging mass. m = _____________________ 7. Calculate the acceleration (a) of the hanging mass. a = α Rpulley 8. Calculate the tension in the string (FT). Since the hanging mass is accelerating, the string tension is not the weight of the mass. Writing Newton’s 2nd Equation for the free-body diagram in Figure 9 yields: ma = mg − FT where g = 9.8 m/s2. FT = _____________________ Figure 9: Free-body diagram of hanging mass 9. Calculate the torque (τ) applied to the pulley by the string. τ = Rpulley FT 10. Calculate the rotation inertia (I) of the pendulum-ball system using the rotational form of Newton’s 2nd Law: τ = Iα I = _____________________ Calculation results Ball velocity =4.2m/s Ball mass =16.8g Pendulum mass = 140.99g The distance, from the center of rotation is210mm trial Max angle Angular velocity 1 307.9 288 2 111.6 211.5 3 109.4 292.5 4 114.8 211.5 5 113.1 292.5 6 111.9 197 average 143.667 248 The first set of data From 0.095s to 0.195s change of angular velocity can be calculated =10degree/s The average angular acceleration is calculated =100 degree/s2 The second data set From 0.405s to 0.595s time there was an angular velocity difference= 9 degree/s From 0.505s to 0.605s time there was an angular velocity difference= 18 degree/s From 0.605 to 0.705s time there was an angular velocity difference= 9 degree/s The average angular acceleration =120 degree/s2 The third set of data From 4.005s to 4.105s time there was an angular velocity difference = 9 degree/s From 4.305s to 4.405s time there was an angular velocity difference =9 degree/s The average angular acceleration = 90 degree/s2 Average angular acceleration =103.3 degree/s2 = 1.8 rads/s2 Radius middle step of the three-step pulley =14.25mm Mass of hanging = 40.09g Acceleration of the hanging=angular acceleration x radius of the pulley =1.8rad/s^2, Linear acceleration = 0.378m/s2 Tension force in the string is ma=mg-F F=0.377N Torque (τ) applied to the pulley by the string =FR=0.391*14.25=0.005383Nm Rotation inertia (I) of the pendulum-ball system was calculated using the rotational form of Newton’s second Law: τ = Iα I=τ/α=0.003Ns^2 Analysis Linear momentum calculation was done about the pendulum pivot. Angular velocity =20rad/s Angular momentum of the ball-pendulum system (L0) immediately after the collision; Ball Kinetic energy (Launch) before the collision; Ball-pendulum system (K0) kinetic energy immediately after the collision; Using the maximum angular displacement of the pendulum (θmax) change in height (hmax) of the ball can be calculated. The max angular displacement of the pendulum is 90degree while the max height is 210mm=0.21m The gain in potential energy of the ball (ΔUball) is calculated by Discussion With linear velocity of pendulum changing constantly, it pauses difficulty in staying focused on this velocity and making the calculations. It is due to this that it is better to make calculations of rotational momentum instead of linear momentum. In this experiment there was comparison of momentum before collision and rotation momentum of pendulum-ball system after collision with their values being 0.07046Kgm/s and 0.0196N*rad*s respectively. The data used in the calculation was picked randomly from the database and may not be free of error. There was also comparison of kinetic energy of the ball before occurrence of collision and the kinetic energy of pendulum-ball system just after the collision with the values being 0.148J and 0.0606J respectively. There was no 100% of energy from the ball to the pendulum-ball system as some energy was lost as thermal energy. Conclusion The conclusion which can be made is that this experiment was a success with the calculation indicating that the momentum before collision being higher than the momentum after collision. In addition, it was found that the maximum potential energy of the ball was more than energy of the pendulum-ball system after collision. Read More

(Note: At this point, if statistics was not already selected on the graph, at this points statistics was turned on and Maximum selected. the maximum displaments were recorded in the table. steps 1 through 6 were repeated several times and then the average maximum displacement θmax. was found from table Finding the Mass and Center of Mass The ball was fired one more time 9without recording data). The pendulum was then stopped near the top of its swing so that it could not swing back and hit the launcher (so as to prevent falling out or shifting).

The pendulum was then removed from Rotary Motion Sensor and the screw was removed from the pendulum shaft. while the ball still in the catcher, the pendulum was placed at the edge of the table with the pendulum shaft being perpendicular to the edge and the counterweight hanging over the edge. the pendulum was pushed out until it barely balanced on the edge of the table and the balance point being taken as the center of mass (See Figure 4) Figure 4: Balancing the Pendulum The distance r from the center of rotation was measured (where the Pendulum was attached to the Rotary Motion Sensor) to the center of mass.

The ball catcher was removed and then the mass of the pendulum without the ball was measured. The mass of the ball was also measured. Analysis The angle found from the data and distance to the center r were used in working out the maximum rise in the system as h = r (1cos(θmax)) (5) The Max h of the system was used in working out the velocity of the ball h = r (1 - cos (θmax)) Figure 5: Finding Height Result: From the experiment Mass of ball mball= 16.8g; Mass of pendulum mpend= 140.

99g and The distance to the center = 367mm The velocity of the ball was worked out by photo gates. This result can be taken as an actual velocity of the ball. In terms of angles, there were 3 data sets of result Angle (1) = 15.5 degree Angle (2) =18.9 degree Angle (3) = 17 degree Calculation: g=9.81m/s2  First data  Second data Third data The average velocity, The actual velocity =5.5m/s Discussion: There was a difference of 3.45% in the velocity which was calculated using height and the velocity caught by photogates with the former being lower.

This could be attributed to fact that the ball velocity was not high enough to bring about accurate calculation. On the other hand, the photogate was able to catch every moment of the system thus resulting in accurate and precise result. The low accuracy in calculation is also due to the assumption that the system was ideal, with no energy loss and the rotational inertia of the pendulum were ignored Conclusion: The actual rate is 5.5 m/s and the theoretical low is 5.3 m/s. After considering all these factor it can be said that our experiment was a successful attempt Experiment: 2(Conservation of momentum and energy) Introduction With a view to comparing the actual kinetic energy and momentum between theoretical results after a pendulum collision with a ball with velocity V by using the law of conservation of energy and momentum AIM Determining the velocity of pendulum using conservation of energy and momentum.

Theory When the system launcher launches a ball with velocity v, due to the conservation of momentum it becomes- m(ball)v=(m+M)V During conversion not all the kinetic energy transfer into the system. The system transfer some of them into thermal energy . However it can be assumed that in an ideal condition there are no energy loss during conversion, all energy transfer into gravitational potential energy. 1/2(m+M) V^2= (m+M)gh V= (m+M)/m (2gh) ^ (1/2) Methods: (From practical manual) The data-taking phase of this experiment has three parts.

In Part 1 you will use photogates to measure the launch velocity of the ball. In Part 2 you will measure the maximum rotational velocity and angular displacement of the pendulum after the collision. In Part 3 you will measure the rotational inertia of the pendulum-ball system.

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