2 Theory 2.1 Momentum Conservation The purpose of this lab is to apply conservat

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2 Theory 2.1 Momentum Conservation The purpose of this lab is to apply conservation laws to predict the behavior of systems of particles. Within an appropriate system, the momentum in a collision is always conserved; however, elastic and inelastic collisions conserve mechanical energy in different ways. In his Principia Mathematica, Newton used the Third Law of motion to derive conservation of momentum, even though the conservation of momentum is in many ways more fundamental than Newton’s Third Law. The conservation of linear and angular momentum are said to be exact laws – that is, they have never been shown to be violated within an appropriately closed physical system. Conservation of linear momentum is therefore obeyed in any collision. Using the tenets of Newton’s Laws one can show that momentum is conserved and that the initial momentum of a closed system is equal to the final momentum of a closed system, ~pix = ~pfx (1) In a closed system, if two masses are considered, each with mass m1 and m2, initial velocities (~vi)1 and (~vi)2, and final velocities (~vf )1 and (~vf )2, (~pi)1 + (~pi)2 = (~pf )1 + (~pf )2 (2) If only the linear or translational momentum is considered along the x-axis, Eq. 2 becomes, m1(~vix)1 + m2(~vix)2 = m1(~vfx)1 + m2(~vfx)2 (3) 1 Again, it is emphasized that this statement of linear momentum conservation is always true for any two particles interacting in a closed system. We define the impulse Jx imparted to an object as the change in the object’s momentum over a period of time, px = Jx = m~vfx m~vix (4) Furthermore, if we plot the force Fx as a function of time t, the impulse Jx is equal to the area under this force curve. J = p = Area Under F(t) (5) This is a result you will use in lab to quantify the change in momentum of an object. 2.2 Energy Conservation The work W a constant force performs on a system is equal to the amount of force Fx applied over a distance x, W = Fxx (6) Applying Newton’s Second Law we can derive the Work-Energy theorem, namely, W = 1 2 mv2 fx 1 2 mv2 ix (7) If we choose the system so that the work done on the system is zero, i.e. W = 0, we find the following result, 1 2 mv2 fx = 1 2 mv2 ix (8) This is rather remarkable as it states that if we choose our system so that no net force is applied to the system, the system’s initial conditions are related to the system’s final conditions in a very particular way. This is yet another statement of a conservation law. We call the term 1 2mv2 i the Kinetic Energy of an object. KE = 1 2 mv2 x (9) In a perfectly elastic collision, no work is done on the system and total mechanical energy T E is conserved. Since we only care about the objects’ behavior before and after an interaction, potential energy need not be considered (that is, P Ei = P Ef = 0). T Ei = T Ef KEi = KEf (10) We may therefore write, 1 2 m1(vix) 2 1 + 1 2 m2(vix) 2 2 = 1 2 m1(vfx) 2 1 + 1 2 m2(vfx) 2 2 (11) 2.3 Conservation Combination Solving this system of Eqs. 3 and 11 for (vfx)1 and (vfx)2 yields: (~vfx)1 = m1 m2 m1 + m2 (~vix)1 + 2m2 m1 + m2 (~vix)2 (12) and (~vfx)2 = m2 m1 m1 + m2 (~vix)2 + 2m1 m1 + m2 (~vix)1 (13) Again, because we may only apply energy conservation in this manner, Eqs. 12 and 13 only apply to perfectly elastic collisions. 2 3 Procedure 3.1 Collision Setup 1. Attach an economy force sensor to each of two Pasco dynamics carts. With a pencil, label these carts “m1” and “m2”. Measure and record the mass of each cart-sensor system and record these values in a spreadsheet. 2. Place the cart-sensor system on an aluminum dynamics track with motion detectors attached to both ends as shown in Fig. 1. Level the track using the adjustable feet so that the carts do not roll when left at rest. You may also use a bubble level to achieve a surface perpendicular to the acceleration of gravity. Remember, in order to achieve an appropriately closed system we must allow no net force to act on our system. 3. Connect the sensors to the appropriate channels on the ScienceWorkshop interface, (a) For the Motion Sensor nearest m1, connect the yellow cable to Digital CH1 and the black cable to Digital CH2. (b) For the Motion Sensor nearest m2, connect the yellow cable to Digital CH3 and the black cable to Digital CH4. (c) For the Force Sensor nearest m1, connect the cable to Analog CHA. (d) For the Force Sensor nearest m2, connect the cable to Analog CHB. 4. Open the “Collisions.cap” file found on Blackboard. This will plot forces F12 and F21 on the same axes and velocities V1 and V2 on the same axes. Note that the coordinate system is fixed so that positive vx is in the direction of m2. Figure 1: Setup Diagram for Conservation Lab 3 3.2 Trial 1 – Collisions – One Stationary Particle, One Moving Particle and m1 m2 1. On a separate sheet of paper, draw a block diagram of the system before and after the collision, clearly labeling m1, m2, (vix)1 and (vix)2 and (vfx)1 and (vfx)2. 2. Screw in the magnet attachments to the force sensors ensuring the alignment pins go into the appropriate holes. You may want to mass these accessories as this adds to both m1 and m2. 3. Perform a practice collision by keeping m2 stationary and slowly pushing m1 by the force sensor toward the stationary cart. Ensure that carts do not collide with either motion sensor, that no wires catch and that the carts do not jump off the track. 4. Zero the force sensors by pressing the “TARE” button on each force sensor. Collect data for V1, V2, F12 and F21 vs. time for a moving cart colliding with a stationary cart. Refer to Fig. 2 for sample data. 5. Select the appropriate data points and use the “Show Coordinates Tool” to find and record values for (vix)1, (vix)2, (vfx)1 and (vfx)2. You can also explore using the “Mean” function from the dropdown menu to find these values. 6. Referring to Eq. 4, find the impulse of each cart on the other cart, (Jx)12 and (Jx)21, by selecting the appropriate data points and finding the area under each force vs. time plot using the “Area” function. Ensure that you only select data from ti to tf that represents the interaction period. 7. Organize your displays so that both force curves display the area under the curve and you are displaying all relevant initial and final velocities. Print this graph and provide a caption. 8. Assuming the collision is perfectly elastic, calculate a predicted (vfx)1 and (vfx)2 using measured values for (vix)1 and (vix)2 and Eqs. 12 and 13. 9. Calculate the percent difference between the change in momentum of p1 and the impulse (Jx)1. 10. Calculate the percent difference between the change in momentum of p2 and the impulse (Jx)2. 11. Using experimental measurements, calculate an initial total momentum, final total momentum and the percent difference between them. 12. Using experimental measurements, calculate an initial total kinetic energy, final total kinetic energy and the percent difference between them. 13. Underneath your block diagram of the system, provide an answer to the following questions: (a) Was momentum conserved in this collision? (b) Was energy conserved in this collision? 4 Figure 2: Sample Collision Data 3.3 Trial 2 – Collisions – One Stationary Particle, One Moving Particle and m2 > m1 1. Measure and record the mass of a 0.5kg weight and place it on the cart corresponding to the stationary cart m2. Repeat the procedure outlined above with your new value for m2, but push the moving cart toward the stationary cart with enough speed so that it rebounds from the stationary cart. Ensure that you: (a) Draw a Before-After block diagram, including all appropriate labels and a caption, (b) Print the V1, V2, F12 and F21 vs. time graph and add a caption, (c) Perform all calculations outlined above, (d) Answer conservation questions. 3.4 Trial 3 – Collisions – One Stationary Particle Sticks To One Moving Particle and m1 > m2 1. Transfer the 0.5kg mass to the moving cart and replace the magnet attachments on the force sensors with one clay attachment and one hook attachment. Repeat the procedure as outlined in Trial 1 ensuring that you update your spreadsheet to reflect the swapping of the masses. You may choose to boost the force sensor sampling rates to 1000Hz so that you might “see” the impulse more readily. Ensure that you: (a) Draw a Before-After block diagram, including all appropriate labels and a caption, 5 (b) Print the V1, V2, F12 and F21 vs. time graph and add a caption, (c) Perform all calculations outlined above (minus the elastic collision assumption), (d) Answer conservation questions. 3.5 Trial 4 – Collisions – Two Convergent Particles and m1 m2 1. Repeat the procedure outlined in Trial 1 using one clay attachment and one hook attachment. Send the carts toward one another so that they come to nearly a complete stop when they collide. You may choose to boost the force sensor sampling rate to 1000Hz so that you can “see” the impulse more readily. Ensure that you: (a) Draw a Before-After block diagram, including all appropriate labels, (b) Print the V1, V2, F12 and F21 vs. time graph and add a caption, (c) Perform all calculations outlined above (minus the elastic collision assumption), (d) Answer conservation questions.

Post-Lab Questions 1. Provide an explanation as to why mechanical energy was or was not conserved in each trial. If mechanical energy was NOT conserved in any one of your trials, has the the conservation of energy been violated? Why or why not?

Explanation / Answer

Some part of Mechanical energy may goes into frictional work, air resistance work hence mechanical energy was or was not conserved in each trial.

No the conservation of energy is not voilated because energy going into frictional work or air resistance work is also form of enrgy.

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