Linear Momentum and Kinetic Energy

Transcription

1 Linear Momentum and Kinetic Energy Introduction The object of this experiment is to investigate the conservation of linear momentum and the conservation of kinetic energy in elastic collisions. We will study collisions between two air track gliders. Velocity measurements are used to study a mechanical event: the collision of two bodies and its effect on their motion. Motions are restricted to a single straight line, and forces other than those of mutual interaction are minimized by the use of the air track. The object is to observe exchange of momentum and the transformation of energy under various conditions of collision. Basis of the experiment The physics of one-dimensional collisions is presented in the reference text. Referring to the system consisting of the two interacting bodies only, the governing principles are: (1) If no forces act on the system, other than the ones that the parts of the system exert on each other, the system's total momentum remains constant and the total energy of the system remains constant (conservation of total momentum and energy for isolated system); this is true for both conservative and non-conservative forces. (2) If the forces acting between bodies of the system are entirely conservative, both the mechanical energy (kinetic plus potential) and the internal energy (heat and chemical forms) of the system remain constant. Principle (1) implies that if the internal forces are non-conservative, the changes in mechanical and internal energy will be equal in magnitude and opposite in sign (i.e. the increase in internal energy will equal the decrease in the mechanical energy). This is for example seen when a body heats up during an inelastic collision (kinetic energy is lost and converted into heat). (3) In a one-dimensional system, the vector momentum reduces to an algebraic number whose sign denotes its direction according to a consistent designated convention. This means that you can choose one direction to have positive momentum and the opposite direction has negative momentum. (4) Since the two bodies of the system do not interact with each other except when in contact, there is no potential energy in the system at times when the objects are separated. When the objects are separated, the system's mechanical energy is its kinetic energy only. 1

2 The different possibilities to which the principles apply are designated by names given to different types of collisions: 1. An elastic collision is one in which the internal forces are conservative, so that both mechanical and internal energy separately stay constant throughout the collision mechanical energy is conserved here. 2. An inelastic collision is one in which at least part of the interaction force is non-conservative, so that both mechanical and internal energies change. An endothermic collision is one in which mechanical energy decreases and internal energy increases, that is, mechanical is converted into internal energy (usually heat). A perfectly inelastic collision is an endothermic one in which the maximum possible amount of mechanical energy is converted into internal energy; it is a collision in which the two bodies stick together and move as a unit after the event. In inelastic collisions momentum is conserved but energy is not. Some of these types of collisions will be investigated in this experiment. In all types of collisions, momentum is conserved. Momentum conservation for a two-body system is expressed in equation (1), where the m s are masses and v s are vector velocities. The subscripts refer to bodies 1 and 2 respectively. The primed velocities are those of the bodies after collision, and the unprimed are velocities before collision. m! v! + m! v! = m! v!! + m! v!! (1) The conservation of kinetic energy (which is conserved only in elastic collisions) gives us 1 2 m!v!! m!v!! = 1 2 m!! v!! m!!! v! (2) In these equations, the lefthand side refers to any time before the collision of bodies 1 and 2, and the righthand side refers to any time after the collision. Once the system is defined for 1-dimensional motion, all velocities in one designated direction are positive numbers in the equations, and all in the reverse direction are negative numbers. The signs of the velocities are essential in equation (1). In equation (2), where only the squares of the velocities occur, the vector direction no longer as the square of positive and negative numbers is always positive. Plan and procedure of the experiment The low friction air track is the same as has previously been used. To observe collisions between unequal as well as equal masses, three gliders are needed - two of about the same mass, and one of a different mass. We will study the following types of collisions: Elastic collisions: Set up the gliders on the track with elastic bumpers facing one another. Inelastic collisions: Set up the gliders on the track with one of the facing ends having a pin attached and the other having a putty attached. 2

3 1. Weigh the gliders and use the same gliders throughout the experiment without changing their weights. Weights should be measured on gliders complete with masks and whatever weight have been attached. 2. To avoid external forces acting on the gliders in the experiment, we need to ensure that the track is perfectly level. Send one glider to the right and note the time it takes to pass each photo gate (operated in gate mode more on operating the photogates below). a. The difference between the two times divided by the time it took to cross the first photogate will give you the fraction of momentum lost due to friction and gained (or lost) due to gravity. 3. Repeat step 2 above by sending the glider to the left. a. If you lost momentum while sending the glider in one direction and gained momentum by sending it in the other, the track is not level, and it should be adjusted. This is critical to the experiment. b. There is always a small amount of friction so if you lose a tiny amount of momentum when sending the glider in both directions, you track is level. 4. To adjust the track (in order to eliminate the effect of gravity on a non-leveled track) there is a screw that acts as a foot at one end of the track. Adjust this screw to raise/lower the end of the track in order to make it level. Consult your TA to make sure your track is level! 5. As a last check: put a glider at the center of the track, make sure that it sits still and let go of it. If it drifts to side, the track is still not level. Go back to step To measure the velocity of the gliders, we will use photogates. The photogate will generate a light beam, and if this beam is interrupted, the photogate system will count the time that the beam is interrupted. When the glider passes the photogate, a flag on the glider of length L!"#$%&!"#$ = 10 cm will interrupt the beam of the photogate for the full length of the flag as the glider passes the photogate. 7. The photogate timer as used here is operated in gate mode; it reads in milliseconds (thousandths of a second) the time during which the light beam of the photogate is interrupted. By measuring the time it takes for the glider to pass through the photogate, T!"#$#%&$', and the length of the flag, L!"#$%&!"#$, the glider's speed, in passing, can be directly calculated v!"#$%& = L!"#$%&!"#$ T!"#$#%&$' (3) 8. There will be practically no acceleration as the gliders pass the photogate, as we will horizontally align the airtrack before performing experiments. 3

4 Figure 1: Airtrack, glider and photogate setup. 9. Two photocell bridges, one on each side of the collision area, are used (see Figure 1). Each is connected to its own timer. Thus each timer, operating in the gate mode, starts when the light of its own photocell is interrupted and stops when the light is resumed. 10. When using the photogates to measure the timing, the gliders will pass the photogates twice; once before collision and once after collision as the glider bounces back. The Pasco ME-9215A timers have a memory feature that can be used to store the second glider transit time. To use this memory feature: a. Set the toggle switch to MEMORY ON, a red light should go on. b. Press RESET button. c. Make the two measurements. The timer will display the first measurement. Record this. d. Flip switch to READ to get the TOTAL of the first and second measurement (i.e the sum of the two timings). e. Subtract the first measurement from the total to get the second measurement. Glider handling care Data Please do not handle the gliders by their masks. Do not disturb the trimming weights. Remove a glider completely from the track to fit or remove the inelastic bumpers. It is highly recommended that all the elastic collisions called for first and then do all the inelastic collisions. This minimizes the time spent in refitting bumpers at the ends of the gliders. Remember that the air cushion is easily broken, which introduces an external force (track friction) and invalidates propositions (1) and (2) on page 1. Therefore, the collisions should be gentle and the initial velocities given to the gliders small. In Appendix I and II you will find a total of four data tables that you can fill out for this experiment. Make sure that you follow the procedure above very carefully. 1. Mass Data We are going to have three gliders: two light and one heavy. 4

5 2. Collision Data The kinds of collisions to be studied are listed below, with the necessary data indicated. For elastic collisions we have: 1. One glider is stationary and the other collides with the stationary glider (one glider is heavy and one is light). 2. One glider is stationary and the other collides with it (both gliders are light). 3. Both gliders have initial velocities, collide and bounce back (one glider is heavy and one is light). 4. Both gliders have initial velocities, collide and bounce back (both gliders are light). Calculations For inelastic collisions we have: 1. Glider 2 is stationary and light, while glider 1 has initial velocity and is heavy. For each collision, calculate the vector momentum (remember the direction of your velocity gives it a sign!) of each body before and after the collision, and the total initial momentum and final momentum of the two-body system. For example, the initial momentum is calculated by so the total initial momentum is p!! = m! v!!, (4) p!! = m! v!!, (5) p! = p!! + p!!. (6) Do this for each of the five sets of data you have taken for each type of collision and for each set calculate the percent between the initial and final momentum: in momentum = p! p! p!. (7) Do the same for the kinetic energy, except that you need to do this only for the data in each type of collision which shows the smallest difference between initial and final momentum. in kinetic energy = E!" E!" E!". (8) There may be large experimental uncertainties, especially where the initial vector momentum is close to zero. We will not carry out an uncertainty calculation here, as it will suffice to simply study the percentage in momentum and kinetic energy, as we are interested in whether momentum and or energy is conserved in our experiments. 5

6 Discussion Discuss your results for the percentage s of momentum and kinetic energy. Keep in mind that we expect momentum to be conserved in every collision, while kinetic energy is conserved for elastic collisions and not for inelastic collisions. Your discussion should go over potential s and the effects of residual friction and gravity. Do your results confirm expectations, given the experimental uncertainties? References Kestin and Tank, University Physics, Vol. 1 (2012), Chapter 7; Physics 0030 Laboratory Measurements. Appendix I: Data tables for elastic collisions Data table 1: Glider masses are equal. Do five measurements with two setups. Glider 2 is stationary Both gliders have initial velocities Run: N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 6

7 Data table 2: Glider masses are different. Do five measurements with two setups. Glider 2 is stationary Both gliders have initial velocities Run: N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 7

8 Appendix II: Data tables for inelastic collisions Data table 3: Glider 2 is stationary. Glider 1 is heavy and glider 2 is light Run: N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A 8