M8 - Newton's Laws and Conservation of Momentum - instructions(1)

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NEWTON'S THIRD LAW AND CONSERVATION OF MOMENTUM Objectives • To study the forces between objects that interact with each other, especially in collisions. • To examine the consequences of Newton's third law as applied to interaction forces between objects. • To formulate the law of conservation of momentum as a theoretical consequence of Newton's third law and the impulse-momentum law. • To explore conservation of momentum in one-dimensional collisions. Risk Assessment Risk assessment is extremely important in engineering. It is your job as engineers to identify and understand the risks present and to suggest measures to prevent harm from happening. Risks are assessed using the matrix below. The two main criteria used are likelihood of occurrences and severity of consequences, outlined in the table below. Risks are mitigated through a hierarchy of controls ,

Part A – Forces Between Interacting Objects Theory There are many situations where objects interact with each other, for example, during collisions. In this investigation we want to compare the forces exerted by the objects on each other. In a collision, both objects might have the same mass and be moving at the same speed, or one object might be much more massive, or they might be moving at very different speeds. What factors might determine the forces the objects exert on each other? Is there some general law that relates these forces? To examine this, three different scenarios will be examined. A1 Same mass, Same Velocity Suppose two objects have the same mass and are moving toward each other at the same speed so that 𝑚 ? = 𝑚 ? and 𝑣 ? ⃑⃑⃑⃑ = −𝑣 ? ⃑⃑⃑⃑ (same speed, opposite direction). In this case, the magnitudes of the forces between object A and object B will be equal and opposite during the collision. A2 Same Mass, Different Velocity Suppose the masses of two objects are the same and that object A is moving toward object B, but object B is at rest. In this case that 𝑚 ? = 𝑚 ? and 𝑣 ? ⃑⃑⃑⃑ ≠ 0, 𝑣 ? ⃑⃑⃑⃑ ≠ 0 . In this situation, the magnitudes of the forces between object A and object B will once again be equal and opposite during the collision. A3 Different Mass, Different Velocity

Now, suppose the mass of object A is much greater than that of object B and that it is moving toward object B, which is at rest. This time, 𝑚 ? > 𝑚 ? and 𝑣 ? ⃑⃑⃑⃑ ≠ 0, 𝑣 ? ⃑⃑⃑⃑ ≠ 0 . Despite the significant difference in the masses, the relative magnitudes of the forces between object A and object B during the collision will once again be the same. This is because, regardless of the masses of objects or their velocities, when two objects collide they will exert an equal and opposite force on one another. This is known as Newton’s Third L aw. Procedure To test Newton’s T hird Law, gentle collisions between two force probes attached to carts, will be observed. Masses will be added to one of the carts to observe the affect that differences in mass has on the forces. The measurements will be performed using a computer-based laboratory system, two force probes with rubber stoppers replacing the hooks, two 1-kg masses to calibrate the force probes, a level ramp, two low-friction carts and a series of masses to vary the mass of the carts. 1. Set up the apparatus as shown in the following diagram. The force probes should be securely fastened to the carts. The hooks should be removed from the force probes and replaced by rubber stoppers, which should be carefully aligned so that they will collide head- on with each other. If the carts have friction pads, these should be raised so that they don't rub on the ramp. 2. Open the experiment file called Collisions (L09Al-1) to display the axes shown

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below. The software will be set up to measure the forces applied to each probe with a very fast data collection rate of 4000 points per second. (This allows you to see all of the details of the collision which takes place in a very short time interval.) 3. Calibrate both force probes for pushes with 1.0-kg (9.8-N) masses balanced on each stopper, or load the calibrations. You may find it easier to calibrate one force probe at a time. (If you are using a Hall effect force probe, you may need to adjust the spacing and check the sensitivity.) 4. Reverse the sign of force probe A, since a push on it is negative (toward the lef t). 5. Use the two carts to explore Situations A-C outlined in Theory section. Sketch the graphs for each collision on the axes provided, or print them and affix them over the axes. Be sure to label your graphs. 6. For each collision use the integration routine to find the values of the impulses exerted by each cart on the other (i.e., the areas under the force-time graphs). Record these values in the spaces below and carefully describe what you did and what you observed. Part B - Other Interaction Forces Theory Interaction forces between two objects occur in many other situations besides collisions. For example, suppose that a small car pushes a truck with a stalled engine, as shown in the picture. The mass of object A (the car) is much smaller than object B (the truck). At first the car doesn't push hard enough to make the truck move. Then, as the driver pushes harder on the gas pedal, the truck begins to accelerate. Finally, the car and truck are moving along at the same constant speed. During all three stages of motion, before the truck starts moving, while the truck is accelerating and when the car and truck are moving at the same speed, the force acting on each object , will always remain equal in magnitude.

If this seems unrealistic, recall that Newton’s Third Law applies to the force exerted by an object on another object, there can be additional forces in play. Consider the second stage of the motion, when the car is causing the truck to accelerate. To understand this see the diagram below. The force of the car acting on the truck (F C → T ), is equal and opposite to the force of the truck acting on the car (F T → C ). But this only makes up part of the total force applied by the car (F car ), the remaining force (F move ) is what causes the truck to move, by Newton’s Second Law ( F=ma ). Procedure 1. Open the experiment file called Other Interactions (L09Al-3) to display the axes that follow. The software is now set up to display the two force probes at a slower data rate of 20 points per second. 2. Use the same setup as in Part A, with the two force probes mounted on carts. Add masses to cart B (the truck) to make it much more massive than cart A (the car) (two or three times the mass). 3. Zero both force probes just before you are ready to take measurements. 4. Your hand will be the engine for cart A. Move the carts so that the stoppers are touching, and then begin graphing. When graphing begins, push cart A toward the right. At first hold cart B so it cannot move, but then allow the push of cart A to accelerate cart B, so that both carts move toward the right, finally at a constant velocity. Try this several times, until clear data is obtained. 5. Sketch your graphs on the axes above, or print them and affix them over the axes.

Part C - Newton's Laws and Conservation of Momentum Theory In Part A and Part B, it was shown that interaction forces between two objects are equal in magnitude and opposite in sign (direction) on a moment by moment basis for all the interactions you might have studied. This is a testimonial to the seemingly universal applicability of Newton's third law to interactions between objects. A consequence of the forces being equal and opposite at each moment, is that the impulses of the two forces were always equal in magnitude and opposite in direction. This observation is the basis for the derivation of the conservation of momentum law , which states that the impulse J A acting on cart A during the collision equals the change in momentum of cart A, and the impulse J B acting on cart B during the collision equals the change in mo mentum of cart B. 𝐽 ? ⃑⃑⃑ = 𝛥𝑝 ? ⃑⃑⃑⃑ 𝐽 ? ⃑⃑⃑ = 𝛥𝑝 ? ⃑⃑⃑⃑ But, as you have seen, if the only forces acting on the carts are the interaction forces between them, then 𝐽 ? ⃑⃑⃑ = −𝐽 ? ⃑⃑⃑ . Thus, by simple algebra 𝛥𝑝 ? ⃑⃑⃑⃑ = −𝛥𝑝 ? ⃑⃑⃑⃑ or 𝛥𝑝 ? ⃑⃑⃑⃑ + 𝛥𝑝 ? ⃑⃑⃑⃑ = 0 i.e. there is no change in the total momentum 𝛥𝑝 ? ⃑⃑⃑⃑ + 𝛥𝑝 ? ⃑⃑⃑⃑ of the system (the two carts). If the momenta of the two carts before (initial-subscript i) and after (final- subscript f) the collision are represented in the diagrams below, then: 𝑝 𝑓 ⃑⃑⃑⃑ = 𝑝 𝑖 ⃑⃑⃑ where 𝑝 𝑖 = 𝑚 ? 𝑣 ?𝑖 + 𝑚 ? 𝑝 ?𝑖 𝑝 𝑓 = 𝑚 ? 𝑣 ?𝑓 + 𝑚 ? 𝑝 ?𝑓

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Procedure In this activity conservation of momentum in a simple inelastic collision between two carts of unequal mass is examined. For this experiment, measurements will be performed using a computer-based laboratory system, two force probes with rubber stoppers replacing the hooks, a motion detector, a level ramp, two low-friction carts, a series of masses to vary the mass of the carts and clay or velcro. 1. Set up the carts, ramp, and motion detector as shown below. Remove the force probes, and place blobs of clay or Velcro on the carts so that they will stick together after the collision. Ad d masses to cart A so that it is about twice as massive as cart B. 1. Measure the masses of the two carts. 2. Open the experiment file called ‘Inelastic Collision’’ to display the required axes. 3. Place Cart A and Cart B on the track with Cart A closest to the motion detector. Fix a small piece of clay or velco to the front of cart A (the opposite side to the motion detector) and the back of Cart B (the side facing cart A) so that when Cart A comes in contact with Cart B they stick together. 4. Begin with cart A at least 0.50 m from the motion detector. Begin graphing, and when you hear the clicks of the motion detector, give cart A a brisk push toward cart B and release it. Be sure that the motion detector does not see your hand. Repeat until you get a good run when the carts stick and move together after the collision. Then sketch the graph on the axes above, or print it and affix it over the axes. 5. Use the analysis and statistics features of the software to measure the velocity of cart A just before the collision and the velocity of the two carts together just after the collision. (You will want to find the average velocities over short time intervals just before and just after-but not during-the collision.) 6. Calculate the total momentum of carts A and B before the collision and after the collision. Show your calculations.