2DC Final Kress

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Iowa State University *

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131L

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Physics

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Apr 3, 2024

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Lab 2DC - 2D collisions Equipment  A table with a glass surface on which two heavy, air-supported pucks can slide about with little friction.  A spark generator that produces sparks with a frequency of 60 Hz (or 60 sparks per second)  Carbon paper  21” × 21” paper  A set of normal pucks Lab 2DC – Page 1

Background Consider a puck that moves on a horizontal, frictionless table. “ The linear momentum of this puck is constant during this motion. In other words, the linear momentum of this puck is a conserved quantity.” True or false? Explain. True because there is no friction on the table, therefore net force is equal to zero and the momentum is conserved. Now assume that we have two such pucks on the frictionless table. The pucks collide around the center of the table, and then each comes out in a different direction. At any given moment, each puck has its own linear momentum: We can define the total linear momentum of the system of two pucks: Prediction Is the total linear momentum a conserved quantity during the collision? Explain. Yes, I think the total linear momentum is going to be conserved because there is no friction and therefore no nonconservative forces acting on the pucks. Your goal today is to test your predictions about the conservation of total linear momentum using the record of a collision between two pucks on a low-friction table. Lab 2DC – Page 2

Experimental setup Before you proceed, make sure that:  The spark generator is turned off (to ensure you don’t get accidentally shocked)  A 21”  21” sheet of plain paper has been placed on top of the carbon paper. The paper does not need to be taped. To avoid abrasions and tears, the pucks should never be placed directly on the carbon paper. Then, use the figure below to identify all the components of your apparatus. Table leveling Use the spirit level and the leveling screws on the feet of the table to make sure that the surface of the table is horizontal. The results of this experiment will be affected by even small deviations, so make sure you are as precise as possible. Air cushion Air flows through the tubing to each of the pucks on provides a cushion of air, so the pucks can slide on the surface with negligible friction. Turn on the air (turn the knob 1 on the pressure regulator CW) to about 8 PSI (pounds per square inch) and check that both pucks can slide freely. 1 You should only need to adjust the knob that feeds air to a couple of tables, NOT one of two main valves in the room. That main valve should be only partially opened, because the maximum pressure in the system is 90 PSI! If the pressure is too high, the tubes will burst! Lab 2DC – Page 3

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Spark generator When the spark generator is turned on AND the pedal is depressed, it periodically produces for a very short time a large electrical potential difference between the two pucks. This causes electric charge to flow from the center of the base of one puck, through the paper in the form of a spark, then along the carbon sheet, and back through the paper to the second puck. Each time this occurs, two small black marks are left on the bottom side of the plain paper, indicating the position of each puck at the time of the electrical pulse. When a puck is moving, a trail of black marks is left on the paper indicating the direction and speed of the moving puck. For today’s labs, the spark generator should be set to 60 Hz. What is the time between two successive sparks? 1/60 of a second. The spark marks are not a perfect record of the positions of the pucks. You will probably notice that the spark marks do not lie exactly along a smooth line, nor are they spaced by the exactly the same distance when the puck is traveling at constant speed. The source of these random errors is the changing direction of the sparks from the puck to the carbon paper, as illustrated in the figure to the right. To minimize these errors, the spark electrode is designed to be very close to the paper. These errors limit the accuracy with which you can determine the true position and speed of each puck. Safety tips While the spark apparatus is not dangerous, the sensation which it can produce is rather unpleasant. To avoid electric shock:  Do not depress the spark generator pedal until you are ready to take a data run.  Turn the power switch off as soon as that data run is finished.  Do not lift a puck from the table while the pedal is depressed.  Handle the pucks only by their insulated stems.  Do not touch the carbon or paper sheets while the pedal is depressed. Be gentle!  Handle the pucks gently and only by their plastic stems.  Handle the carbon paper gently. A carbon sheet should last for many classes. Tears may be repaired with ordinary transparent tape applied to the underside of the carbon paper (not the side that prints!) Lab 2DC – Page 4

Practice runs  Take some practice runs with spark recording. o Review the safety tips! o Holding the two pucks by their plastic stems, depress the foot pedal (to turn on the sparks) and give the pucks an initial push so they collide somewhere near the center of the table. o After the pucks have traveled well away from the collision point, release the foot pedal and turn OFF the power switch of the spark generator. o Inspect the spark record on the underside of the paper. Some of the marks may be faint. This is just fine as long as they are visible.  Don’t waste paper! Several test runs can be recorded on the same side of a data sheet, and both sides of the sheet can be used!  Practice giving the pucks speeds such that there is about 1 cm between spark marks. Lab 2DC – Page 5

Example of a good collision. Data collection Collision record Some collisions make the analysis a bit easier. In order to obtain good results, we recommend using a collision record such that: Each of the traces before and after the collision has several dots in a straight line. If the trajectories when the pucks are moving freely are not straight, think what can be the cause and correct it.  None of the pucks is moving too slowly or too fast at any given point.  It is not a head-on, 1D collision with almost the same directions before and after the collision. Draw arrows to record the direction of motion of the pucks along the traces. Once you have a nice record of a collision, turn off the equipment. Lab 2DC – Page 6

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Measurements Organization is key:  Work with pencil , so you can erase and keep things neat.  Label all the quantities carefully. Making mistakes is a lot easier when the calculation is a mess. Also, you will have to turn in a photograph of your work at the end. Magnitude of linear momentum Let the trace in figure A below be the trajectory of puck 1. The puck starts at the top right corner, moves with velocity , and collides with another puck (whose trajectory is not pictured), and comes out with velocity . Figure B shows how to measure the magnitude of these speeds. For each segment, we measure a stretch with 6 marks, or 5 intervals, starting with the first mark after the collision. Let these distances be d 1 and . Since the sparks are produced at a rate of 60 sparks per second, each of the intervals between sparks corresponds to 1/60 s. Therefore, the magnitude of the velocities of puck 1 before and after the collision are: Lab 2DC – Page 7

Important details:  We are measuring these distances 1) as close as possible to the collision, and 2) for a small number of points. This is because the system does have a little friction.  You should avoid including the “corner point”. It is often difficult to know if it belongs to before or after the collision! Once you have the velocities (properly labeled and showing the calculations on the paper), we should compute linear momentum. The mass is written on it on each puck. Record the masses of your pucks below: Puck 1: 522.8 g Puck 2: 528.0 g Direction of the linear momentum Select a coordinate system. We only really need to draw the x axis. The y axis is of course at 90  , you just need to decide which way is positive. Lab 2DC – Page 8

Draw long lines along the direction of each branch, making sure the line intersects your x axis. You will need to measure the 4 angles shown below. Lab 2DC – Page 9 In case you need a reminder about how to use a protractor: Draw lines for the sides for the angle you want to measure. The lines should be longer than the radius of the protractor. Align one of the sides with the 0  line, making sure that the vertex of the angle is at the center of the protractor. Read angle that coincides with the second side, making sure you use the correct scale (the angle below is 40  , not 140  )

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Write all your measurements on the paper, clearly labeling each quantity. Then, enter your measurements below: Before the collision After the collision Puck 1 Puck 2 Puck 1 Puck 2 d (cm) 3.6 3.3 2.6 3.7 v (cm/s) 43.2 39.6 31.2 44.4 p (g cm/s) 2.26 x 10 4 2.09 x 10 4 1.63 x 10 4 2.34 x 10 4  (  ) 55 72 73 61 Take a photograph of your collision, with all your markings and measurements. Insert the image here. Lab 2DC – Page 10

Uncertainty The two major sources of uncertainty are: 1) The inherent unpredictability of the path of the spark. The spark marks do not lie exactly along a smooth line, nor are they spaced by the exactly the same distance when the puck is traveling at constant speed. The source of these random errors is the changing direction of the sparks from the puck to the carbon paper, as illustrated in the figure to the right. To minimize these errors, the spark electrode is designed to be very close to the paper. These errors limit the accuracy with which you can determine the true position and speed of each puck. 2) The precision of our measurements with rulers and protractors. The uncertainty of the spark generator frequency and of the puck masses (which were measured with an electronic scale) is much smaller and thus negligible. A proper treatment of uncertainty would demand that we estimate the uncertainty of the distances and angles, and propagate that throughout the entire calculation. The errors one obtains for each momentum are usually between 2% and 4%. For simplicity, we will instead assume a flat 3% uncertainty in each of the linear momentums, and which approximately results in an uncertainty of 6% in the total linear momentum of the system (either before or after the collision.) Lab 2DC – Page 11

Data analysis Our goal is to see if total linear momentum is conserved. 1. Before the collision Use your measurements to compute the total linear momentum before the collision, . Remember that this is a vector sum! P1 x = 2.26x10^4 x cos(55) = 1.30 x10^4 g*cm/s AND p 1y = 2.26 x10^4 x sin(55) = 1.85 x10^4 g*cm/s p 2x = 2.09 x10^4 x cos(72) = -6.48 x10^3 g*cm/s AND p 2y = 20.9 x10^4 x sin(72) = 1.99 x10^4 g*cm/s Before x = 1.30 x10^4 – 6.48 x10^3 = 6.52 x10^3 g*cm/s Before y = 1.85 x10^4 + 1.99 x10^4 = 3.84 x10^4 g*cm/s Find the magnitude of the total linear momentum before the collision |p before | = √ ( 6.52 x 10 3 ) 2 +( 3.84 x 10 4 ) 2 = 3.89 x10^4 g*cm/s As argued above, this value could be off by 6%. Find the absolute uncertainty in ( i.e ., what is 6% of ?) 2.34 x10^3 g*cm/s is the uncertainty. 2. After the collision Compute the total linear momentum after the collision, P1’ x = 1.63 x10^4 x cos ( 73 ) = -4.77 x10^3 g*cm/s AND p1’ y = 1.63 x10^4 x sin ( 73 ) = 1.56 x10^4 g*cm/s P2’ x = 2.34 x10^4 x cos ( 61 ) = 1.12 x10^4 g*cm/s AND p2’ y = 2.34 x10^4 x sin ( 61 ) = 2.04 x10^4 g*cm/s After x = -4.77 x10^3 + 1.12 x10^4 = 6.43 x10^3 g*cm/s After y = 1.56 x10^4 + 2.04 x10^4 = 3.60 x10^4 g*cm/s Find the magnitude of the total linear momentum after the collision |p after | = √ ( 6.43 x 10 3 ) 2 +( 3.60 x 10 4 ) 2 = 3.66 x10^4 g*cm/s This value could be off by 6%. Find the absolute uncertainty in Lab 2DC – Page 12

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2.19 x10^3 g*cm/s is the uncertainty. 3. Change in the magnitude of total linear momentum Find the difference 3.66 x10^4 – 3.89 x10^4 = |2.3 x10^3| Using the uncertainties in each total momentum, estimate the final uncertainty in this difference: by how much could be off? The final uncertainty could be off by 12% because each momentum, before and after could each be off by 6%. 4. Change in the direction of total linear momentum We are trying to test if total linear momentum (a vector!) is conserved. Our analysis above compares the magnitudes. In principle, we should now also verify that the direction hasn’t changed significantly, either. Find the polar angle for the total linear momentum before and after the collision: Θ (before) = 80.4 o Θ (after) = 79.9 o Each of your angle measurements could be off by about 1  , due to the limitations of the protractor and the unpredictable nature of the sparks. Using this for a quick estimate for the uncertainty of the two angles above, we can then expect them to be off by 2-3  . Is this the case? Yes, our angles are very close but not exactly the same. Our angles are even closer than the 2-3 degree uncertainty but it is an accurate estimate. Conclusions Did your data confirm your prediction? Your explanation must include the final uncertainties in magnitude and angle. Before momentum = 3.89 x10^4 +/- 2.34 x10^3 g*cm/s = [3.66x10^4, 4.12x10^4] Lab 2DC – Page 13

After momentum = 3.66 x10^4 +/- 2.19 x10^3 g*cm/s = [3.44x10^4, 3.88x10^4] I think our data does prove that momentum was conserved because the value of the momentum after the collision is contained in the uncertainty of the momentum before the collision. The polar angles that we calculated also show that the linear momentum direction didn’t change very much because the values are contained in the ranges for both of the angles. 79.9 is contained in the range from [78.4,82.4] and 80.4 is contained in the range from [77.9,81.9]. Lab 2DC – Page 14

2DC Final Kress

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