Physics 141 2 nd Air Table Experiment Conservation of Angular Momentum Introduction In this experiment, you will investigate angular momentum, and learn something important about the conditions under which angular momentum is conserved. The definition of angular momentum, L, of a particle located a distance r away from some well-defined origin having mass m moving with velocity v (as shown in the figure to the right) is: m (1) (Note that this definition applies only to the angular momentum of a single particle. In class, you will also discuss the angular momentum of a collection of particles or a rigid body, but we only need to worry about this single particle definition for this experiment.) As you can see from equation (1), angular momentum is a vector, which means it has both magnitude and direction. The magnitude is given by (2) L = mvr sin and the direction is given by the right-hand-rule for cross products (you should review vector cross products if this is fuzzy for you ) Now, you remember from studying dynamics that one way of writing Newton s second law is F = dp/dt. Let s use our definition of torque to find an important interpretation of angular momentum: (3) What does this equation tell us? If the net torque on a particle is equal to the time rate of change of its angular momentum, then what will happen if there is no net torque on the particle at all? Then dl/dt = 0, which means that both the magnitude AND direction of L must not change with time! In this unique situation, we say that L is conserved. But remember, for L to be conserved, it is essential that there be no net torque on the object. Using the picture at the top of the page for reference, list a couple of ways for there to be no net torque on that black object. What must be true about any forces acting on it?
Procedure For this experiment, you will once again be using the air tables and spark timer (shown below). Please remember that when the spark timer is on, you must not touch any metal part of the puck! It is only safe to touch the insulator. You will use two magnetic pucks, one fixed and one free to move. The pucks consist of one or two ceramic ring magnets mounted on plastic disks. Please be aware that these disks are very fragile, so it is important that they never be dropped or banged together. Begin by leveling the air table with one of the non-magnetic pucks. Then, with a piece of white paper underneath, setup the pucks as shown in the image to the right (top). Tape the upper puck to the white paper with a couple of pieces of masking tape. Attach the air hoses to both pucks and practice launching the movable puck a couple of times towards the fixed puck so that it follows a trajectory similar to the one shown to the right (bottom). The moving puck should be deflected about 90º. After practicing, make a spark trace. Fixed Moveable Notice on the diagram that three points along the trajectory have been labeled (A, B and C). For those 3 points, draw arrows on this page that indicate what you predict to be the direction of the magnetic force (not the velocity!) acting on the moveable puck due to the fixed puck. Draw the arrows to scale, as well. A larger force should receive a longer arrow. Now, let s define the origin of our coordinate system to be the center of the stationary puck. For 6-8 points at regular time intervals of 0.05 seconds, measure and record t, r,, v, x and the z-component of L in the table on page 4. For an example of how to calculate some of these quantities, refer the sample trace on page 3. Also measure the mass of the moveable puck and record it here: A B C
t (sec) r (m) (deg) x (m) v (m/s) L z (kg m 2 /s) On a separate sheet of graph paper, create a plot of angular momentum vs. time. Is it reasonable to say that angular momentum is conserved about this origin? Explain how you know. Now, select a new point far from the center of the stationary puck to be the new origin of our coordinate system. (Remember, a coordinate system is a completely human-made thing, and we can pick it to be wherever we want!) Repeat the calculations and fill in a new table for this new origin (your trace will be quite messy at the end of all this!): t (sec) r (m) (deg) x (m) v (m/s) L z (kg m 2 /s) On a separate sheet of graph paper, create a plot of angular momentum vs. time.
Questions 1. Did your calculations of L for the two origins agree (circle one)? Yes / No 2. If they didn t, why not? 3. What was special about the first origin that caused L to be conserved? 4. Was there a net torque on the puck about the first origin (circle one)? Yes / No 5. Was there a net torque on the puck about the second origin (circle one)? Yes / No 6. If your answers to #4 and #5 were different, explain why. 7. What is the physical interpretation of the slope of your second graph? 8. At what point during the puck s motion was the slope of the second graph the largest? Why do you suppose that was?