Conservation of Angular Momentum

Size: px

Start display at page:

Download "Conservation of Angular Momentum"

Stanley Underwood
5 years ago
Views:

1 In analogy with linear momentum, one can define in rotational motion the quantities L, the angular momentum, and K, the rotational kinetic energy. For the rotation of a rigid body about a fixed axis, the vector nature of angular momentum is accounted for by simply choosing clockwise rotations as positive and counter clockwise rotations as negative (or vice versa, if one prefers). Notice the correspondence: Translation Rotation Momentum p mv L I Energy 2 T 1/ 2mv 2 K 1/ 2I Here I is the moment of inertia and the angular velocity. In this experiment we will test the law of conservation of angular momentum. For this purpose we use an air table, which reduces friction by providing a cushion of air on which disks can rotate (see Figure 1). Aluminum disk E Steel disk D Valve V Air Figure 1 Air escaping through the six holes A supports the steel disk D on an air cushion

2 Air passing through the small nozzle B also passes through holes in the upper part of the steel disk and provides a cushion upon which a second disk E (of aluminum) can rotate, practically independently of the lower disk s motion. Depressing the rearmost button on the air table closes valve V, cutting off the air through B. Disk E promptly collapses upon D - which itself can still rotate freely and solid to solid friction very quickly brings the two disks to the same value of angular velocity. In short, a perfectly inelastic rotational collision has occurred compare to the perfectly inelastic linear collision when the bodies stick together after contact. THREE METHODS OF CALCULATING ANGULAR VELOCITY I. OSCILLOSCOPE METHOD There is no acceleration involved except at the moment of contact. Since the angular velocities we want to observe should therefore be constant, we can use a photocell technique to observe these velocities. We reflect light off the edges of the disks into the photocells. Highly reflective white strips are attached to the rim of each disk at 360/200 degree intervals. Thus as the disk turns, the photocell sees the light whenever a strip passes in front of it and does not receive any light at other times. There is one photocell for the bottom disk and another one for the top disk. The photocell signal is then fed into the vertical input of an oscilloscope. On the oscilloscope screen, we see an upward spike when the photocell starts receiving light, i.e., when the white strip enters the photocell s viewing angle, and a downward spike when the white strip leaves its viewing angle. Since the white strips are spaced at 1.8 degree intervals, the time t between two consecutive upward spikes corresponds to the time the disk takes to turn through 1.8 degrees. Hence the time, T, for a complete revolution (the period) is T = 200 t Since we know the sweep time s of the oscilloscope (i.e., the time the electron beam takes to travel one centimeter on the screen), we can find t by measuring the distance 1 between two consecutive spikes, since t =ls Finally, we can find the angular velocity of the disk by using the period T, since

3 II DIGITAL DISPLAY METHOD 2 /T Also note that when the disks are spinning the LED s on top of the display housing will flicker on and off. Each LED comes on when the optical reader senses a black bar and goes off when it detects a white bar. The display counts the number of black bars that are detected by the optical reader per second. The measurement is made and the display updated every 2 seconds. The maximum count rate for the optical readers is 700 Hz. Any reading above 700 Hz may not be reliable. Angular Velocity of the Disk This angular velocity is simple to obtain on the rotation tables, because the disks used contain alternate black and white bars completely around the circumference. The table includes an optical detector that produces a pulse each time a black bar passes it. The pulses are counted for a fixed amount of time (one second in our case) and then displayed digitally. Thus, the counter reads the number of black bars per second passing the detector, which can be used to derive the angular velocity of the disk, as discussed below. The digital display is updated each second, when a new count of black bars passing is completed. In the conditions of the current experiment, we would expect only a small change (decrease) in the bar count caused by friction, since no other forces act after collision. To obtain an angular velocity from the bar frequency" reading shown on the counter, we use the fact that the black and white bars are each one millimeter wide. Therefore, one count on the digital display corresponds to two bars (one black and one white) or two mm/sec of disk circumference sweeping past the optical reader. More generally, a bar frequency of N showing on the counter corresponds to the disk circumference sweeping past the detector at a linear speed s, equal to 2N mm/sec. But this linear speed of the circumference is simply s=r, where R is the radius of the disk and omega is the angular velocity. Therefore f s R 2N radians/sec R where N is the counter reading and R the disk radius. Note that the radius should be measured in millimeters to use the relation in this form (because the 2 in the numerator has millimeter units)

4 III COMPUTER INTERFACE METHOD The output signals from the stereo phone plugs are TTL compatible, so with the proper interface hardware and software you can monitor disk rotation (upper and lower disks) with a computer. The Science Workshop 750 interface and Data Studio software (available on all lab computers) allow one to easily graph disk rotation angle versus time in real time. The slopes of the graphs can be computed to give angular velocity and angular acceleration. Procedure:( Oscilloscope Method) Level the air table very accurately, using the corner screws. This can be done more easily if one of the four screws is raised and only the remaining three are used for leveling. Set up the disks as in Figure 1. The nozzle B is rather fragile. Disk D should not be dropped onto the air table, but centered and then carefully lowered into place. Connect the stereo phone plug from the optical reader into a stereo plug to BNC adaptor and connect the BNC part of the adaptor to the oscilloscope. If the switch is in the bottom position, you will see the bottom disk signal on the scope; in the top position you will see the top disk signal on the scope. Set the sweep time of the scope to 10 msec/cm and the vertical sensitivity to 100 mv/cm. Make sure that the verniers are at their calibrated positions. Turn the AC/DC switch to the AC position. Turn on the air blower and spin the disks. Observe the top and bottom signals on the scope. If you get a noisy signal rather than sharp spikes, that is most likely due to bad light focusing. Readjust it. Three cases of collision can be studied: CASE I: One disk initially stationary Then hold the bottom disk stationary and spin the top disk. After seeing the spikes on the scope, adjust the trigger level such that the sweep starts with an upward spike. Measure the distance between spikes. Then close the valve, V to bring the disks in contact. Again measure the distance between spikes. CASE II: Both disks initially rotating in the same direction Now spin both disks in the same direction. In this case first measure the top signal,

5 then measure the bottom signal and then close the valve, V for collision. The air cushions are not perfect. Hence any sizeable delay between consecutive measurements will mean that the disks will slow down by an unknown amount before collision. Thus work quickly. CASE III: Disks rotating in opposite directions If you have time repeat the experiment spinning the disks in opposite directions. Check for conservation at angular momentum and the kinetic energy of rotation in each of the cases studied. Note that the moment of inertia of a cylinder about its axis is given by(1/2)mr 2 Questions: 1.Investigate estimated uncertainties in the measurements and compare these with discrepancies in your results. 2.If the aluminum disk had a defect in the form of a cylindrical cavity of radius a, symmetric around the axis and almost as high as the disk, how could you in principle detect this cavity? About how large would a have to be to detect this cavity, taking into account the discrepancies at question 1? 3.Calculate for all cases of rotational- collisions measured, the fraction of initial kinetic energy lost in the collision. Added Note: The rotational dynamics apparatus manual contains further technical information about the unit and the experiment described above, other experiments are also described. The manual is available in the lab and online

Angular Momentum. Brown University Physics 0030 Physics Department Lab 4

Angular Momentum. Brown University Physics 0030 Physics Department Lab 4 Angular Momentum Introduction In this experiment, we use a specially designed air table on which we cause the collisions of a ball and a disk, and so observe the consequence of angular momentum conservation.