Experiment 2 Rotational Dynamics

Transcription

1 Experiment 2 Rotational Dynamics Background You may find it useful to review chapters from your introductory physics textbook dealing with rotational motion, torque and angular momentum. The average angular velocity of a rotating object can be calculated using ω = Δθ (1) Δt where Δθ is the angle through which the object rotates during a time Δt. We usually express Δθ in radians, and in fact for many of the following equations, angles MUST be expressed in radians. If the angular velocity is constant, then equation (1) also represents the instantaneous angular velocity. For a rotating object experiencing constant angular acceleration, α, θ =θ o +ω o t αt 2. If it starts at rest, this simplifies to Δθ = 1 2 αt 2. (2) Equation (2) can be used to calculate the angular acceleration of a rotating object. Angular acceleration is determined by the net torque acting on the object. First, torque is defined through the equation τ = r F (3) In the case of a force applied at right angles to the lever arm, which is the case when a string is wound around a pulley, the magnitude of the torque is given by τ=rf, (4) where R is the radius of the pulley and F is the tension in the string. Moment of inertia is a measure of the resistance of an object to changes in its rotational motion. The value of the moment of inertia depends on the axis of rotation and on the distribution of mass in the object. For some simple situations we have 1

2 I=mr 2 I=(m/2)(R 2 1 +R 2 2 ) I=(1/2)mR 2 I=(1/12)mL 2 A point mass m a distance r from the axis of rotation A hoop or ring of mass m, inner radius R 1, and outer radius R 2, rotated about an axis perpendicular to it and passing through its center A uniform solid disk of mass m and radius R rotated about an axis perpendicular to it and passing through its center A thin rod of mass m and length L rotated about an axis perpendicular to it and passing through its center Moments of inertia of more complex objects can sometimes be found by breaking the object down into components and adding up the individual contributions to the moment of inertia. The moment of inertia relates torque to angular acceleration: τ net = Iα (5) The moment of inertia also shows up when calculating angular momentum: L=Iω (6) In the absence of external torques on a system, the angular momentum is conserved. This means that in a collision between rotating objects, to a good approximation the total angular momentum of the objects just prior to the collision is equal to the total angular momentum of the objects just after the collision. During such a collision, friction acting between the objects will allow the objects to achieve a common final angular velocity. While this friction does result in a decrease in the kinetic energy of the system (that is, the collision is inelastic), the angular momentum is NOT affected by the friction between the two objects since internal torques do not change the total angular momentum of the system. However, if there is friction external to the system, then the total angular momentum will not be constant over time. 2

3 Apparatus Pasco Rotational Dynamics Platform: As configured for the first part of this experiment, this device consists of a heavy black metal base with a freely rotating vertical metal rod. Mounted on the rotating rod is a horizontal metal platform (it looks like a fancy ruler). The rotating platform has masses whose positions can be varied. You will also need a stopwatch and a mass set. Why do we use a precision length measuring instrument (Vernier calipers) but a crude time measuring instrument (handheld stopwatch) in this experiment? Some lengths you will be measuring are small (around a centimeter) and so having an uncertainty of 1 mm (which would be the case when using a conventional ruler) would introduce a 10% uncertainty in that measurement and thus create at least a 10% uncertainty in all related calculations. The measured times, though, will typically be around 10 s. Using a hand held stopwatch results in a few tenths of a second uncertainty typically, or an uncertainty of a few percent. This is down near the level of the uncertainty introduced by approximating the rotating platform as a thin rod, neglecting friction, etc., so there is no need to go to the next level of timing precision by using a photogate timer. Procedure Part 1: Varying torque 1. There are two adjustable masses mounted on the top of the horizontal rotating platform. One may have coins glued to it. Do not remove the coins since they are present to balance the weights. The weights should be centered at 15 cm on either side of the middle of the rod. A hole through one edge of the block allows you to see through to the scale below. Important: DO NOT OVERTIGHTEN THE SCREWS (the tops break off easily). Tighten the screws just enough to keep the blocks from sliding. SAFETY GOGGLES MUST BE WORN WHILE THE PLATFORM IS ROTATING. These are provided for you in the lab. 2. Slide the apparatus towards the edge of the table so that the plastic pulley mounted on the horizontal rod extends past the edge of the table. Directly underneath the rotating platform, mounted on the vertical rotating rod, you will see three other pulleys ( platform pulleys ). There should be a black thread tied to that system of pulleys. Wind the thread on the uppermost platform pulley. Drape the free end of the thread over the other pulley (the guide pulley ) that hangs over the edge of the table. Hook a 50 g mass onto the free end of the string. When released, the hanging weight pulls on the string that in turn applies a torque to the platform pulley, causing the rotating platform to undergo angular acceleration. You will determine this acceleration with the help of equation (2). 3. Line the rotating platform up with the horizontal rod on which the guide pulley is mounted. Release the platform from rest at the same time you start the stopwatch. 3

4 Equation (2) will only apply if you start the watch at the moment you release the rotating platform. Stop the watch when four full revolutions have been completed. Record this time and repeat until you have three complete measurements. In between each measurement, rewind the thread, being careful that it stays on the uppermost platform pulley. Each time you need to wind the hanging mass back up to a high enough point that the thread will not get unwound completely (and start winding back up in the other direction) while you are timing the four revolutions. 4. Change the hanging mass to 70 g by hooking a 20g mass to the bottom of the 50 g mass, and repeat step Change the hanging mass to 100 g and repeat step 3. You should see an obvious trend in your data indicating that applying more tension to the string produces a greater angular acceleration. 6. Now repeat steps 3-5 for the middle platform pulley and then the bottom platform pulley. You should see an obvious trend in your data indicating that increasing the lever arm (i.e., the pulley radius) increases the angular acceleration. In the end, you should have taken 27 measurements (three measurements for each of three hanging masses for each of three pulleys). It is not necessary to re-tie the thread to a different pulley, but you do need to be careful each time you rewind it that it is going on the correct pulley. Also, particularly for the bottom pulley, you will need to be careful to start the mass at a high enough position so that it does not reach the bottom before the four revolutions are complete. Also watch carefully to make sure the string does not come off the bottom pulley during the timed interval. 7. Detach the rotating platform by loosening the black set screw underneath the platform. Measure the length of the platform, and then measure the mass of the rotating platform and the two sliding masses separately. As you pull the sliding masses off the platform, pay attention to how they are attached so that you can remount them later. The black base at the center of the rotating platform has a mass of 70.9g. It contributes a negligible amount to the moment of inertia of the platform since that mass is concentrated near the center of the platform. When you go to calculate the moment of inertia of the platform, subtract this 70.9g mass from the total platform mass you measured on the scale. 8. Use a Vernier caliper to measure the diameter of the three platform pulleys and remember to convert these numbers to radii for your calculations. Make sure you are measuring the radius of the pulley out to the point at which the thread makes contact with the pulley (not past that to the lip of the pulley). If you have not used calipers before or it has been a while since you have used them, there are instructions on the supply table. I would encourage you to do the following simple exercise to make sure you are lining things up properly: adjust the caliper so that one tip is on the 10 cm mark on a meter stick and the other tip is halfway between the 11.5 and 11.6 cm mark. The distance between the tips should then be about 1.55 cm. Verify that you get about this result when you read the number off the caliper scale. Note that there are two pairs of tips on this device. One 4

5 is particularly convenient for measuring the inside diameter of an object while the other is convenient for gripping an object from the outside to measure an exterior dimension. Part 2: Varying moment of inertia 1. Reattach the rotating platform to the shaft, being careful to line up the set screw with the flat side of the shaft. Wind the string around the middle platform pulley. Adjust the sliding masses so that they are each centered 3cm from the middle of the rotating platform. DO NOT OVERTIGHTEN THE SCREWS. Remember when positioning the adjustable masses to make sure that the scale-viewing hole in the mass is on the scale side of the platform and that the position is as read through that hole. Hang a 50 g mass from the thread and time the platform through 4 revolutions starting from rest. Repeat to obtain three measurements. 2. Repeat step 1, continuing always to use the 50 g mass on the middle pulley and moving both masses successively through the following positions : 8 cm, 13 cm, 18 cm, 23 cm. In each case, these positions refer to the center of each mass. In the end, you will have taken 15 measurements (three measurements for each of five mass positions). Return the adjustable masses to their 15 cm locations so they will be ready for the next students to use. Part 3: Conservation of angular momentum 1. Take the hanging mass off of the string and then loosen the set screw underneath the rotating platform. Be careful not to allow a knot to form in the thread it has a tendency to twist up. Slide the platform off, setting it aside in a safe location. Wind the thread completely around the pulley and use a small piece of tape to secure it to the pulley to keep it from getting tangled. You will not need the thread for this part of the experiment. 2. You should see a nearly solid black disk with a circular groove in it. Mount it where the rotating platform was. If this disk does not slide onto the shaft smoothly, you do not have the flat side of the hole lined up properly with the flat side of the shaft. You should also see a black hollow cylinder that is made to fit in the groove. Spin the solid disk. Hold the hollow cylinder just above the groove on the disk and practice dropping the cylinder onto the disk while the disk is rotating, so that the hollow cylinder lands in the groove. Make sure you let go of the cylinder completely before it makes contact with the rotating disk. If your hand is still touching the hollow cylinder at the same time that the hollow cylinder is making contact with the solid disk, then angular momentum will not be conserved during the collision (since your hand will be applying an external torque during the collision). 3. Remove the hollow cylinder from the disk. Give the disk a spin so that it is taking roughly 0.5s to 1.0s to complete a revolution. Time the disk for ten consecutive revolutions. As soon as possible after timing that (so that friction does not have a chance 5

6 to slow the disk down), drop the cylinder onto the groove and then immediately time another 10 revolutions. Repeat this process for a total of five pairs of measurements. With the help of equation (1), these times will allow you to calculate the angular velocity before and after the collision. 4. Remove both the hollow cylinder and the solid black disk. Remove the tape that was securing the thread and remount the rotating platform. This is a lab courtesy step which gets the apparatus ready for the next group. 5. Measure the mass of the hollow cylinder and of the solid disk. 6. Measure the diameter of the solid disk and of the hollow cylinder. For the hollow cylinder, measure both the inner and the outer diameter (or one of those two and the thickness) using the caliper. The solid disk is too large to measure the diameter with the caliper. There are several possible approaches to determining the diameter of the solid disk I suggest you try two different approaches since each approach has its own potential for introducing error. One approach is to first measure the circumference by measuring the length of a thread that is wrapped around the outside. The radius is this length divided by 2π. Alternatively, you may be able to get an accurate reading by using the meter stick to measure the diameter on one side of the disk there are two small screw holes that can be used to help locate a diameter. A third possibility is to use the caliper to measure the distance from the middle of the center hole to the outer edge of the disk. However you do this, make sure you estimate the uncertainty generated by the experimental procedure (which is likely to exceed the instrument uncertainty). When done, return the solid disk and the hollow cylinder to the table with the rest of the rotational dynamics apparatus. Data Analysis and Writing the Results Section This section will help guide you through writing up the Results section of your report. A complete Results section should show all the raw data (in tabular form when you have a lot), calculated results with uncertainties, sample calculations of each type, an explanation of the data and calculation, and a discussion of the results. Make sure that this section includes sufficient text to be readable. That is, do not just show a data table but introduce it with a sentence or two indicating what data you are showing (e.g., The following data were acquired to test the principle of conservation of angular momentum. ). Put another way, the Results section is not just a collection of numbers and calculations; it is TEXT interspersed with data and calculations. As always, consult the sample lab report online for further information and keep in mind the comment on significant digits in the Experiment 1 write-up. For lab reports (as opposed to submission of raw data a week before the report is due), you should NOT submit a spreadsheet (i.e., an xls or xlsx file). All of your results must be embedded into your report document. In this and subsequent experiments, when you need a value for g, you should use the accepted value for this course, including the provided uncertainty. This value will be 6

7 posted in the lab, and will likely be a value determined by student results from Experiment 1. Part 1 For each of the pulley and mass combinations, calculate the angular acceleration using equation (2). Use the variation in measured times to estimate an uncertainty in each angular acceleration, as discussed in the Uncertainty Analysis presentation. A fair question to ask is, should the uncertainty in Δθ be included too? If you assume that there is only random error (as opposed to systematic error) in this value, then the uncertainty in Δθ will show up as an uncertainty in Δt, since it will only affect when you stop the watch. Thus the uncertainty in Δθ is accounted for in the uncertainty in Δt. If this is the approach you plan to take, then you should state so explicitly in your Results section. In general, I will be looking at each input into your calculation for an uncertainty. If you have not included the uncertainty, then you need to justify why it is not needed. Use equation (4) to calculate the applied torque, taking the tension in the string to be approximately mg for the hanging mass. Don t forget to determine the uncertainty in the calculated value of the torque! Using a spreadsheet program, plot the applied torque (vertical axis) vs. the angular acceleration (horizontal axis) for all nine mass/pulley combinations. If you have correctly analyzed the data and equation (5) is correct, the result ideally would be a straight line passing through the origin. Add a trend line showing a linear fit. Under options, do not specify that it passes through the origin but do have it display the equation. The best-fit equation will likely not pass through the origin. As you discuss your results in this section, discuss the physics behind this non-zero intercept and discuss what the slope of the trend line represents. (This should serve as a reminder that discussion DOES belong in the results section!) Remembering that both the calculated torque and the calculated acceleration have uncertainties, figure out how to get the spreadsheet to display representative error bars (this is possible in Excel highlight the data series and then pull up the format data series menu). Make sure the final plot has all appropriate labeling. Part 2 Knowing the mass and length of the rotating platform, you can estimate its moment of inertia if you treat it as a thin rod. Add to this the contributions from the two adjustable masses, treating them like point masses. You should express everything in kilograms and meters. Calculate the approximate applied torque as above (which should be the same for all trials in this part) and then for each mass position you can predict an angular acceleration from equation (5): α pred = τ appl I Compare this value to the measured angular acceleration, calculated the same way you did in Part 1. Each of these accelerations (both predicted and measured) should have an 7

8 uncertainty calculated. If the uncertainty intervals overlap (e.g., if α pred =2.3+/-0.4 and α meas =2.1+/-0.3), then you have verified the theoretical prediction to within the uncertainties in your experiment. If the uncertainty intervals do not overlap (e.g., 2.3+/- 0.4 and 1.7+/-0.1), then you have not verified the theoretical prediction. Among the possible reasons are (1) you underestimated the uncertainties; (2) the theory has been incorrectly applied because a significant effect was overlooked (such as friction); and (3) the theory is incorrect. Your discussion of the results should include your analysis about whether the uncertainty intervals overlap or not. You can postpone conclusions from these observations to the Conclusions section (which, fortunately for you, is not required for this experiment). Part 3 Treating the solid disk as uniform and the hollow cylinder as a hoop (see the moment of inertia table on page 2), calculate their moments of inertial. We expect that during the disk collision L i = L f ω i = ( + I cylinder )ω f ω f ω i = + I cylinder Δθ Δt f Δθ Δti = Δt i Δt f = + I cylinder + I cylinder The ratio on the right hand side can now be calculated and should be the same for each trial if in fact angular momentum was conserved (which was the starting assumption in the calculation). The ratio on the left hand side needs to be calculated separately for each trial. Note that is does not make sense to average the Δt s first since you do not expect them to be the same for each trial. Furthermore, since angular momentum should be conserved separately for each trial, we expect agreement in each trial to within expected uncertainties in this last equation. Your analysis should include a table showing the calculated values for each trial of the left hand side of the last equation compared to the predicted value (the right hand side of the above equation), and an indication for each about whether they are within expected uncertainties. In Parts 1 and 2 you developed a good feel for expected uncertainties when timing rotational motion. You can use an estimate from those numbers for your uncertainty in Δt here. You should also estimate uncertainties in the mass and the radius for your moment of inertia calculations. This will allow you to determine the uncertainty in the predicted ratio (the right hand side of the equation above). 8

9 What is due? For this lab, you should turn in a Title Page and the Results Section. The Title Page should include the name of your lab partner (or No Lab Partner if you had none), the date you turned the report in, and an Abstract. The Abstract should describe the nature of the experiment in a few sentences and briefly summarize the results. This is not a mystery novel you are writing it is expected that you reveal the essential results in the Abstract. For instance, specify whether your results were consistent with conservation of angular momentum in the collisions. See the introductory material posted on the web site for a sample lab report (showing all sections, not just those required for this lab). Check this introductory material on the web site every time you are learning how to write a new section. About half the points that are lost by students on these first few reports typically arise because they did not read the introductory material outlining expectations for the report, and they did not consult the sample report. A word on including numerical results in abstracts: In your previous experiment, your goal was to measure the local value of g. This result has some general interest to others who might be doing experiments in Ypsilanti and to those interested in how g varies geographically. For that reason, an abstract for that experiment could usefully include the numerical value of g, along with its uncertainty. In this experiment, however, you are doing things like calculating the moment of inertia for a particular configuration of a particular apparatus. That value will not have any general use outside the context of your experiment so it should NOT be quoted in the abstract. Instead, you might include statements like, We found that we were able to calculate angular accelerations to within an uncertainty of x%. We then compared them to the predicted values (known to within y%) determined by dividing the torque by the moment of inertia. We found these two calculations did/did not agree. That is, rather than quoting any actual values of acceleration or moments of inertia, which are not important, you should focus on whether you verified the key equation, τ=iα. Grading comment Pay careful attention to the way sections of lab reports are graded (see the introductory material posted on the web site). In particular, the uncertainty analysis multiplier is a grading device designed to ensure that you take uncertainty analysis seriously. Your score on all sections related to the results is multiplied by a number ranging from zero to one. The multiplier will be 1.0 if you do a credible (not necessarily perfect) job of estimating uncertainties and other sources of error, and of carrying these estimates through numerically in all of your calculations. If you skip this, your uncertainty analysis multiplier will be 0, meaning you will get no credit for any of the relevant sections. 9