Rotational Dynamics. Theory of Rotational Motion. Rotational Dynamics 1

Transcription

1 Rotational Dynamics 1 Rotational Dynamics Overview: In this lab you will set a disk into rotational motion by pulling on a string wrapped around a spindle at its center; this will allow you to determine the relation between torque and angular acceleration. You will also test conservation of energy and angular momentum in collisions of rotating objects. Physics principles: Moment of inertia (rotational inertia) Angular velocity and angular acceleration Torque Conservation of angular momentum New lab skills: Mastering the photogate timing software Equipment needed: Mounted hub with fixed disk and removable disk Rectangular wooden block Steel weights connected by aluminum rod Two short wooden dowels that fit over center spindle on disk Pin to hold dowels in place Vernier caliper Ruler or meter stick Balance (with counterweights) String (about two meters) Spring scale (20 N range) Computer with ScienceWorkshop interface Photogate Theory of Rotational Motion The current location of a point on a rotating wheel or other rotating object can be specified by an angle, θ, which we usually measure in radians. The rate of rotation can then be specified by computing the change in θ per unit time, ω = θ t, (1) measured in radians per second. This quantity is called the angular velocity. The change in this quantity per unit time is called the angular acceleration, α = ω t, (2)

2 Rotational Dynamics 2 and is measured in radians per second squared. Just as the linear acceleration of an object is determined by the sum of all the forces acting on it, the angular acceleration of an object (about a fixed axis) is determined by the sum of all the torques acting on it. Written as an equation, τ = Iα (3) all (analogous to F = ma). Each torque τ is associated with a force F, acting at a distance r from the axis of rotation. When the force is directed tangentially, the torque is simply the product, τ = rf (when force is tangential). (4) Thus a given force, acting farther out from the axis, produces more torque. Torque is measured in newton-meters. The quantity I in Equation (3) is called the moment of inertia. It represents an object s resistance to angular acceleration, just as the mass of an object represents its resistance to linear acceleration. An object s moment of inertia is determined by its total mass and also by how the mass is distributed: The farther the mass is from the axis of rotation, the more it contributes to I. The precise formula is I = m i ri 2, (5) pieces where the sum runs over all parts of the object. (A continuous object such as a round disk can be mentally divided into pieces, e.g., concentric rings, in order to carry out the sum. In your textbook you can look up specific formulas for moments of inertia for various shapes, derived by converting the sum to an integral and using calculus.) The unit of moment of inertia is kg-m 2. In collisions between rotating objects, the quantity that is most often conserved is the angular momentum, L = Iω, (6) analogous to linear momentum, p = mv. The precise rule is this: If there are no external torques acting on a system of objects, then the total angular momentum of the system (that is, the sum of the angular momenta of all the objects) will not change over time. Angular momentum is measured in kg-m 2 /s. In some collisions, mechanical energy may also be conserved. The kinetic energy of a rigid object rotating about a fixed axis is given, fundamentally, by the sum of mv 2 /2 for all its pieces. It s not terribly hard to prove that this is equivalent to the formula As usual, kinetic energy is measured in kg-m 2 /s 2, or joules. Instructions K = 1 2 Iω2. (7) In this experiment you will do three things: Compute moments of inertia for three types of objects; investigate the relation between torque and angular acceleration; and study collisions between rotating objects.

3 Rotational Dynamics 3 Part 1: Moments of inertia First you ll compute the moments of inertia of three types of objects, by weighing and measuring them. The three objects are a uniform disk, a rectangular block, and a pair of steel weights connected by an aluminum rod. Make the necessary measurements and fill in Part I of the Report. (Hints: For the disk and the rectangular block you can look up relevant formulas in your textbook. For the rod with weights, neglect the rod itself (which is very light) and treat each weight as a point mass located at its center. For all three moments of inertia it s a good idea to check your results with another group.) Setting up the computer In the remaining parts of this experiment you ll be using a photogate to determine the angular velocity of the rotating disk as a function of time. You ll also use the photogate timing software to produce a graph of ω vs. t. Plug a single photogate into port 1 and open the Rotation document in the Lab Files folder on the computer s desktop. You will see a data table that is set up to measure time and angular velocity ω, and a graph prepared to plot Omega vs. Time. The Data window displays two pieces of information: Timer 1 (s), which is set to measure the time between two successive blockings of the photogate; and a formula that calculates Angular Velocity ω, measured in radians per second. Make sure each person in your group understands where this formula comes from and why before you go any further. Give the disks a good spin and click Start to begin collecting data. Data is recorded in the table and simultaneously plotted on the graph. Click Stop after you have at least 10 data points. (If necessary, adjust the graph s axes by running the mouse over the numbers (the cursor becomes squiggly arrows). Then click and drag the numbers to accommodate your data.) The software can tell you interesting statistics about this data. Click on Fit at the top of the graph and select Linear from the drop down menu. (If you have multiple data runs, you can select the run you want by clicking the desired run s icon in the Y-axis label (Omega (rad/s).) The software draws the best-fit line and displays information such as slope, Y intercept, etc. You can also find the slope by clicking and dragging over the points of interest (the points become highlighted) and selecting the Slope Tool in the graph s toolbar. Make sure that all members of your lab group understand the computer setup procedure. While taking data, give everyone a chance to operate the computer. When you are ready to move on, select Delete ALL Data Runs from the Experiment menu and click Yes to remove your previous data. Alternatively, you can highlight the label of any of your data and hit the delete key. Part 2: Torque and angular acceleration Next you ll exert various torques on a rotating disk, and measure the resulting angular acceleration. Use the two combined disks, since it s desirable to have a fairly large I. First give the disks a good (but not violent) spin and use the photogate and computer to measure the angular velocity ω as a function of time. Click Start and let the computer collect 15 or 20 data points, and then stop the collection. You ll find that ω slowly decreases,

4 Rotational Dynamics 4 due to friction in the bearings. Notice that the computer not only plots the data, but also draws the best straight line through the points and computes the slope of this line (which is α). Print a copy of this graph for each person and label the graph Frictional Deceleration. Enter the value of α into the Report, then multiply it by I to obtain the frictional torque being exerted on the disk by the bearings, and enter this into the Report as well. Delete ALL Data Runs before moving on. Figure 1: Applying a torque to the spindle. Now wrap a string (one or two meters long) around the metal spindle sticking up above the disks. Wrap it as smoothly as possible, making at least 15 turns. Then pull the string using a spring scale, doing your best to exert a steady force of 3 N. (Practice once or twice before you begin taking data.) While pulling on the string, use the computer to measure ω as a function of time, in order to determine α. You need not print the graph of ω vs. t; just write down the computer s determination of the slope α in your Report. You need to measure the diameter of the spindle and use this information to compute the torque. It is difficult to make this measurement with a ruler, so you will instead measure the diameter by placing the spindle between the jaws of the vernier caliper. Examine the caliper carefully to see how it works, and have your instructor or lab aide show you how to read the scale. (You may be using the caliper and other vernier scales in other experiments, so make sure that everyone learns how to read it.) Repeat with forces of 6 N, 9 N, and 12 N, giving each person a chance to pull on the scale. Then affix each of the two wooden dowels to the spindle (to increase r) and do it again, both times with a force of 3 N. Enter all data in the table provided (or set up a spreadsheet). Be careful to look at each column in this table and think through what measurement and calculation you need for each piece of data. Then, make a plot of the angular acceleration (vertically) vs. the net torque (horizontally). (If you use a computer to make the plot, print a copy for each person.) Because the force of your pull was not perfectly steady, each point on this graph

5 Rotational Dynamics 5 has an associated uncertainty in the horizontal direction. It is conventional to represent this uncertainty by drawing a horizontal error bar, extending to each side of the point to show the range of possible values. Make a very rough estimate of the uncertainty in the torque, and draw an appropriate error bar through each point on the graph. Ask your instructor or lab aide for assistance if necessary. According to the equation τ = Iα, your graph of angular acceleration vs. net torque should be a straight line. Because of experimental uncertainties, though, you should not expect the points to lie on a perfect straight line. However, you should be able to draw a straight line that passes through, or very close to, every error bar. If you can do so, then you have confirmed the prediction that the relation between net torque and angular acceleration is linear. Draw the best-fitting straight line through the points on your graph, answer the questions in the Report, deselect the Linear option from the Fit menu, and Delete ALL Data Runs to prepare for the next activity. Part 3: Collisions In this final part of the experiment, you ll drop several non-rotating objects onto a rotating disk, and test whether angular momentum and mechanical energy are conserved in these collisions. In the photogate timing program, bring the graph window to the front and turn off the Regression Line and Statistics options. (In this part of the experiment the angular velocity is not expected to change linearly.) Get ready to restart the timing. Lift the removable disk an inch above the fixed disk, then give the fixed disk a good spin and start the timing program. After several turns (at least five), drop the removable disk onto the spinning disk and continue timing for at least five more turns. Print a copy of the graph for each person, then graphically determine the angular velocity immediately before and immediately after the collision. (Please mark the points on the graph that you use for these determinations.) Compute the initial and final angular momenta to see whether angular momentum was conserved. Compute the initial and final kinetic energies to see whether mechanical energy was conserved. Answer the questions in the Report. Repeat the experiment, instead dropping the rectangular block onto the fixed disk. Repeat again, dropping the rod-with-weights on the fixed disk. Label the three graphs according to the type of object that was dropped.

6 Report: Rotational Dynamics Rotational Dynamics 6 Name Partners Lab Station Date Part 1. Calculating Moments of Inertia Moment of inertia of weights connected by rod = Explanation and calculations: Moment of inertia of removable disk = Explanation and calculations:

7 Rotational Dynamics 7 Moment of inertia of two combined disks = Explanation and calculations: Moment of inertia of rectangular block = Explanation and calculations: Moment of inertia of combined disks = Part 2. Torque and Angular Acceleration Angular acceleration due to friction = (Be sure to attach graph showing how you determined α.) Frictional torque = Diameter of metal spindle = Radius = Diameter of small dowel = Radius = Diameter of large dowel = Radius =

8 Rotational Dynamics 8 Table Torque and Angular Acceleration (You may either use this table or set up a facsimile on a spreadsheet.) spindle applied frictional net angular force radius torque torque torque acceleration (N) (m) (N-m) (N-m) (N-m) (rad/s 2 ) Question: With what accuracy do you think you were able to keep the average force equal to the desired value? State your answer as a percentage, and explain briefly. (For example, if instead of a 3 N force you think it might have been as little as 2.5 N, the uncertainty would be 17%.) Attach a graph of angular acceleration vs. net torque, including uncertainty bars drawn according to your answer to the previous question. You may either draw the graph by hand or produce it with the computer. If you do the latter, print out a copy for each person. Question: Taking uncertainties into account, does your data support the prediction that the relation between net torque and angular acceleration is linear? Explain briefly.

9 Rotational Dynamics 9 Measured slope from graph = Predicted slope (from the relation τ = Iα) = Question: Given the uncertainties in this experiment, are the measured and predicted slopes in agreement? Explain briefly. Part 3. Collisions and Conservation Laws (Be sure to include printouts of all three graphs, clearly labeled, showing how you determined the initial and final angular velocities. One set of graphs per lab group is sufficient.) a. Uniform disk. Initial angular velocity = Final angular velocity = Initial moment of inertia = Final moment of inertia = Initial angular momentum = Final angular momentum = Percentage change in angular momentum = (change divided by initial value 100) Based on your data, was angular momentum conserved in this collision? (Explain fully, taking experimental uncertainties into account.) Would you expect angular momentum to be conserved in this collision? Why or why not?

10 Rotational Dynamics 10 Initial kinetic energy = Final kinetic energy = Percentage change in kinetic energy = Based on your data, was kinetic energy conserved in this collision? (Explain fully, taking experimental uncertainties into account.) Would you expect kinetic energy to be conserved in this collision? Why or why not? b. Rectangular block. Initial angular velocity = Final angular velocity = Initial moment of inertia = Final moment of inertia = Initial angular momentum = Final angular momentum = Percentage change in angular momentum = Initial kinetic energy = Final kinetic energy = Percentage change in kinetic energy = c. Weights connected by rod. Initial angular velocity = Final angular velocity = Initial moment of inertia = Final moment of inertia = Initial angular momentum = Final angular momentum = Percentage change in angular momentum = Initial kinetic energy = Final kinetic energy = Percentage change in kinetic energy =