Rotational Dynamics, Moment of Inertia, Torque and Rotational Friction

Transcription

1 Rotational Dynamics, Moment of Inertia, Torque and Rotational Friction Junaid Alam, Imran Younas, Waqas Mahmood, Sohaib Shamim, Wasif Zia, Muhammad Sabieh Anwar LUMS School of Science and Engineering Version 2; July 20, 2014 When you push the throttle to accelerate a moving car, all you are doing is using the engine to apply more force. But why is there no acceleration after a certain speed is reached, even when the engine is still applying the force? Why do you nd it dicult to stop a rotating wheel even if there is no translational motion? Why are the brake linings of formula 1 cars made dierently from ordinary cars? Why is it easy to accelerate a bicycle having smaller wheels but hard when it has large wheels? How are the rotations of a wheel of a car translated into the linear speed of the car? Why is the lower gear capable of imparting more acceleration than the higher ones? Study of the quantities like moment of inertia, torque, angular speed and speed dependent friction may help us nd the answers to these questions. This is what we will be dealing with in this experiment. The design of very fundamental mechanical devices, such as gears, ywheels and mechanical centrifuges, requires a clear understanding of the concepts of rotational mechanics and mechanical friction that depends on angular speed. The relationship between linear and rotational motion has to be dealt with in many situations. This experiment will help grasp these concepts. KEYWORDS Rigid Body Angular Momentum Angular Velocity Angular Acceleration Moment of Inertia Torque 1 Conceptual Objectives In this experiment, we will, 1. learn to appreciate the similarities, dierences and relationship between rotational and translational motion; 2. investigate energy loss due to friction; 1

2 3. realize that there exist dierent ways of measuring a physical quantity with dierent accuracy and precision; 4. t experimentally observed curves with mathematically modeled solutions; and 5. see how errors propagate from measured to inferred quantities. 2 Experimental Objectives The experiment is divided into two sections: determining the frictional losses and measuring the moment of inertia of a circular disc. In the rst section, height loss measurements will be done by correlating the rotation of a disc with the loss in the height of a mass attached to a thread wound over a pulley on the disc. After having established the nature of the relationship between angular speed and frictional losses taking place in the system, a value of the moment of inertia of the disc will be determined. 3 Theoretical Introduction This experiment introduces you to the concepts of rotational motion. We shall touch upon a number of topics and discuss how a large complex object can be considered to be composed of a large assemblage of ideal particles. We will elaborate that a full description of a body's motion must include linear as well as rotational motion. Furthermore, we will discuss torque as it applies to our experiment. 3.1 Angular Momentum We can consider the provided circular disks (rigid bodies) to be made up of small innitesimal particles of masses m 1 ; m 2 ; m 3 ; : : : m i ; : : :. Their placement may be dened with the position vectors r 1 ; r 2 ; r 3 ; : : : r i ; : : : and when rotating, their instantaneous velocities may be dened as v 1 ; v 2 ; v 3 ; : : : v i ; : : :. The index i shows one of the many particles. Figure 1 illustrates the ith particle rotating about the z axis. The angular momentum of the particle about z axis is given by, J i = m i v i r i ; (1) where denotes the vector or cross product. For a particle rotating with an angular velocity! about z axis, we can say that, v i = r i!: (2) 2

3 z r i m v i i J= m i v i r i Figure 1: A representative particle rotating about the z-axis; m i v i is the linear momentum and J is the angular momentum. Consider a circular disk rotating about the z axis. The disk itself can be considered to be composed of with all its innitesimal elements in the xy plane. Using Equation (1) and Equation (2) we can write for the ith particle, J i = m i r 2 i!: (3) z J 1 = m 1 v 1 r 1 J 2 = m 2 v 2 r 2 R J 3 = m 3 v 3 r 3 Figure 2: Disk can be considered to comprise particles. The individual angular momentums of these particles will all add up resulting in the total angular momentum. The total angular momentum of a disk about an axis is simply the sum of all the angular momentums for the innitesimal particles, J = N m i v i r i : (4) i=1 3.2 Moment of Inertia The cross product for a disk rotating about the z axis with its components in the xy plane can be expanded as, N J z = (m i r 2 i )!: (5) i=1 3

4 Here I = m i r 2 i is a constant (irrespective of the angular velocity of the disk) and is known as the moment of inertia of the disk. Therefore Equation (5) becomes, J z = I!: (6) In the present experiment, we will investigate the rotational kinematics of a disk. It will also be helpful to know the moment of inertia for a circular disk, which is, where M is the mass and R is the radius of the disk. I = 1 2 MR2 (7) Q 1. Show that the kinetic energy of a rotating disk is given by, KE = 1 2 I!2 : (8) (HINT: Kinetic energy for a particle is given by, K = 1 2 mv 2. Sum for all the particles and use the fact, I = m i r 2 i.) Moment of inertia is analogous to inertia in linear kinematics. However, since in rotational motion, we always nd ourselves dealing with moments, we call the inertia in circular motion as moment of inertia. Moment of inertia of a particular body is dened with respect to a particular rotation axis and is dierent for a body when it is rotating about x, y or z axes. Table 1 provides a brief comparison of linear and rotational motions and their characteristics. Table 1: Comparison between Linear and Rotational Motion. Concepts and quantities Linear Motion Rotational Motion Position x Velocity v! Acceleration a = v =! t t Motion Equations x = v t =!t Newton's 2 nd Law F = ma = I Momentum p = mv J = I! Work F x 1 Kinetic Energy 2 mv I!2 Q 2. A circular disk of mass 0:2 kg and radius 0:1 m is rotating at 10 revolutions per second. Calculate the angular frequency, moment of inertia and kinetic energy of this disk. 3.3 Turning Eect Q 3. Dene torque. 4

5 Mathematically, torque ( ) is, = r F; (9) where r is the displacement between the line of action of force and the particle and F is the force applied. Expanding the cross-product we get, where is the angle between F and r. = r F sin(): (10) In a gravity driven system, we may replace F using Newton's second law and express the equation as, = mgr sin(); (11) where m is the mass used to drive the mechanism and g is the acceleration due to gravity. 3.4 Angular Acceleration Q 4. Dene angular acceleration. Mathematically, angular acceleration is given by = d! dt : (12) You may want to refer to Table 1 to become more comfortable with this seemingly new term which is just an equivalent of linear acceleration adapted for rotational motion. In other words it is the gradient of angular velocity versus time graph. Newton's second law for rotational motion states that, = I (13) where is the applied torque, is the angular acceleration and I is the moment of inertia the rotational equivalent of mass. Note its similarity to Newton's law for linear motion F = ma, establishing the moment of inertia as the analogue of mass and torque as the analogue of force. If we substitute Equation (11) and Equation (12) in Equation (13) we get, mgr sin() = I d! dt : (14) 5

6 4 Introduction to the Apparatus The apparatus (Figure 4) consists of the following components: 1. Rotational Motion Apparatus This apparatus, ME-9341 was procured from PASCO Scientic. The contents of the apparatus are in Figure 3. Base Bushing of the bearing assembly Spindle Screw Rod Leveling Support Bubble Level Main Platter Auxiliary Platter Super Pulley (SP) Holder Step Pulley Photogate Smart Timer (ST) PASCO Count: M anual SMART TIMER `Vernier `LabPro Motion Sensor Figure 3: The components of the rotational motion apparatus. Note the arrows showing the particulars of the components. 2. Smart timer, photogate and super pulley: The photogate sends a narrow beam of infra-red radiation from one arm which is detected by a detector in the opposite arm. When the beam is blocked, the LED at the back of the photogate lights and a signal is sent to the smart timer (ST ). A super pulley is placed such as its spokes block the photogate beam. As the super pulley has 10 spokes, in one revolution, it blocks the photogate beam 10times. Using these devices, we will measure the speed as well as the number of rotations of the platter. We use the smart timer in the fence mode to record the time between ten successive interruptions of the photogate. The timing begins when the photogate beam is rst blocked and stops when it has been blocked ten times. Using the Select Measurement key, user can recall the cumulative readings of the durations for 10 consecutive interruptions of the beam. 6

7 Base Main platter Step pulleys Holder super pulley 1 levelling screws Figure 4: Arrangement of the experimental set up (side view). Spindle Main platter Base Base 5 Experimental Method 5.1 Preparation NOTE: Record every measurement and the associated uncertainty in your lab notebook with suitable commentary wherever required. 1 Place the bubble level on the base. See if the bubble is in the inner ring. If not, screw or unscrew the three adjustable supports underneath, to bring the bubble in the inner most ring. 2 Find the outer diameter of the main platter (D mp ) and the supper pulley (D p ), and the inner diameter of the pulleys on the main platter using vernier callipers. 3 Find the mass of the main platter using a mass balance. Note the uncertainty in your reading and comment about your choice of a particular value for it. 4 Slide the spindle into the bushing of the bearing assembly. Then slide the main platter atop with the step pulleys facing up. 7

8 super pulley 1 Vernier LabPro and Motion Detector USB Cable to computer 5 Clamp the super pulley with a clamping screw to the base. 5.2 Frictional Losses In every real life situation, a mechanical system involves losses due to friction. In order to measure the moment of inertia of the disc accurately, we have to quantify the losses, so that a balanced energy transfer equation can be written and solved. In this section you will measure the frictional losses by relating friction to the speed of rotation of the disc. Besides nding out the qualitative dependence of frictional loss on angular speed, you will be able to establish a quantitative relationship as well. To measure frictional losses, we will be using an ultrasonic motion detector. It measures the distance of a moving object from its sensor in the units of meters. To power the motion detector and write the distance measurements from it to a computer we use a Vernier LabPro, which allows such sensors to communicate with the computer. Using the data from the motion sensor, we can measure the energy losses. In order to be able to proceed, follow the steps below. 1 Take a thread of suitable length e.g. 85 cm. Tie one end of the thread to the screw protruding from the smallest pulley on the main platter. Then pass the thread from the holes 8

9 right beside the screw. Pass it through only as many holes as suitable for the pulley that you choose for this part of the experiment. Screw To hanger Holes Fasten the free end of the thread to the hanger and put two 100 g masses in it. 2 Connect the motion detector to Vernier LabPro, using the DIG/SONIC2 port. With the USB cable of the latter, connect it to the computer. Place the motion detector on the oor, with its sensor exactly below the mass-hanger so that distance measurements are not erroneous. Also, when the data is being recorded, don't let anything come between the mass hanger and the detector, or you'll end up with wrong height measurements. 3 Position the super pulley (1) such that it is exactly in front of the axle of the disc. Pass the thread over the pulley and wind the thread on the chosen step pulley until the mass hanger is at a suitable height, e.g. 65 cm above the oor. `Vernier LabPro Motion Sensor Figure 5: Experimental arrangement for frictional loss measurement. 4 Start the Motion.vi program in LabVIEW. Release the mass-hanger from its maximum height and click on the 'Run' button in LabvIEW. The program will start to record the varia- 9

10 tions in the height of the mass-hanger. After you have recorded the data for up-down trips of the mass-hanger, STOP the program and import your data into MATLAB. Once plotted, your data should look like gure (6). Notice that the horizontal axis should have the units of time and its scale will depend on the sampling time used in the LabVIEW program Height (m) Time (s) Figure 6: Height loss of the mass-hanger with time. Using the \Marker", measure the maximum height achieved by the mass-hanger in every trip. Tabulate these measurements and calculate the dierences between subsequent maxima to nd out the height loss in every trip. 5 Using the diameter value of the step pulley, you can calculate the number of revolutions of the main platter that took place during every round trip of the mass-hanger. HINT: The distance relationship between rotational and linear motion will be useful in this matter. 6 From the plotted data calculate the instantaneous speed of the main platter every time the mass-hanger reaches its minimum height. You can do this my measuring the distance between two time instants near the minimum position of the mass-hanger. This value represents the maximum speed achieved by the main platter in the respective cycle. To calculate the average speed of the main platter during the whole cycle of the mass-hanger, the following formula is used:! av g = 0 +! max 2 =! max 2 : (15) 7 Once you have measured the height losses for every cycle and the average angular speed, you can construct a table that resembles the one in Table (2) Here, h, E, N and E R represent, respectively, the corresponding height loss, energy loss 10

11 Cycle h h N E E R! av g `n' (m) (m) (J) (J) (rad/s) Table 2: Sample data table for calculating frictional loss. ER ωavg Figure 7: Average angular speed vs. energy loss per rotation. in a respective cycle (= mgh), number of rotations of the main platter that occurred during a round trip of the mass hanger and the energy loss per rotation. 8 Plot the energy loss per rotation against the average angular speed, with! av g on the x-axis. Ideally, you should get a straight line, as shown in gure (7), that shows a linear proportion between the two quantities. Q What does the relationship between E R and! av g signify? 5.3 Determination of Moment of Inertia I As frictional losses can be calculated for a range of angular speeds, we are now in a position to measure the moment of inertia of the main platter. Using a reasonable mass in the hanger, you will drive the main platter to a certain speed and nd out the corresponding energy loss per rotation due to friction from the graph you have plotted in the previous section. Once you can write a complete energy equation, the value of the moment of inertia can be calculated. 11

12 Super pulley 2 Screw rod super pulley 2 Holder Holder super pulley 1 Photogate Smart Timer Holder 1 Attach the screw rod to the second super pulley and fasten it into the holder. Then slide the holder into one of the holes on the sides of the base and use a rubber band to keep the rim of the super pulley touching that of the main platter. 2 Place the photogate such that the spokes of the horizontal super pulley block its beam. Check the connections of the photogate (P 1 ) with the smart timer (ST ). 3 Check if ST beeps when you switch it ON using the power switch on the side. Press 1 to select the quantity to be measured and then 2 to choose the measurement mode. Try taking a reading in the GATE timing mode. 4 Give a push to the main platter and press 3 from ST. 5 Explain the value that has been returned by ST. 6 What is the uncertainty in your measurement? 7 Unfasten the thread from the step pulleys and the mass-hanger. Take a new thread of a suitable length, e.g. 120 cm approximately, and attach it to the mass-hanger. Load a mass of 150 g in the hanger. Wind the free end of the thread around the pulley that you have been using. Make sure that the vertical super pulley is in line with the thread. 8 Measure the height of the mass-hanger using a meter rule. Try to make a relevant judgment about the uncertainty, which can be greater than the least count of the meter rule in this case. 9 Set the smart timer to measure time in the ONE GATE mode. 12

13 Photogate Smart Timer Holder 10 Release the mass and let it drive the main platter. As soon as the mass hits the ground, press the START button on the smart timer. Using this measurement, determine the instantaneous and average angular speed of the main platter during the period when the mass-hanger was descending towards the ground. 11 To determine the energy losses, refer to the graph you plotted in the previous section and nd out the energy loss per rotation that corresponds to now measured value of average angular speed. 12 From the height of release of the mass-hanger and the circumference of the step pulley used, calculate the number of revolutions of the main platter that occurred during the time of the mass-hanger's descent. Total frictional loss will be the product of energy loss per rotation and the number of rotations of the main platter. 13 Write down an energy equation for the experiment that includes the energy lost and retained by the mass-hanger as well as the frictional losses. 14 Solve the equation to nd out a value of the moment of inertia. You are expected to give a reasonable estimate of uncertainty in the calculated value. 15 Discuss the main sources of error in your measurement. 6 Experience Questions 1. Why do we feel vertigo after a spin on a merry-go-round? 2. Does the inertia of our body increase in a swimming pool? 3. If mass m is right on top of center of gravity, does that mean there is no value of I for that mass? 4. In an accident, the body may be stopped by the seat belt and air bags. Where does all the momentum go? Can this cause bodily injury? 5. Why do cats always fall feet rst on the ground? 13

14 7 Idea Experiments 1. Find your body's moment of inertia. 2. Use dierent masses to check if the relationship between the frictional losses and energy delivered is linear. 3. Drop the auxiliary platter on the main platter when rotating to verify the conservation of momentum. 14