Lab 9 Rotational Dynamics

Lab 9. Rotational Dynamics Goals • To calculate the moment of inertia of two metal cylindrical masses from their measured dimensions and their distance from the axis of rotation. • To use the principle of conservation of mechanical energy (the sum of kinetic and potential energies) to derive an expression for the moment of inertia in terms of other measurable quantities for a system involving both translational and rotational motion. • To determine the moment of inertia of two brass masses from the motion of the rotating system using the Rotary Motion Sensor to determine the angular speed of rotation and using appropriate graphical techniques for analyzing the data. • To compare quantitatively the values of the moments of inertia determined by these two very different methods. Are they equal within the limits of expected uncertainties? Introduction In rotational motion the moment of inertia, I, plays a role similar to that of mass in translational (or linear) motion but with some important differences. If a point mass m is located a perpendicular distance r away from its axis of rotation, its moment of inertia is just mr2 . If more than one point mass is present, then the total moment of inertia is just the sum of the individual moments of inertia. If there are just two masses, labeled A and B, the total moment of inertia is: ITotal = IA + IB (9.1) If there are a large number N of point masses, labeled i = 1, 2, ...N, rotating about a common axis, and each mass m j is located a perpendicular distance rJ from their common axis, the total moment of inertial is: N ITotal = N ∑ Ij = j=1 ∑ m j ri2 (9.2) j=1 For an extended object like a solid sphere, the moment of inertia can be computed by dividing the object into small (infinitesimal) masses and summing the moments of inertia of each mass. Since 88 CHAPTER 9. ROTATIONAL DYNAMICS 89 the moment of inertia of an object depends on the distances r j , changing the axis of rotation usually changes the moment of inertia. The apparatus consists of a Rotary Motion Sensor with a bar attached. Additional cylindrical masses can be attached to the bar. As the rod rotates, the masses move in a circle around the axis of the apparatus. The moment of inertia (for a single mass) corresponding to this motion is md 2 , where d is the distance from the center of the mass to the axis of rotation. If this were a point mass, this would be its entire moment of inertia. However, the cylinder is extended and rotates about its own center-of-mass. This rotation increases the torque required to rotate the apparatus. The Parallel Axis Theorem states that the total moment of inertia of an object located a distance d from the axis of rotation and rotating rigidly around it is: ITotal = md 2 + ICoM (9.3) where ICoM is the moment of inertia of the object (here, a cylinder) about an axis through its center-of-mass (CoM) and parallel to the axis of rotation (here, the axis of the Rotational Dynamics Apparatus). The torque required to accelerate the Rotational Dynamics Sensor is provided by a string with a small mass attached to the end and wrapped around a pulley. Initially the system is at rest with the mass, mh , suspended by the string some distance above the floor. When the mass is released, it speeds up as it falls to the floor, and the Rotary Motion Sensor spins correspondingly faster. The goal is to determine the moment of inertia of the two cylindrical masses that clamp on the bar attached to the sensor (ICyl = IA + IB ). This will be done in two ways: (1) calculating the moment of inertia using a geometric expression for moment of inertia, and (2) by analyzing its motion due to the forces and torques acting on the system. The nylon screws are easily broken if overtightened or if the masses are dropped. Please use appropriate care when you handle the masses. Calculating the moment of inertia from the system geometry The moment of inertia for an object depends on the mass distribution of that object, and so shape as well as composition must be considered for any object being calculated. In our case, we have a uniform density (all one material) thick walled cylindrical tube, as shown in Figure 9.1. Equation 9.4 show the formula for such an object: r12 + r22 h2 + ICoM = m 4 12   (9.4) Choose the axis of rotation perpendicular to the length of the cylinder—called the transverse axis. Use this formula to calculate ICoM for the brass masses. Ignore the error introduced by including the nylon screw as part of the mass. When you weigh the masses, be sure to include the mass of the screw as well. Then assume that each mass is simply a solid cylinder with a hole through the center along the axis of the cylinder. Use the calipers to measure the necessary dimensions of the CHAPTER 9. ROTATIONAL DYNAMICS 90 Figure 9.1. Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m. Image from wikimedia. cylinder. You can assume that the dimensions of both cylinders are the same, but the masses must be measured separately. Choose the value of d (0.12 m ≤ d ≤ 0.18 m) (from Equation 9.3) that you will use in the experiment, then calculate the moment of inertia of the brass masses from the system geometry using the Parallel Axis Theorem. Use a ruler to measure the distance d and electronic balances to measure the masses m of each cylinder. Remember that all of the distance/size measurements you make have some inherent uncertainties. You will need to use these uncertainties later to find the overall uncertainty in your calculated value of ICyl . Determine the appropriate uncertainties for each measurement and record them for future reference. Relating the moment of inertia to the system dynamics Find an expression for the moment of inertia, INoCyl , of the rotary motion sensor itself, the attached pulley, and the bar (without the cylindrical masses) by setting the initial potential energy of the hanging mass, mh g, equal to the final kinetic energy of the system just at the instant before the hanging mass hits the floor. Remember that the linear speed of the string, v, is related to the angular speed of the pulley, ω (in radians per second), by the relation v = ωR where R is the radius of the pulley. Show that your expression is equivalent to: " INoCyl = mh R2 2gh v2NoCyl ! # −1 (9.5) where vNoCyl is the speed of the hanging mass just before it strikes the floor with no cylindrical masses on the rod, and h is the distance traveled by the hanging mass from its point of release to the floor. Derive a similar expression for the case when the two cylindrical masses are clamped on the rod. We will denote the moment of inertia of the two cylindrical masses by themselves as ICyl . CHAPTER 9. ROTATIONAL DYNAMICS 91 " INoCyl + ICyl = mh R2 2gh ! v2w/Cyl # −1 (9.6) where vw/Cyl is the speed of the hanging mass just before it strikes the floor with both cylindrical masses clamped to the rod. Subtract Equation 9.5 from Equation 9.6 to find an expression for ICyl alone. This expression can be further simplified by using the relationship between the linear speed of the hanging mass and the angular speed of the system (v = ωR). Your expression should reduce to ICyl = 2ghmh R2 1 v2w/Cyl − 1 v2NoCyl ! = 2ghmh 1 2 ωw/Cyl − 1 2 ωNoCyl ! (9.7) where ωw/Cyl is the angular velocity of the Rotary Motion Sensor with the cylinders attached just before the hanging mass hits the ground. Likewise, ωNoCyl is the angular velocity of the Rotary Motion Sensor without the cylinders just before the hanging mass hits the ground. The subtraction in Equation 9.7 is valid only when the value of the hanging mass, mh , is the same in both expressions. Equation 9.7 allows us to calculate the moment of inertia on the basis of system dynamics and the concept of energy. Measuring the final angular velocities To obtain reliable values for the maximum angular speeds it is necessary that time between angular speed measurements be small. When you set up the Rotary Motion Sensor in the “Hardware Setup” tool, navigate to the “Properties” menu and increase the sensor resolution from “Low” to “High: 1440 counts per revolution.” Before acquiring data, verify that the sample rate is 20 Hz in the tool bar along the bottom of the Display Area. Use Capstone to measure the values of both ωNoCyl and ωw/Cyl for each hanging mass value from 5 to 30 g in 5 g increments. Since it is difficult to attach the cylinders precisely at distance d from Sensor’s axis of rotation, the best results are obtained if you do this in two steps: work through all the hanging mass values with the cylinders attached, then remove the cylinders and work through the hanging mass values again. Make sure that the hanging mass actually falls from rest at precisely the same height for each run. If you are careless here, the values for the angular speeds at the bottom will not be consistent. Check them for consistency by taking multiple measurements at each mass. Use the mean values for your calculations. The Rotary Motion Sensor is sensitive enough that the paper clip attached to the end of the string makes a measurable difference in the torque. Therefore you should account for the mass of the paper clip itself in the value of mh you use in your calculations. For instance, the mh value with a 10 g mass on the string will actually be 10 g plus one paper clip mass. Extra paper clips identical to the one on your string are available near the electronic balances in the back of the lab. Average the mass of several paper clips to reduce the round-off error of the balances. CHAPTER 9. ROTATIONAL DYNAMICS 92 Calculate ICyl for each hanging mass value, mh in an Excel spreadsheet. Are your computed values the same, assuming reasonable uncertainties, or do they show a trend as the mass value changes? Any trend points to some systematic error that may be fixable. Look carefully at your results. Finishing the calculations in this paragraph before proceeding to the next. Every bearing has some friction associated with it, so that a sufficiently small mass on the end of the string will not produce any acceleration. If the friction due to the bearings is constant, the torque applied by friction will equal the torque applied by some small, constant value of hanging mass. For instance, if the frictional force is equivalent to 1 g on the end of the string, and if I put 10 g on the end of the string, only 9 g is “left over” to cause rotational acceleration. In the Newton’s Second Law laboratory, we analyzed the acceleration versus force data to determine a “frictional mass equivalent” that reflected the magnitude of the frictional force. A similar trick is useful here. Solve Equation 9.7 for mh . Plot whatever is necessary to get ICyl as a slope of a linear graph with mh on the y-axis and whatever else remains on the x-axis. The intercept should give the “frictional mass equivalent” and the slope should give an unadulterated (without the effects of friction) value for ICyl . Uncertainties in both the slope and the intercept should be determined and recorded. The “frictional mass equivalent” is the amount of hanging mass required just to overcome the friction of the pulley and Rotary Motion Sensor bearings. Since the mass of the paperclip alone is sufficient to overcome the force of friction on the Sensor, it is especially important to include the paperclip mass in the values of mh used in making this graph. Summary Summarize your findings. Compare the moment of inertia value you calculated on the basis of system geometry with the value determined from your measurements of angular velocity. How big is the difference relative to the uncertainty of the difference? If the two values differ by more than three times the expected uncertainty, carefully examine your calculations and procedures for possible sources of error. Figure 9.2. Setup guidance for Rotational Apparatus CHAPTER 9. ROTATIONAL DYNAMICS SL.A.b Is able to identify the hypothesis for the experiment proposed Labs: Is able to determine hypothesis validity Labs: Is able to explain steps taken to minimize uncertainties and demonstrate understanding through performance where able. Expectation Scientific No deliberately identified hypothesis is present in the first half page or so of notes An attempt is made to state a hypothesis, but no clearly defined dependent and independent variable, or lacking a statement of relationship between the two variables A statement is made as a hypothesis, it contains a dependent and independent variable along with a statement of relationship between the two variables. This statement appears to be testable, but there are some minor omissions or vague details. The hypothesis is clearly stated and the direct link to the experiment at hand is apparent to any reasonably informed reader. No deliberately identified attempt to use experimental results to validate hypothesis is present in the sections following data collection. A statement about the hypothesis validity is made, but it is not consistent with the data analysis completed in the experiment A statement about the hypothesis validity is made which is consistent with the data analysis completed in the experiment. Assumptions which informed the hypothesis and assumptions not validated during experimentation are not taken into account. A statement about the hypothesis validity is made which is consistent with the data analysis and all assumptions are taken into account. No explicitly identified attempt to minimize uncertainties and no attempt to describe how to minimize uncertainties present No explicitly identified attempt to minimize uncertainties is present, but there is a description of how to minimize experimental uncertainty. An attempt is made and explicitly identified for minimizing uncertainty in the final lab results, but the method is not the most effective. The uncertainties are minimized in an effective way. No sketches present and no descriptive text to explain what was observed in experiment Sketch or descriptive text is present to inform reader what was observed in the experiment, but there is no attempt to explain what details of the experiment are not accurately delivered through either representation. Sketch and descriptive text are both present. The sketch and description supplement one another to attempt to make up for the failures of each to convey all observations from the experiment. There are minor inconsistencies between the two representations and the known reality of the experiment from the week, but no major details are absent. Sketch and description address the shortcomings of one another to convey an accurate and detailed record of experimental observations adequate to permit a reader to place all data in context. 3, 4, 7-9, 11 CT.A.a Is able to compare recorded information and sketches with reality of experiment Labs: Progressing 3, 4, 6, 7, 9, 12 SL.B.c Labs: No Effort 3, 4, 6, 7, 9, 12 SL.A.c 94 2-7, 9, 12 CHAPTER 9. ROTATIONAL DYNAMICS CT.A.b Is able to identify assumptions used to make predictions Labs: No Effort Progressing Expectation Scientific No attempt is made to identify any assumptions necessary for making predictions An attempt is made to identify assumptions, but the assumptions stated are irrelevant to the specific predicted values or apply to the broader hypothesis instead of the specific prediction Relevant assumptions are identified regarding the specific predictions, but are not properly evaluated for significance in making the prediction. Sufficient assumptions are correctly identified, and are noted to indicate significance to the prediction that is made. Multiple experimental trials lack predictions specific to those individual trial runs. Predictions made are too general and could be taken to apply to more than one trial run. OR Predictions are made without connection to the hypothesis identified for the experiment. OR Predictions are made in a manner inconsistent with the hypothesis being tested. OR Prediction is unrelated to the context of the experiment. Predictions follow from hypothesis, but are flawed because relevant experimental assumptions are not considered and/or prediction is incomplete or somewhat inconsistent with hypothesis or experiment. A prediction is made for each trial set in the experiment which follows from the hypothesis but is hyper-specific to the individual trial runs. The prediction accurately describes the expected outcome of the experiment and incorporates relevant assumptions. descworst An attempt is made to identify experimental uncertainties, but many sources of uncertainty are not addressed, described vaguely, or incorrect. Most experimental uncertainties are correctly identified. But there is no distinction between random and experimental uncertainty. All experimental uncertainties are correctly identified. There is a distinction between experimental uncertainty and random uncertainty. No equations are presented in algebraic form with known values isolated on the right and unknown values on the left. Some equations are recorded in algebraic form, but not all equations needed for the experiment. All the required equations for the experiment are written in algebraic form with unknown values on the left and known values on the right. Some algebraic manipulation is not recorded, but most is. All equations required for the experiment are presented in standard form and full steps are shown to derive final form with unknown values on the left and known values on the right. Substitutions are made to place all unknown values in terms of measured values and constants. No attempt is made to analyze the data. An attempt is made to analyze the data, but it is either seriously flawed, or inappropriate. The analysis is appropriate for the data gathered, but contains minor errors or omissions The analysis is appropriate, complete, and correct. 3, 4, 6, 7, 9, 12 CT.A.c Is able to make predictions for each trial during experiment Labs: 3, 4, 6, 7, 9, 12 CT.A.d "Is able to identify sources of uncertainty " Labs: 3, 4, 7-9, 11 No attempt is made to identify experimental uncertainties. QR.A Is able to perform algebraic steps in mathematical work. Labs: 2-4, 6-12 QR.C Is able to analyze data appropriately Labs: 95 1-12 CHAPTER 9. ROTATIONAL DYNAMICS No Effort IL.A Is able to record data and observations from the experiment Labs: Is able to construct a force diagram Labs: Progressing Expectation Scientific "Some data required for the lab is not present at all, or cannot be found easily due to poor organization of notes. " "Data recorded contains errors such as labeling quantities incorrectly, mixing up initial and final states, units are not mentioned, etc. " Most of the data is recorded, but not all of it. For example measurements are recorded as numbers without units. Or data is not assigned an identifying variable for ease of reference. All necessary data has been recorded throughout the the lab and recorded in a comprehensible way. Initial and final states are identified correctly. Units are indicated throughout the recording of data. All quantities are identified with standard variable identification and identifying subscripts where needed. No force diagrams are present. Force diagrams are constructed, but not in all appropriate cases. OR force diagrams are missing labels, have incorrectly sized vectors, have vectors in the wrong direction, or have missing or extra vectors. Force diagram contains no errors in vectors, but lacks a key feature such as labels of forces with two subscripts, vectors are not drawn from the center of mass, or axes are missing. The force diagram contains no errors, and each force is labelled so that it is clearly understood what each force represents. Vectors are scaled precisely and drawn from the center of mass. 1-12 IL.B 1-12 96 CHAPTER 9. ROTATIONAL DYNAMICS 97 Print this page. Tear in half. Each lab partner should submit their half along with the lab report and then retain until the end of semester when returned with evaluations indicated by TA. Lab 9 Rotational Dynamics: Name:     Lab Partner: EXIT TICKET: Quit Capstone and any other software you have been using. Restore Rotary Aparatus to position indicated in Figure 9.2, with the string wound around a single axis, and masses positioned so the arm will not rotate freely. Straighten up your lab station. Put all equipment where it was at start of lab. Required Level of Effort.  Complete the pre-lab assignment  Arrive on time  Work well with your partner  Complete the lab or run out of time SL.A.b SL.A.c SL.B.c CT.A.a CT.A.b CT.A.c CT.A.d QR.A QR.C IL.A IL.B Lab 9 Rotational Dynamics: Lab Partner: Name:     EXIT TICKET: Quit Capstone and any other software you have been using. Restore Rotary Aparatus to position indicated in Figure 9.2, with the string wound around a single axis, and masses positioned so the arm will not rotate freely. Straighten up your lab station. Put all equipment where it was at start of lab. Required Level of Effort.  Complete the pre-lab assignment  Arrive on time  Work well with your partner  Complete the lab or run out of time SL.A.b SL.A.c SL.B.c CT.A.a CT.A.b CT.A.c CT.A.d QR.A QR.C IL.A IL.B