The Center of Mass of a Homogeneous L Bracket

Transcription

1 L07 Rotational Motion and the Moment of Inertia 1 Lab Section: L07 prelab 1 NAME The Center of Mass of a Homogeneous L Bracket Consider a homogeneous L bracket of mass (M) and symmetric length L. The bracket can be chopped into two segments, one of mass 4m with r 1 as a position vector to its cm and the other of mass 3m with r 2 as its position vector to its cm point. Find the center of mass vector R cm of the entire bracket using the origin O which is at the center of the middle square. Find numerical values for fractions f and f in the expression: R cm = f L i + f L j. From your results, draw the center of mass point on the drawing.

2 L07 Rotational Motion and the Moment of Inertia 2 L07 prelab 2 The Moment of Inertia About the C.M. Point of a Homogeneous L Bracket Find an expression for the moment of inertia about the center of mass point of a homogeneous L bracket of total mass, M and symmetric length L. As you did in the previous calculation, break up the bracket into two pieces, one of mass 4m with its cm at pt 1 and find its moment of inertia I 1cm at pt 1. Then use the Parallel Axis Theorem to find this segment s moment of inertia at point 3, the cm point of the whole bracket, I 1@3. In your calculation you will need c 1, the distance between points 1 and 3. You can find it by using your knowledge of the vector addition of r 1 + c 1 = R cm. Do the same for the other segment and then add the two moments I 1@3 and I 2@3. At the end of your calculation, express the moment of inertia of the entire bracket about the center of mass in terms of M and L only, i.e. I cm = f ML 2 where f is some fraction. Show all work here. m 1 = 4m, I 1 cm = ( 1/12 )( 4m )[( L 2 +( L/4 ) 2 ] = ( 1/3 ) m ( 17/16 ) L 2 = ( 17/48 ) ( M/7 )L 2 = M L 2 since r 1 + c 1 = R cm, c 1 = R cm r 1 = (3/14) L i +( 3/14 ) L j -( 3/8) L j = ( 3/14 ) L i - ( 9/56 ) j c 1 2 = [ (3/14 ) L ] 2 + [ ( -9/56 ) L ] 2 = L 2 I 3 = I 1cm + ( 4m ) c 2 1 = M L ( M/7 ) ( L 2 ) = M L 2

3 L07 Rotational Motion and the Moment of Inertia 3 Name: Lab Section Number: Pre-Lab Exercises (the time you spend doing these will decrease the time you spend in lab this week): 1. Using the 1 st worksheet, calculate the center of mass point of a uniform L bracket of 6 inch length and 1.5 inch width. Express your answer as R cm = f L i + f L j 2. Using the 2 nd worksheet calculate the moment of inertia of the bracket at its center of mass point. Express your answer as I cm bracket = f ML 2 3. Consider an aluminum disk (density 2.7 g/cm 3 ) which is free to rotate about an axis as shown above. It is attached thru a rotary motion sensor to a small shaft and a pulley. The rotary motion sensor can measure the shaft s rotation as a function of time. A light cord is wrapped around the pulley and is attached to a hanging hook weight. The hook weight which is held at rest and then is let go, moves downward with a constant acceleration.

4 L07 Rotational Motion and the Moment of Inertia 4 Introduction: This experiment deals with some of the characteristics of rotational motion. The normal linear quantities, distance (x), velocity (v), acceleration (a) and force (F) have rotational equivalents that refer to motion around an axis, rather than along a line. The angular quantities for the linear ones just mentioned are angular displacement (θ), angular velocity (ω), angular acceleration (α) and torque (τ). You will be using the relations between these linear and angular variables, as well as the angular equivalent of Newton s Second Law (τ = I α) throughout this experiment. In doing so, you will determine the moment of inertia (which is represented by the letter I) of an L bracket both experimentally and by theoretical calculation. You will also determine the moment of inertial of the pulley + shaft. Note: Calculations for this lab may be done by hand, as they are the purpose of part of the experiment. Procedure: Materials Pasco Rotary Motion Sensor with pulley Aluminum Disc 50 g hanger and 50 g mass.. 6 L bracket Meter stick Balance, Calipers 1. Setting up the computer: 1.1 Connect the Science Workshop interface to the computer, turn on the interface and then turn on the computer. (this may have already been done for you) 1.2 Open the Data Studio software package by double clicking on the icon from the desktop menu. Select Create Experiment. 1.3 In the Experiment Setup screen double-click the Rotary Motion Sensor picture. This will add it to the interface box diagram, showing the proper connections. Connect the leads as indicated. It will also yield sensor properties. 1.4 On the tab: General: set the sample rate to 500 Hz (fast) On the tab: Measurement: Click on Angular velocity rad/s On the tab: Rotary Motion Sensor: on Div/rotation, click Close the experiment setup window. 1.6 Under the display window on the lower left, select the type of display you would like. For this experiment we will use a graph of angular velocity verses time data. 2. Setting up the lab equipment: 2.1 The aluminum disk and rotary motion system should be assembled for you when you enter the lab. Wrap a length of cord on the pulley and connect the other end to the mass hanger.

5 L07 Rotational Motion and the Moment of Inertia 5 3. Theory Calculations: 3.1 Weigh the mass of the rotating aluminum disk. Confirm your measurement by calculating the mass of the disk from its volume and its density (aluminum 2.7 g/cm 3 ). You must show the actual calculation, not just an answer! 3.2 Measure the mass of the L bracket. You ll need it to calculate your value of the theoretical moment of inertia that you found in your second worksheet. 3.3 a) Draw all the forces acting on the extended free body diagram below for the disk shaft - pulley system and on the hook weight mass point M H. b) From the FBDs above, write Newton s 2 nd Law for the hook weight (eq 1) and one for the system (eq 2). c) Express the linear acceleration of the hook weight in your eq 1 equation in terms of α, the angular acceleration of the disk shaft pulley system.

6 L07 Rotational Motion and the Moment of Inertia 6 d) Add eq 1 and eq 2 equations together, eliminating the tension T, and then solve for the moment of inertia, I of the system. Your equation for I should only be a function of the hanging mass, M H, the radius of the pulley, R P, the angular acceleration and g. 4. Taking data: Record the mass of the hanger and the mass provided. (Remember to use the mass in kilograms for all your calculations.) Place the empty hook weight on the string hanging off the pulley, ensure that it is stationary and start collecting data for a second before letting the mass fall. Stop collecting data sometime after about fifteen seconds. The first portion of your graph, indicate the initial descent of the mass. Why does the mass move this way after it is released? 4.4 Select the sloping straight line on the angular velocity vs. time graph and perform a linear fit on it using the fit/linear command in the menu immediately above the graph. Note all the fit parameters, especially the slope parameter. Include them in your lab report along with proper units. 4.5 The slope represents the angular acceleration of the disk. Label it α Also collect motion data for the hanger loaded with one addition mass of 50g. As this is your second run, label this second angular acceleration α 2. Repeat all the steps in section 4 with the L bracket fastened to the aluminum disk so that the bracket s center of mass lies on the axis of rotation as shown below. With runs 3 and 4, label the angular acceleration: α 3, and α 4.

7 L07 Rotational Motion and the Moment of Inertia 7 5. Data analysis: 5.1 Print the angular velocity vs. time graphs each with the best straight line fit parameters. 5.2 Calculate the moment of inertia from your equation in 3 d) for each of your 4 runs and fill in the table below. (Show your complete calculation for one case, and the answers for the remaining three.) 5.3 From your 1 st two runs, find an average moment of inertia of the system + disk. 5.4 From your 2nd two runs, find and average moment of inertia of the system + disk + L bracket. 5.5 From 5.3 and 5.4, find an experimental I cm of the L bracket 5.6 How does your theoretical moment of inertia of the L bracket compare to your experimental value? (Use a percent difference calculation to answer this question, and show the calculation. Use the theoretical moment of inertia as the accepted value. 5.7 What physical reasons might there be for the difference between your theoretical calculation for the moment of inertia and your experimental result? 5.8 From your calculation of the moment of inertia of the aluminum disk, describe how you can calculate the percentage of the moment of inertia of the system disk that is just the system (pulley + axle)? What is this percentage?