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ROTATIONAL DYNAMICS Name: ____________________ Pre-Lab Questions Page Roster Number: _____________ Class:_____________________ Instructor:__________________ 1. Write the symbolic representation and one possible unit for angular velocity, angular acceleration, torque and rotational inertia. angular velocity = _____ _____ angular acceleration =_____ ______ torque = ______ ______ rotational inertia = ______ _______ 3. Describe a situation in which ω < 0 and ω and α are antiparallel. 4. In the figure 1, a force whose magnitude is 55.0N is applied to a door. However, the lever arms (moment arms) are different in the three parts of the drawing: (a) d = 0.80 m, (b) d = 0.060 m and (c) d = 0. Find the magnitude of the torque in each case. THIS PAGE LEFT BLANK.......................... Name:_______________________ Date:________________________ Lab Partners:__________________ __________________ ________________________________________________ ROTATIONAL DYNAMICS OBJECTIVE: To determine the rotational inertia (I) of a disc and cylindrical axle by measuring its angular acceleration and applying the rotational form of Newton's laws of motion to the rotating disc. This value is compared to the geometrically computed value for a disc. (Optional: compared to computed value for disc and cylindrical axle) APPARATUS: Rotational inertia apparatus 1-meter string 50g mass holder Assorted masses (1g - 100g) PC and ULI timer Meter Stick Vernier caliper Photogate Flag ULI INTRODUCTION: The amount of torque(τ) applied to the axle of the rotational Figure 1 dynamics apparatus is varied by adding additional mass to the string attached to that axle. The resulting angular accelerations (α) are measured by timing the flag as it passes through the photo gate. From the plots of angular velocity(ω) vs time the angular accelerations are found. The values of the applied torque are computed from τ = rF sin θ = Fd and a graph of applied torque vs angular acceleration is prepared. From this graph an experimental dynamic value of the rotational inertia and the frictional torque are determined. In the last equation, d, is known as the lever arm(moment arm). PROCEDURE: 1. Check to be sure the apparatus is connected as shown in Figure 1. 2. Make sure the computer is on. Double click on the “Physics Lab” folder and select <Rotational Dynamics> icon. 3. Rotate the disc so that the string is wrapped around the axle as shown in figure 1. Hang the 50g mass holder from the end of the string. 4. 5. Click on the Start button, then give the rim of the disc a small push to start Stop the rotation. After 10 rotations click on the button. If the angular velocity was nearly constant the mass attached provides a torque just sufficient to overcome the friction in the axle. Observe the angular speed column of the table. If the angular velocity increases, lower the amount of mass attached to the string and repeat the timing run. If the angular velocity decreases, add additional mass and repeat the timing run. Repeat this step until the mass required to overcome friction is determined to within 1g and record this value. Start Add 50g, rewind the string, click on and release the mass from rest . Stop Click after 10 rotations. Check the angular velocity in the table to see if the results are reasonable. 6. If your data is reasonable then go to the <View> menu and select “Auto Scale Once.” 7. The computer can find the best straight line fit for a set of data points but it calls it a “linear fit” line. Do this by clicking and holding on the first data point and then drag to the last data points. Let go and the box will remain there. 8. Next, go up to the <Analyze > menu and select “Linear Fit.” 9. Go to the <View> menu, select Graph options and make sure the following are selected “Point Protector Every”, “Graph Title” and “Grid.” At the bottom of the page in the box below “graph title” type “Velocity vs. Time”. Click “ok” 10. Save this data on your diskette and record the name you assigned to the saved file on your data sheet. To do this select Save As from the <File> menu. Save the file as a MBL file. For example, “50 grams added.mbl” 6. Repeat steps 5 through 10 for at least 4 additional times with attached masses in increments of 50 grams. 7. Note and record the measured mass of the disc with the axle. This value is marked on the apparatus along with its implied uncertainty. 8. Use the most precise tool available to measure (1) the diameter of the axle, (2) the distance that the axle protrudes on each side of the disc, (3) the thickness of the disc and (4) the diameter of the disc. Record these values with their uncertainties. Record this information in a data chart. (Check out the Vernier Caliper from the instructor. Return it when you are finished.) CALCULATIONS: Combine Newton's second law for rotation applied to the disc and Newton's second law for translation applied to the attached mass to derive Equation 1. Refer to figure 2. 2 Iα = mgr − mr α − τ f Equation 1 Where m is the suspended mass, r is the radius of the axle, g is the acceleration due to gravity,α is the angular acceleration of the disc and τf is the frictional torque. (Refer to figure 2) Equation 1 can be rearranged as: mgr = (I + mr ) α + τf 2 Equation 2 Because applied masses and axle radius are so small using this apparatus, mr2 is negligibly small compared to ( I ) this equation can be written as: Fig. 2 mgr = Iα + τ f Equation 3 M 1. Print out your graphs from your experiment using the computers in SM252 by selecting Logger Pro from the“Physics Lab” folder. Open your first graph of data. r R FT 2. Print graphs of your angular velocity in rad/s vs time as FT usual for each trial. Be sure to include the regression and have the vertical intercept shown on the graph. Your name should be printed in the header or footer. To do this Go to the <File> menu and select “printing options” then FW = mg type your name and roster#. In the comment section type the amount of mass that was suspended(be sure to include the mass holder). Next, click “page setup” and select “landscape”. 3. From these graphs determine the angular acceleration for each trial. 4. Compute mgr (the torque applied by the weight force) for each trial. Be sure to calculate mgr for the case with no angular acceleration when the torque was applied only to overcome the friction in the axle. 5. Use Graphical Analysis to make a graph of mgr vs angular acceleration. 6. From the regression box of the graph from step 5 determine dynamic values for the moment of inertia (I) and the frictional torque ( τ f ). Hint: the equation for a linear situation is y = mx + b and also see equation 3. 7. Compute a geometric value for the moment of inertia of the rotating system according to: (a) Use the mass (total of large disc and axle) marked on the apparatus and the equation for the moment of inertia of a disc (I = 1 MR 2 ). 2 (b)* Compute the moment of inertia of the large disc and axle by adding the separate moments of inertia (Itotal = Idisc + Iaxle ). *First find the volume of the large disc and the axle separately using the formula 2 for the volume of a cylinder ( V = πr h ), where h is the thickness and r is the radius. Find the combined volume. Because the density is uniform and the same for both the disc and its axle the density is: ρ= M total M disc M axle = = Vaxle Vdisc Vtotal to find the mass of the disc alone: ⎛M ⎞ Mdisc = ⎜ total ⎟ Vdisc ⎝ Vtotal ⎠ to find the mass of the axle alone: ⎛M ⎞ Maxle = ⎜ total ⎟ Vaxle ⎝ Vtotal ⎠ Treat each as a cylinder and find the combined moment of inertia: Itotal = Idisc + Iaxle 8. Compare the dynamic values of I and τ f , from the graph, to each computed(geometric) value. A E x100 Recall: %error = , where E = experimental value and A = calculated A value. QUESTIONS: 1. Was your experimental value for rotational moment of inertia (I) greater than or less than the calculated value? Is this what you expected? What affect did friction play, if any? 2. Was your experimental value for the frictional torque (τf) greater than or less than the calculated value? Is this what you expected? 3. If torque is the product of the moment of inertia and rotational acceleration, calculate the torque supplied by a motor that rotates a circular blade from rest to an angular rad speed of 660 in 306 rev. Suppose that the moment of inertia for the blade is 1.61 x s 10-3 kg m2.