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 Mass holder Assorted masses (1g ‐ 100g) Computer Meter Stick Vernier caliper INTRODUCTION: The amount of torque(τ) applied to the axle of the rotational 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). 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) 2
Equation 1 can be rearranged as: mgr = (I + mr
) α + τf 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: mgr = Iα + τ f Equation 3 ROTATIONAL DYNAMICS PROCEDURE: 1. Double click on the Physics Lab folder and select Rotational Dynamics icon. 2. Rotate the disc so that the string is wrapped around the axle as shown in figure 1. Add 25 grams to the 5 gram mass holder. Start
3. button and then give the rim of the disc a small push to start the rotation. The computer Click on the will stop taking data after a certain number of points. 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 on the data table. 4. Add 50g (record this value on the data table), rewind the string, click on Start
and release the mass from Stop
rest. Click after 10 rotations. Check the angular velocity in the table to see if the results are reasonable by selecting page 2 from the drop down menu on the top right of the page. 5. 6. 7. 8. 9. 10. 11. 12. If your data is reasonable then go to the <View> menu and select “Auto Scale Once.” 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. Next, go up to the <Analyze > menu and select “Linear Fit.” The slope of each graph will be your angular acceleration α. Record this value on the data table. Go to the <View> menu, select Graph options and make sure the following are selected “Point Protector Every” and “Graph Title”. Label the graph “Velocity vs. Time”. Save this data on a flash drive and record the name you assigned to the saved file on your data sheet. To do this select Save As from the <File> menu. For example, “50 grams added”. DO NOT PRINT THESE GRAPHS
Repeat steps 4 through 9 for at least 4 additional times with attached masses in increments of 50 grams. Record on the data table the measured mass of the disc with the axle. Use the most precise tool available to measure (1) the diameter of the axle, (2) the diameter of the disc. Record this information in a data table. (Check out the Vernier Caliper from the instructor. Return it when you are finished.) ROTATIONAL DYNAMICS LAB SHEET 1 DATA TABLE: INCLUDE UNITS
m α Mass used to overcome friction Mass used to overcome friction
Mass of Disk Diameter of Disk Diameter of Axle CALCULATIONS: SHOW ALL WORK RECORD ALL ANSWERS ON RESULTS TABLE 1. Calculate mgr (the torque applied by the weight force) for each trial. 2. Use Graphical Analysis to make a graph of mgr vs α and perform a linear fit. Print this graph. Include values from the linear fit box on the results table. These values are the experimental values of the moment of inertia (I) and the frictional torque (τf). 3. Calculate the actual moment of inertia of disk (treating it like one solid disk) using I = 1 MR 2 . Refer to figure 2. 2
4. Compare the actual and the experimental values of I and (τf) using the percent error formula. RESULTS TABLE: INCLUDE UNITS mgr Experimental I Experimental α CONCLUSION: TURN IN: Rotational Dynamics Lab Sheet 1, Graph of mgr vs α Actual I Actual α Percent Error