Rotational Inertia Physics 231 At the end of the lab experiment please clean your table and wait for the instructor to check you out! All the group partners must be present. Thank you. Equipment: You will use the Pasco rotary motion sensor (similar to the picture below and you will measure the angle of a rotating axle as a function of time. Note: the rotary sensor used in this lab has a white plastic disk instead of black. In addition to the rotary sensor, you will need: universal table clamp super pulley on clamp set of small weights (blue box) silver disk thin black rod with long screw long steel rod 3-tier pulley with string 2 weights for on thin black rod ruler 850-UI Introduction: In this lab, you will investigate Rotational Inertia and its dependence on mass and the distribution of that mass. You will make use of the Pasco rotational sensor. Background: Just as inertia is the tendency of an object to continue motion in a straight line (or to continue in a state of rest if it is already at rest), rotational inertia (or moment of inertia) is the tendency of an object to continue a rotational motion (or to continue in a state of non-rotation if it is already not rotating). The smaller the rotational inertia, the easier it is to get the object rotating and the easier it is to stop it from rotating. The larger the rotational inertia, the harder it is to get the object rotating and the harder it is to get it to stop rotating once it has begun. Additionally, as it takes the application of a force to change the state of linear motion of an object, it takes the application of a torque to change the state of rotational motion of an object. A significant difference between linear and rotational inertia, however, arises in their dependencies on the mass of an object. Linear inertia depends only on the object's mass, while rotational inertia depends not only on the object's mass, but also on the distribution of that mass. Two objects in rotation of the same mass may have different rotational inertias if the mass is distributed differently. In general, the closer the mass is to the axis of rotation, the smaller the object's rotational inertia. As the mass becomes distributed farther away from the axis of rotation, the rotational inertia will increase.Angular velocity w is related to angle as W = 40/At(6) and that the angular acceleration a is a = Aw/At(7) Setup: 1. Secure the universal table clamp to the edge of the table. Be careful you do not overtighten the screw, it need only be tight enough to hold it in place. 2. Insert the steel rod into the rod holder on the universal table clamp and let the end of the rod rest on the floor. 3. Look at your rotary motion sensor and in the bag - find the screw that belongs with the sensor (it should be screwed into the axle of the sensor.) If it's in the axle of the sensor remove it from the axle and lay it in the bag for later. 4. Take the 3-tier pulley, wind the string around it if it hasn't already been done. Look inside the center hold and find the raised line. 5. Line up the raised line in the center hole of the 3-tier pulley with the groove running down the axle of the rotary motion sensor and slide the 3-tier pulley onto the axle with the small, square end facing up. This should be a snug fit, but not need to be forced onto the axle. ."If the pulley won't go onto the axle without forcing it, ask your professor for help!* * 6. Slide the rod holder on the end of the rotary motion sensor onto the top end of the rod, and secure the sensor so the rod is at/very slightly above the black plastic of the rod holder, as shown in the picture.7. Look at the silver disk and find the recessed square in the center of one side. Place the silver disk on the 3-tier pulley so that recessed square sits snugly on the raised square at the top of the 3-tier pulley. Secure it with the screw that you located in step 3. 8. Attach the super pulley onto the front end of the rotary sensor (the end with the silvery cord), with the screw underneath the end of the sensor, as shown in the picture. O 9. Route the string from the 3-tier pulley over the super pulley, using the groove on the wheel. You may need to slightly angle the pulley clamp where it's attached to the rotary motion sensor to keep the string in the groove. 10. The string on the 3-tier pulley should have a small loop in the end of it, hook one of the weight hangers from the weight set in the loop. 11. Check the positioning of your setup and make sure the string and weights will clear the edge of the table when the weight falls. 12. Raise the rod in the rod holder so the pulley is at a height where the weight and hanger will not hit the floor. 13. Remove the 850-UI from the box and set it up like you have in previous labs. *Remember: look at all plugs before you plug them in and make sure they are turned the right direction.* *14. Plug the rotary motion sensor into the PasPort1 spot. **Remember: make sure the plug is turned the right direction before you plug it in."* 15. Turn on the 850-UI and open the Capstone software. 16. Select table & graph for your display. 17. Add a second graph and set them as angle versus time and angular speed versus time. 18. Set the frequency for the rotary motion sensor (at the bottom of the Capstone screen) to 10Hz. 19. You should now be ready to conduct the experiments. Procedures: Silver disk 1. Release the 10 gram weight and immediately click on RECORD in Pasco. STOP recording before the weight hits the floor. 2. Repeat the experiment twice more. You will need to reset the string in place after the weights hit the floor. 3. Create graphs: angle versus time, and angular speed versus time in Capstone. Weighted rod 1. Remove the silver disk from the rotary motionsensor. 2. Remove the 3-tier pulley from the rotary motionsensor, turn it to the other side, and then replace it. ""Remember: line up the raised line in the center hole on the pulley with the groove in the axle. "*There will be two raised grooves on this side of the pulley that the rod will fit into to hold it securely onto the disk. 3. Attach the rod without weights to the rotational sensor, using the longer screw found in the rod and collect the data the same way as in case of silver disk. 4. Reset the string and attach the weights to the rod, and move them out to the ends of the rod. Repeat the experiment. Measuring components 1. Find the mass (in kg) of the silver disk, the thin black rod and the individual weights on the thin black rod. 2. Measure the length of the thin black rod (in meters). 3. Measure the diameter of the silver disk (in meters).Analysis: 1. Based on the measurements you made, calculate the rotational inertias for the silver disk, thin black rod with the weights at either end, and thin black rod without weights. 2. Calculate the net torque on the system t = RF. R is the radius of the pulley from which the string unwraps, and F is the tension in the string. 3. Using the calculated net torque and rotational inertias, calculate the angular accelerations for each of the three experiments. 4. Plot the graphs of angle vs, time and angular velocity vs. time for each of your three experiments. 5. Using a smooth section of data on the angular velocity vs. time graphs, plot a best fit line for each experiment and find the slope of the line. This is the experimental value of angular acceleration. 6. Calculate the percent difference between your calculated angular acceleration and what you found from the best fit line for each of the three experiments. Conclusions and Discussion: 1. Discuss the results of your experiment. Did the calculated and measured values for angular acceleration match up well? 2. In this experiment, we assumed no friction in the pulleys or in the rotary motion sensor. Were these assumptions valid? 3. In this experiment, we assumed no rotational inertia in the pulleys or in the rotary motion sensor. Were these assumptions valid? 4. What are possible sources of error in the experiment? (No "human error", please!)Rotational Inertia - Data Sheet Physics 231 Name: Date: Basic Measurements 1. Mass of Silver disk (kg) Diameter of Silver disk (m) 2. Mass of Thin rod (without weights) (kg) 3. Mass of each weight on Thin Rod (kg) 4. Length of Thin Rod (m) 5. Calculated net Torque on system (N.m)_ Calculated Rotational Inertias 1. Silver disk (kg.m") 2. Thin rod (weights at both ends) (kg.m?) 3. Thin rod (no weights) (kg.m?) Calculated Angular Accelerations 1. Silver disk (rads/sec*] 2. Thin rod (weights at both ends) (rads/sec?) 3. Thin rod (no weights) (rads/sec?)_ Measured Angular Accelerations 1. Silver disk (rads/sec ) 2. Thin rod (weights at both ends) (rads/sec ) 3. Thin rod (no weights) (rads/sec) % Differences between Calculated and Measured Angular Accelerations 1. Silver disk 2. Thin rod (weights at both ends) 3. Thin rod (no weights).Rotational Inertias The rotational inertia for a point particle is given as 1 = mr (1) where I = rotational inertia m = mass r = distance from axis For a group of particles, the total rotational inertia is the sum of all the individual rotational inertias. Solid objects have rotational inertias that depend on the shape and mass. The rotational inertia of a solid disk rotating about its central axis is given by 1 = > mr (2) where m = mass r = radius of the disk Interestingly, this is also the same rotational inertia for a solid cylinder rotating about its central axis. The rotational inertia does not then depend on the height of the cylinder. The rotational inertia of a thin rod about its center (perpendicular to the length of the rod) is 1/12 ml (3) where m = mass of rod L = length of the rod For rotating objects, we replace the standard linear variables for position, velocity and acceleration (x, v, and a) with their rotational counterparts of angle, angular velocity and angular acceleration (0, w, and a). Recall that the rotational form of Newton's Second Law is: [= la(4) where t = torque I = rotational inertia a = angular acceleration From the other side, torque T = RF(5) where R is the distance between the line of force, F and axis of rotation