Lab05_Rotational Kinematics

Physics 211: Lab Rotational Kinematics Reading: HRW: Chapter 10, sections 1-5 Knight: Chapter 4, sections 5-7 Introduction: The translational motion of any point particle can be described in terms of standard Cartesian coordinates ( x, y, z ). For rotational motion, however, it is simpler to use Polar coordinates where the position of the particle can be specified by a radius r and the angular position,  , rather than by x and y . The radius of a particle undergoing circular motion is always a constant. The angular position  , however, will change with time depending on the motion of the particle. Since only the angular position changes with time its behavior is exactly analogous to the behavior of the position in one-dimensional motion that was studied previously. Thus, the angular equivalent of the kinematic quantities for one-dimensional motion can be defined as follows: Arc distance traveled s = r Δθ Angular Displacement Linear (tangential) velocity v t = rω Angular Velocity Linear (tangential) acceleration a t = rα Angular Acceleration The relationships between the angular position, velocity, and acceleration are exactly the same as the relationships previously determined for one-dimensional motion. For example, for motion with constant angular acceleration,  : ω = α . t + ω o θ = 1 2 α . t 2 + ω o t + θ o An object experiencing rotational motion is also experiencing an inwards radial/centripetal acceleration of magnitude: Physics 211R: Lab – Rotational Kinematics S (x, y) = (r cos  , r sin) X Y r  1

a c = v 2 r = w 2 r For circular motion that is not uniform (constant w ), the total acceleration (magnitude and direction) of the object, at every moment, is determined by adding the centripetal (radial component) and linear (tangential component) accelerations together. Note: a c and a t are vectors that form a right triangle when added together because they are perpendicular to each other. ⃗ a total =⃗ a c +⃗ a t Remember that tangential acceleration is a consequence of any change in the linear (and therefore, angular) speed of the object. Centripetal acceleration is a consequence of the continuous change in direction of the object’s linear velocity. Conceptual Understanding Goals: By the end of this laboratory, you should be able to: (1) Relate the angular position  (t), the angular velocity w (t), and the angular acceleration  (t). (2) Determine the linear acceleration – both radial and tangential – of a point on a rotating object in terms of the point’s radius, angular acceleration, and angular velocity. (3) Use Newton’s Second Law and the equation for static friction to relate the static friction coefficient to the acceleration at which the maximum static friction is exceeded in a system. Laboratory Skill Goals: By the end of this laboratory, you should be able to: (1) Use the rotary motion sensor to measure the rotational motion of an object. (2) Design an experiment to calculate the static friction coefficient between two objects. Equipment List: Computer with PASCO Capstone™ and PASCO 550 Universal Interface Rotating Platform with attached Rotary Motion Sensors Stickers located at two different radii (4 cm and 10 cm) Pulley String Hanging mass and hanger Penny Ruler Physics 211R: Lab – Rotational Kinematics Figure 1 2

Physics 211R: Lab Report Template Rotational Kinematics (Type in this document and print these pages at the end of the laboratory) NOTE: Remember to SAVE this template as a Word document before starting the lab. Always remember to save your template periodically throughout the lab.  A maximum of three students will be allowed per group without prior instructor permission.  All the members of the group must participate in the activity. If a student is not participating (even when present) s/he may receive a score of zero in the activity. Students arriving 10 minutes or more past start will not be admitted.  This activity must be returned at the end of the lab period. All the students completing the activity must be present when handing this to the laboratory instructor; a student not present at this time may not get credit for the activity. Writing the name of a person not present is not permissible and may result in a potential academic integrity violation being processed.  After you receive the graded report back, you should make a copy of the front page (this page) and keep it for you records. This will serve as evidence of your grade for this activity.  You are responsible for checking your grade (in the course website) and report any mistakes to your laboratory instructor within two weeks after the activity. Date: 10/17/23 Enter your name as it appears in your PSU registration, no nicknames please. Name: Ian Davis Section # 38R Name: Jocelyn Marciniak Section # 38R Name: Kiranmai Attaluri Section # 37R Clean Up Check: After you finish working and completing the lab report, you need to clean and organize your working area. Then call one of your laboratory instructors who will check your area, initialize below and take the lab report. All the members of the group must be present at that time. If you leave the lab before your laboratory instructor performs the check up, you will be deducted 5 points from your score for this lab report. Laboratory Instructor Initials: _______ Score: _______ Physics 211R: Lab – Rotational Kinematics 4

Lab Activity 1: Kinematics of Circular Motion – Free Rotation [20 minutes] For this first activity, you just need to spin the disc while making sure that the attached string does not get caught on anything. You want free rotation, so once you spin it up, nothing, other than some small friction at the axle that we will neglect for now, affects it). Q1. Predict what the  (t), w (t), and  (t) graphs would look like for a disc that is freely rotating in the positive direction of rotation [while CCW is most commonly used for positive, many of the rotary motion sensors use CW for positive] – draw on your printed sheet (the last pages in this file). Q2. Explain how the  (t), w (t), and  (t) graphs are related in general. Slope of  (t) gives us angular velocity Slope of w (t) gives angular acceleration Area under w (t) gives angular displacement Area under  (t) gives angular velocity Now, spin your disc as described above and do this experiment with your disc (press Start just after you spin up the disc). Q3. Check the relationships you described in Q2 – do they hold? In particular, do a linear fit to your  (t) data: How can we determine the angular velocity w from the fit parameters (m & b)? Does the value for w you get from your fit to  (t) match the value in your w (t) graph? Linear fit: f(t) = mt + b Everything discussed in Q2 stands. The slope (m) of the  (t) graph is 4.24, and the angular velocity is roughly 4.24. The angular velocity is found from the slope of the  (t) graph. Q4. A graph of a different experiment is shown below (the actual experiment ran from 0.5-3.5 sec). How could the presentation of this graph be improved? ( This is an important question since effective presentation of graphs is a valuable skill to learn – and your grade on this lab will depend on how well you present your graphs .) The graph should be shortened to only include the experiment time and the scale of the y axis. The line of best fit should also be added. Q5. Attach your graph showing  (t) and w (t), with your linear fit to  on it. Be sure to take the time to crop and present your graph properly (unlike the graph in Q4 above). Physics 211R: Lab – Rotational Kinematics 5

Lab Activity 2: Kinematics of Circular Motion – Torqued Disc [20 minutes] Now we’re going to let something change the rotational speed of the disc. Carefully wind the string around the base of the turntable. Place the string over the pulley and attach the hanging mass (use 100 grams or 150 grams) to the other end of it as shown in the picture above (Figure 1 at the equipment list). Q6. Predict what the  (t), w (t), and  (t) graphs should look like for a disc that starts from rest but is being pulled by the string so that it begins rotating in the negative direction of rotation. Draw on your printed sheet (the last pages in the file). Now do the experiment! Q7. What function (constant value, linear, or quadratic) best fits the w (t) graph? If not constant, do the appropriate fit in Capstone and use the fit values to determine  . Explain what fit parameter you used ( m/b for linear, A/B/C for quadratic) to find  and state the value you obtained. Is  approximately constant or is it varying? The linear function best fits the angular velocity graph. The slope of the angular velocity is the angular acceleration. The slope is 0.608 and the angular acceleration is 0.614. We used the slope (m) of the angular velocity graph to find angular acceleration. It is a constant value. Q8. What function (constant value, linear, or quadratic) best fits the  (t) graph? Do the appropriate fit in Capstone and use the fit values to determine  . Explain what fit parameter you used to find  and state the value you obtained. How does it compare to your value in the previous question? How does it compare to your value in your  graph? The quadratic function best fits the  (t) graph. We used the A term in the quadratic function with 2 derivatives to find the angular acceleration. The value we got was 0.606, which is off by .002 to the slope of the angular velocity and 0.008 off by the constant angular acceleration value. Q9. Paste your  (t), w (t) and  (t) graphs below with the fits you did shown on the graphs. Be sure to crop your graph appropriately (remember the rules for well-presented graphs). Physics 211R: Lab – Rotational Kinematics Linear fit: f(t) = mt + b Quadratic fit: f(t) = At 2 + Bt + C 7

Q10. A point on the disc experiences both radial (inwards) a c and tangential a t acceleration. Which component of the acceleration is more important (larger) when: (a) the disc just begins rotating? The tangential acceleration is more because at the beginning, the angular velocity is zero, so the radial acceleration is also zero. (b) the disc is rotating its fastest in your experiment? The magnitude of the radial and tangential acceleration are the same when the disc is rotating its fastest because the radial acceleration is proportional to the square of the angular acceleration and the radial acceleration is proportional to the angular velocity itself. Justify your reasoning! Q11. Describe how you could obtain the w (t) graph shown below using your experimental apparatus. You could obtain this graph by having the disc rotating at a decreasing speed, stopping, and rotating in the opposite direction. Check with an instructor at this point! Lab Activity 3: An a- m s -ing Experiment [30 minutes] Physics 211R: Lab – Rotational Kinematics 8

Physics 211R: Lab Report Template Rotational Kinematics PRINT THESE PAGES (p. 1-3) WHEN YOU BEGIN AND INCLUDE IN YOUR REPORT Q1. Predict what the  (t), w (t), and  (t) graphs would look like for a disc that is freely rotating in the positive direction of rotation [generally CCW, but may be CW for your disc] – draw on your printed sheet . Q6. Predict what the  (t), w (t), and  (t) graphs should look like for a disc that starts from rest but is being pulled by the string so that it begins rotating in the negative direction of rotation. Draw on your printed sheet . Q12. On the diagram below, draw arrows showing the direction of a c , a t , and a = a c + a t . (Assume both components of acceleration are approximately equal.) Q13. Above, draw all forces in the plane of the page just before it starts sliding. To the right of the diagram, write Newton’s Second Law for the object. How can you relate the acceleration a = | a| at the instant the penny slips to the coefficient of static friction m s ? Physics 211R: Lab – Rotational Kinematics  t w t  t  t w t  t w 10

Q14. With the penny at radius R o from the center of the disc (but not on a sticker), release the disc starting from rest. Using your hand, stop the turntable at the very moment the penny slips from the surface . (Of course, collect data in Capstone during this time.) Since there is some error in this process, do this four times and average your values for w &  at the instant the penny slips. Quantity Run 1 Run 2 Run 3 Run 4 Average w (rad/s) 9.338 9.861 9.036 9.604 9.459  (rad/s 2 ) 0.496 0.437 0.501 .509 .486 Q15. In the table on your printed sheet , calculate the acceleration at the moment of slippage and use that to find the coefficient of static friction m s . Quantity Result Explanation of how Result was obtained and/or show calculation R o = Radius (meters) 0.045 This radius was measured using a ruler. a t = Tangential Accel. (m/s 2 ) .022 R  a c = Centripetal Accel. (m/s 2 ) 4.026 R w 2 a total = Total Acceleration (m/s 2 ) 4.032 a t 2 + a c 2 = a total 2 m s = Coefficient of Static Friction 0.411 a/g Which component of acceleration (radial/centripetal or tangential) is more important in this experiment? Could you reasonably neglect one of them without introducing much error? Centripetal acceleration is more important in this experiment. You could neglect tangential acceleration and only have a 2% difference in the final acceleration vector. Q16. Record your calculations for making your prediction for w below. Quantity Result Explanation of how Result was obtained and/or show calculation R i = Radius (meters) 0.045 Measured using a ruler. a total = Total Acceleration (m/s 2 ) 4.032 a t 2 + a c 2 = a total 2 a t = Tangential Accel. (m/s 2 ) 0.022 R  a c = Centripetal Accel. (m/s 2 ) 4.026 R w 2 Physics 211R: Lab – Rotational Kinematics 11