This preview shows page 1. Sign up to view the full content.
Unformatted text preview: UC Berkeley, Fall 2010, GSI: Aaron Alpert
Question 1. The rope shown in the ﬁgure is wound ¯ around a cylinder of mass 4 kg and I = .02 kg · m2 , about the cylinder’s axis. If the cylinder rolls without slipping, what is the linear acceleration of the center of mass? What is the frictional force? [Hint: Use an axis of rotation through the CM.] Question 2. Sarah is standing motionless on a turntable, holding a bicycle wheel horizontally directly over her head, as shown in the ﬁgure. The wheel’s angular velocity is 6 rad/s in the +k direction. Suddenly, Sarah twists the wheel so that now, the wheel’s CM is still directly over her head, but the wheel is vertical. In what direction, and at what speed, does Sarah rotate? Treat the bicycle wheel as a ring with mass 10 kg and radius .3 m. Treat Sarah as a cylinder with mass 60 kg and radius .25 m. For a ring Iz = mR2 and Ix = 1 mR2 , where Iz is the moment of inertia about the central axis, 2 and Ix is the moment of inertia about a vertical axis. For a cylinder, Iz = 1 mR2 . 2 Physics 7A: Quiz 5 UC Berkeley, Fall 2010, GSI: Aaron Alpert
Question 1. The rope shown in the ﬁgure is wound ¯ around a cylinder of mass 4 kg and I = .02 kg · m2 , about the cylinder’s axis. If the cylinder rolls without slipping, what is the linear acceleration of the center of mass? What is the frictional force? [Hint: Use an axis of rotation through the CM.] Question 2. Sarah is standing motionless on a turntable, holding a bicycle wheel horizontally directly over her head, as shown in the ﬁgure. The wheel’s angular velocity is 6 rad/s in the +k direction. Suddenly, Sarah twists the wheel so that now, the wheel’s CM is still directly over her head, but the wheel is vertical. In what direction, and at what speed, does Sarah rotate? Treat the bicycle wheel as a ring with mass 10 kg and radius .3 m. Treat Sarah as a cylinder with mass 60 kg and radius .25 m. For a ring Iz = mR2 and Ix = 1 mR2 , where Iz is the moment of inertia about the central axis, 2 and Ix is the moment of inertia about a vertical axis. For a cylinder, Iz = 1 mR2 . 2 Physics 7A: Quiz 5 Physics 7A: Quiz 5, Solutions
UC Berkeley, Fall 2010, GSI: Aaron Alpert
Question 1. We have three unknowns: the CM acceleration a, the angular acceleration α, and ¯ the frictional force f . We have two dynamics equations: F = ma and τ = I α. The third equation comes from the noslip condition, which tells us a = Rα. We solve the three equations ¯ simultaneously. Note in the equations below I decided that “to the left” and “counterclockwise” would be positive, and I arbitrarily guess that friction is to the left. F = 20 N + f = (4 kg )¯ a (1) τ = (20 N )(.1 m) − f (.1 m) = (.02 kg · m2 )α (2) a = (.1 m)α ¯ (3) I substitute (3) into (2). I multiply (2) times 10 and add it to (1), which gives me: 20 + 20 = (4 + 2)¯ a → a = 6.67 m/s2 ¯ (4) Now, I just plug in the known acceleration into (1) and solve for f . 20 + f = 4(6.67) Question 2. The initial angular momentum is
2 L0 = I ω = mw Rw (6k) = (10 kg )(.3 m)2 (6k) = 5.4k → f = 6.67 N (5) (6) Angular momentum in the k direction must be preserved because there are no external torques about k. However, afterwards, both Sarah and the wheel have the same k component of rotation, so we simply add the moments of inertia about the vertical axis. 1 1 2 2 Lf,z = L0,z = 5.4 = If ωf,z = (Iwheel + ISarah )ωf,z = mw Rw + mS RS ωf,z 2 2 1 1 = · 10 · (.3)2 + · 60 · (.25)2 ωf,z = 2.325ωf,z (7) 2 2 Solving gives us ωf,z = 2.232 rad/s in the +k direction (i.e., counterclockwise).
Note: The indicated number of points show how this problem might have been graded on an actual exam, to indicate how prepared you are for the midterm. However, as per the syllabus, your score for this quiz was based on a holisitic (easier) scale.
These notes ©2010 by Aaron Alpert. 2 ...
View
Full
Document
This note was uploaded on 05/04/2011 for the course PHYSICS 7A taught by Professor Lanzara during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Lanzara
 Physics, Acceleration, Mass

Click to edit the document details