1. Chapters 10 & 11 – Rotational motion, torque, and angular momentum
PHY 113 C General Physics I
11 AM - 12:15 PM MWF Olin 101
Plan for Lecture 12:
Chapters 10 & 11 – Rotational motion, torque, and angular momentum
Torque
Angular momentum
Problems 1.1,1.6,1.10,1.11
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PHY 113 C Fall Lecture 12

2. PHY 113 C Fall Lecture 12
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3. angular “displacement”  q(t) angular “velocity” 
Angular motion
angular “displacement”  q(t)
angular “velocity” 
angular “acceleration” 
“natural” unit == 1 radian
Relation to linear variables: sq = r (qf-qi)
vq = r w
aq = r a s
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PHY 113 C Fall Lecture 12

4. angular “displacement”  q(t) angular “velocity” 
Rotational motion
q(t)
w(t)
angular “displacement”  q(t)
angular “velocity” 
angular “acceleration” 
Special case of constant angular acceleration: a = a0:
w(t) = wi + a0 t
q(t) = qi + wi t + ½ a0 t2
( w(t))2 = wi2 + 2 a0 (q(t) - qi )
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PHY 113 C Fall Lecture 12

5. Webassign Assignment #10:
A centrifuge in a medical laboratory rotates at an angular speed of 3800 rev/min. When switched off, it rotates 48.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.
Special case of constant angular acceleration: a = a0:
w(t) = wi + a0 t
q(t) = qi + wi t + ½ a0 t2
( w(t))2 = wi2 + 2 a0 (q(t) - qi )
Recall:
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PHY 113 C Fall Lecture 12

6. Review of rotational energy associated with a rigid body
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PHY 113 C Fall Lecture 12

7. Which moment of inertia is the smallest? (A) i (B) j (C) k
Note that for a given center of rotation, any solid object has 3 moments of inertia; some times two or more can be equal j d d m m i k
iclicker exercise:
Which moment of inertia is the smallest?
(A) i (B) j (C) k
IB=2md2
IC=2md2
IA=0
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PHY 113 C Fall Lecture 12

8. From Webassign:
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9. Digression – use of rotational energy in energy storage
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10. Beacon Power company (went bankrupt in 2011) Continuing efforts –
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11. Physics of rolling -- CM CM 10/08/2013
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12. Rolling motion reconsidered:
Kinetic energy associated with rotation:
Distance to axis of rotation
Rolling:
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PHY 113 C Fall Lecture 12

13. Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first? C B A
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PHY 113 C Fall Lecture 12

14. How can you make objects rotate?q
How can you make objects rotate?
Define torque:
t = r x F
t = rF sin q r
F sin q q F
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PHY 113 C Fall Lecture 12

15. Another example of torque:
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17. X Example form Webassign #11 iclicker exercise t3
When the pivot point is O, which torque is zero?
A. t1?
B. t2?
C. t3? t3 X t2 t1
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PHY 113 C Fall Lecture 12

18. Newton’s second law applied to center-of-mass motion
Newton’s second law applied to rotational motion ri mi di Fi
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PHY 113 C Fall Lecture 12

19. In this case I = ½ m R2 and t = FR
An example:
A horizontal 800 N merry-go-round is a solid disc of radius 1.50 m and is started from rest by a constant horizontal force of 50 N applied tangentially to the cylinder. Find the kinetic energy of solid cylinder after 3 s. R F
K = ½ I w t = I a w = wi + at = at
In this case I = ½ m R and t = FR
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PHY 113 C Fall Lecture 12

20. Re-examination of “Atwood’s” machine
T1-m1g = m1a
T2-m2g = -m2a
t =T2R – T1R = I a = I a/R R I T2 T1 T1 T2
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PHY 113 C Fall Lecture 12

21. Another example: Two masses connect by a frictionless pulley having moment of inertia I and radius R, are initially separated by h=3m. What is the velocity v=v2= -v1 when the masses are at the same height? m1=2kg; m2=1kg; I=1kg m2 ; R=0.2m . h m1 m2 v1 v2
h/2
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22. Example from Webassign #10Tl
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23. Note that rolling motion is caused by the torque of friction:
Newton’s law for torque: F fs
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PHY 113 C Fall Lecture 12

24. Bicycle or automobile wheel:fs
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PHY 113 C Fall Lecture 12

25. What happens when the bicycle skids? Too much torque is applied
iclicker exercise:
What happens when the bicycle skids?
Too much torque is applied
Too little torque is applied
The coefficient of kinetic friction is too small
The coefficient of static friction is too small
More than one of these
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26. Vector cross product; right hand rule
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27. For unit vectors: k j i
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28. More details of vector cross products:
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29. What is the point of vector products To terrify physics students
iclicker exercise:
What is the point of vector products
To terrify physics students
To exercise your right hand
To define an axial vector
To keep track of the direction of rotation
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PHY 113 C Fall Lecture 12

30. From Newton’s second law:
iclicker exercise:
Is this
Wrong?
Approximately right?
Exactly right?
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PHY 113 C Fall Lecture 12

31. From Newton’s second law – continued – conservation of angular momentum:
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32. Torque and angular momentum Define angular momentum:
For composite object: L = Iw
Newton’s law for torque:
In the absence of a net torque on a system,
angular momentum is conserved.
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PHY 113 C Fall Lecture 12

33. The student will remain at rest.
iclicker exercise:
A student sits on a rotatable stool holding a spinning bicycle wheel with angular momentum Li. What happens when the wheel is inverted?
counterclockwise
The student will remain at rest.
The student will rotate counterclockwise.
The student will rotate clockwise.
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PHY 113 C Fall Lecture 12

34. More details:
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35. Other examples of conservation of angular momentumw1 w2 m m m m d1 d2 d1 d2
I2=2md22
I1=2md12
I1w1=I2w2 w2=w1 I1/I2
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PHY 113 C Fall Lecture 12

36. What about centripetal acceleration?
ar = v2/R = w2 R
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PHY 113 C Fall Lecture 12

37. Webassign problem:
A disk with moment of inertia I1 is initially rotating at angular velocity wi. A second disk having angular momentum I2, initially is not rotating, but suddenly drops and sticks to the second disk. Assuming angular moment to be conserved, what would be the final angular velocity wf?
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PHY 113 C Fall Lecture 12

38. Conserved quantity Necessary condition
Summary – conservation laws we have studied so far
Conserved quantity
Necessary condition
Linear momentum p
Fnet = 0
Angular momentum L
tnet = 0
Mechanical energy E
No dissipative forces
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PHY 113 C Fall Lecture 12