1. HW sets 12, 13, and 14 now available
Announcements
HW sets 12, 13, and 14 now available
Mid-term grades – will be calculated from your first exam only
Today’s lecture
Rotations continued
Torques continued
Rotational energy
Conservation of angular momentum
12/5/2018
PHY Lecture 12

2. Newton’s second law applied to center-of-mass motion
Newton’s second law applied to rotational motion ri mi di Fi
12/5/2018
PHY Lecture 12

3. Kinetic energy associated with rotation:
Distance to axis of rotation
Rolling:
12/5/2018
PHY Lecture 12

4. In this case I = ½ m R2 and t = FR
Example problem:
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. F R
K = ½ I w t = I a w = wi + at = at
In this case I = ½ m R and t = FR
12/5/2018
PHY Lecture 12

5. 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 Lecture 12

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PHY Lecture 12

7. Peer instruction question
Three objects of uniform density – a solid sphere (a), a solid cylinder (b), and a hollow cylinder (c) -- are placed at the top of an incline. If they all are released from rest at the same elevation and roll without slipping, which object reaches the bottom first?
(a) solid sphere (b)solid cylinder (c)hollow cylinder
12/5/2018
PHY Lecture 12

8. Rolling without slipping
vCM w
w =vCM/R R F
t = R fs
fs =F
12/5/2018
PHY Lecture 12

9. Torque and angular momentum Define angular momentum:
For an extended object: L = Iw
Newton’s law for torque:
 If ttotal = 0 then L = constant
In this case: I1w1 = I2w2
12/5/2018
PHY Lecture 12

10. Ktot = KCM + Krot (c) (b) (a) Review Ia = 2/5 MR2 Ib = 1/2 MR2
Ic = MR2
Three objects of uniform density – a solid sphere (a), a solid cylinder (b), and a hollow cylinder (c) -- are placed at the top of an incline. If they all are released from rest at the same elevation and roll without slipping, which object reaches the bottom first?
Krot = ½ I w2  For fixed w, (c) has the largest Krot
 For fixed Krot, (a) has the largest w
Ktot = KCM + Krot
12/5/2018
PHY Lecture 12

11. If Ftrack=0, then ½ mvt2=½ mgR
-Ftrack-mg=-mvt2/R h v 2R
If Ftrack=0, then ½ mvt2=½ mgR
Without rolling: Wnet = Kf – Ki  mg(h-2R)= ½ mvt2
h=5/2R
With rolling:
12/5/2018
PHY Lecture 12

12. Newton’s law for torque:
F-fs= M aCM
Rfs=I a=I aCM/R fs
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PHY Lecture 12

13. Bicycle or automobile wheel:fs
12/5/2018
PHY Lecture 12

14. Torque and angular momentum Define angular momentum:
For composite object: L = Iw
Newton’s law for torque:
 If ttotal = 0 then L = constant
In the absence of a net torque on a system,
angular momentum is conserved.
12/5/2018
PHY Lecture 12

15. Peer instruction question
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 Lecture 12

16. More details:
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PHY Lecture 12

17. 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
12/5/2018
PHY Lecture 12

18. What about centripetal acceleration?
ar = v2/R = w2 R
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PHY Lecture 12

19. T
T-mg= - maCM
RT=Ia=IaCM /R mg
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PHY Lecture 12

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PHY Lecture 12