1. Conservation of Momentum
A system is a pair or group of objects that are interacting in some way.
If a system has no external forces acting on it (eg no friction), then the system is isolated.
Total momentum in an isolated system is said to be conserved.

2. Angular Momentum (L) unit: kgm2s-1
Any rotating object has angular momentum…

3. Angular Momentum Linear momentum DL = t Dt DP = FDt Conservation of L
Any rotating object has angular momentum…
Angular Momentum
Linear momentum
DL = t Dt
DP = FDt
Conservation of L
ΣLbefore = ΣLafter
(if no net outside torques)
Conservation of P
ΣPbefore = ΣPafter
(if no net outside forces)
Compare…
Eg “friction torques”
Eg “friction forces”

4. What is the angular momentum of a 20 g, 11
What is the angular momentum of a 20 g, 11.8 cm compact disc spinning at 500 rpm? (use I = ½mr2)
m = kg
r = m
ω = 500 rev/min = 52.4 rad/s
L = Iω = (1/2)mr2ω
L = (1/2) (0.059)2 · 52.4
L = kg·m2s-1

5. Which has more angular momentum if both objects have the same rotational velocity?
A. B. C. The same D. Can’t be determined

6. Identical objects are spinning at the same speed, but in opposite directions. Which measurements are the same for both objects?
Angular Momentum
Rotational Kinetic Energy
Both A & B
Neither A nor B

7. Applications of

8. What could you calculate using this L vs t graph to determine the Net Torque?
Slope of the line
Area between the line and the x-axis
Y-intercept
None of these

9. A merry-go-round initially spinning clockwise traveling experiences a positive applied torque from the uncle as shown in the graph. Clockwise torques are considered are positive. While traveling the full 0.9 sec, the merry-go-round’s rotational speed:
Torque
Time (seconds)
1000 mN
0 mN
first increases and then decreases.
first decreases and then increases.
continuously increases.
drops to zero at 0.9 sec.
None of the above.

10. A merry-go-round initially spinning clockwise traveling experiences a positive applied torque from the uncle as shown in the graph. Clockwise torques are considered are positive. While traveling the full 0.9 sec, what is the merry-go-round’s change in angular momentum?
Torque
Time (seconds)
1000 mN
0 mN
0 kg(m2)/sec
10 kg(m2)/sec
300 kg(m2)/sec
450 kg(m2)/sec
900 kg(m2)/sec

12. Conservation of Momentum
See merry go round Applet

17. Two disks—A and B—on frictionless axles are initially at rest
Two disks—A and B—on frictionless axles are initially at rest. Disk A has twice the moment of inertia as Disk B. Now you exert twice as much Torque on Disk A as you do on Disk B, both for 1 second. One second later, which measurements of Disk A and Disk B are the same?
Change in Angular Momentum
Angular Acceleration
Change in Angular Velocity
A & B
B & C
A, B & C 2τ A τ B 17

18. A constant torque is exerted on a disk that is initially at rest on a frictionless axle. The torque acts for a short time interval and gives the disk a final speed. To reach the same speed using a torque that is half as big, the torque must be exerted for a time interval that is
Four times as long.
Twice as long.
The same length.
Half as long.
A quarter as long. τ 18

19. Two disks—A and B—on frictionless axles are initially at rest
Two disks—A and B—on frictionless axles are initially at rest. Disk A has twice the moment of inertia as Disk B. Now you exert the same constant torque on both disks for 1 second. One second later, the momentum of disk A is:
Twice the angular momentum of Disk B
The same as the angular momentum of Disk B
Half the angular momentum of Disk B
Not enough information to determine τ A τ B 19

20. Two identical disks, A and B, initially are spinning on frictionless axles. The initial rotational velocity of Disk A is twice as that of Disk B. You then exert the same constant torque on the two disks over 1 second. One second later, the change in angular momentum of Disk A is: 2ω
Non-zero and twice the change in angular momentum of Disk B
Non-zero and the same as the change in angular momentum of Disk B
Zero.
Non-zero and half the change in angular momentum of Disk B
Not enough information to determine τ A ω τ B 20

23. See rotation_joe_ball

24. A child (the blue dot) with is running at a constant velocity v
A child (the blue dot) with is running at a constant velocity v. Which of these line segments would be used to calculate the child’s angular momentum about point P?
D. A child running in a straight line can’t have angular momentum because they are not going in a circular path. P A B C

25. A child (the blue dot) with is running at a constant velocity v
A child (the blue dot) with is running at a constant velocity v. What is the direction of the child’s angular momentum about point P?
Clockwise
Counter-Clockwise
Can’t be determined P

26. Into Out of the the page page Left Right rod
The non-spinning axle of a rapidly spinning wheel is attached to a non-spinning rod supported by a pole on one end. The direction of spin is such that a point on the wheel is coming at you when it is at the bottom of the wheel and moving away from you when it is at the top. What is the direction of the wheel’s angular momentum? (Use the right-hand rule)
Into Out of the the page page
Left Right
rod
wheel
pole
Into the page
Out of the page
Up (on the page)
Down (on the page)
Left (on the page)
Right (on the page)

27. Into Out of the the page page Left Right rod
The force of gravity on the wheel plus rod is downward. What is the direction of the resulting torque on the wheel-plus-rod system about the pointed support? (The wheel is spinning very rapidly.)
Into Out of the the page page
Left Right
rod
wheel mg
pole
Into the page.
Out of the page
Up (on the page)
Down (on the page)
Left (on the page)
Right (on the page)

28. Because of the gravitational force/torque, the wheel-plus-rod system begins to move. Viewed from above, what does it do?
rod
wheel mg
Left Right
rod
wheel mg
pole
Fall down
Rotate clockwise
Rotate counterclockwise
Not enough information to determine

29. Conservation of Momentum problems
with worked solutions…

30. Linear Motion can provide Angular
Momentum, example:
Bubba and Betty-Sue go to the play-ground.  Bubba runs to the spinny-ride and jumps on at the edge.  The ride is at rest.  Bubba's mass is 81kg and he runs at 3.7m/s.  The spinning ride has a diameter of 3.3m.
(a)  Calculate the angular momentum of the ride just after Bubba jumps on.

31. Next to the playground is an iced-over pond
Next to the playground is an iced-over pond.  Betty-Sue has brought her ice-skates and goes out and starts to spin while staying in the same place.  In this orientation her rotational inertia is 0.064kgm2, her mass is 59kg and she is spinning at 39rpm.
(b) Calculate her angular momentum.
(c) She then brings her arms inwards (towards her body). 
Explain what happens and why.

32. (c) She then brings her arms inwards (towards her body). 
Explain what happens and why.
As she brings her arms towards her chest she will speed up.
 Since more or her mass as near the centre of the spin, she has decreased her rotational inertia.
Since the angular momentum is conserved, assuming no external torque from friction, the L = Iω with arms out must equal L = Iω with arms in.
(d) Her new rotational inertia is 0.014kgm2. Calculate her new angular velocity

33. (d) Her new rotational inertia is 0. 014kgm2
(d) Her new rotational inertia is 0.014kgm2. Calculate her new angular velocity.

35. Bubba has a model airplane
Bubba has a model airplane.  Each set of propellers has a rotational inertia of kgm2 and are spinning at ~1800 rpm.
(e) Explain to Bubba why each propeller spins opposite directions.
If there was only one propeller the body of the airplane would try to rotate in the opposite direction as the propeller because of the conservation of angular momentum, L.
Each propeller spins in opposite circles so that they cancel out their angular momentum, L, and the body of the airplane does not have any torque and does not try to spin.
This cancelling out of each propellers angular momentum assumes each propeller has identical rotational inertia, I and angular velocity, ω.

36. (f) Calculate the total angular momentum of both propellers if the left propeller spins at 1848rpm clockwise while the right propeller spins at 1735rpm anticlockwise?

37. (g) If it takes 1.45s for the propellers to reach a speed of 1840rpm starting from rest, calculate the angular displacement and the angular acceleration of a single propeller.

38. (h) If each blade on the propeller is 12
(h) If each blade on the propeller is 12.5cm long, calculate the linear velocity of the tip of each blade while at this 1840rpm.