1. Warm-up Write a formula for Newton’s second law.
Identify and describe the quantities in the formula.

2. Torque and Angular Momentum
Vectors of rotation

3. Objectives Determine the torque applied by a force about an axis.
Predict an object’s response to a torque.
Calculate an object’s angular momentum.

4. Question You push on a door. It will open easiest if you push
opposite the hinge.
at the center of the door.
near the hinge.

5. Torque Turning force = torque = radius  force = r  F
Units: Nm (not J)

6. Vector Cross Product Operation symbol 
Another way to multiply two vectors
Product is a vector!
Direction of AB is perpendicular to both A and B

7. Cross Product Magnitude
A  B = ab sin q A q
Maximum for q = 90°
Zero for q = 0°, 180° B b

8. Magnitude Geometricallyq B b
AB = area of parallelogram

9. Cross Product Direction
Curl right-hand fingers in direction of q
Right-hand thumb points in direction of cross-product
Not commutative A B a b q
AB = –(BA)

10. Radius Vector Radius from reference point to application of force
Strictly, reference point must be specified to determine a torque (about the point)
Torque depends on your choice of point!

11. Adding Torques Net torque about fulcrum is zero here
Torques are vectors

12. Whiteboard Work
A 10,000-N truck is stalled 1/4 of the way across a 100-m bridge. What torque does its weight apply about the far (right) support? r

13. Whiteboard Work
What upward force must the near (left) support provide to cancel the truck’s torque about the far support? r F

14. Whiteboard Work
What upward force must the far support provide to support the weight of the truck?
Hint: Several ways will work:
canceling forces on the bridge
canceling torques about the near support r F

15. Lever Arm Component of r perpendicular to Fq F
r  F = area of parallelogram
= lF = area of rectangle

16. Question
Force P is applied to one end of a lever of length L. What is the magnitude of the torque about point A?
PL sin q.
PL cos q.
PL tan q.
PL sec q.
PL cot q.
PL csc q.
Source: Young and Freedman, Test Your Understanding §10.1

17. Center of Gravity
Where weight vector would be applied to duplicate torque of weight
In a uniform gravitational field, same place as center of mass

18. Angular Momentum Torque is rotational force
Angular momentum is rotational momentum
L = r  p
Angular momentum is a vector
Direction by right-hand rule

19. Units of Angular Momentum
L = r  p
r units = m p units = kg m/s L units = kg m2/s

20. Extended Object Angular momentum L = Iw I = moment of inertia
rotational inertia

21. I of a Point Mass
Iw = r  p
What is I?

22. SCI 340 L24 Torque and angular momentum
Moments of Inertia
Usually expressed in the form I = cMR2
c depends on the shape (mass distribution) of the object
Formulas for a few shapes are given in Cutnell & Johnson Table 9.1, on p. 244

23. SCI 340 L24 Torque and angular momentum
Moments of Inertia
Source: Young and Freedman, Table 9-2 (p. 291).

24. Scenario A B
Two wheels in contact rotate with the same tangential speed at their rims. Wheel A has half the radius of the wheel B.
Which wheel’s rim has the greater centripetal acceleration?

25. Group Work A B
Two wheels in contact rotate with the same tangential speed at their rims. Wheel A has half the radius of the wheel B.
Which wheel’s rim has the greater angular momentum?

26. Misconception Alert
Angular momentum and centripetal acceleration are different!
a = v2/r
decreases as r increases
proportional to v2
L = r  p
increases as r increases
proportional to v (not v2)

27. Newton’s Second Law Force is the rate of change of momentum Dp F = Dt
Torque is the rate of change of angular momentum Dt
t = DL

28. Conservation of Angular Momentum
SCI 340 L24 Torque and angular momentum
Conservation of Angular Momentum
If no outside torque, L = r  p is constant.
If r decreases, p increases!

29. Question
The moon revolves around the Earth with a period of days. If the moon were to move farther away from the Earth but maintain its angular momentum, how would its period adjust?
Its period would become shorter than days.
Its period would become longer than days.
Its period would remain days.

30. Conservation of Angular Momentum
Nothing can apply a torque to itself.
Any change in one object’s angular momentum is accompanied by an opposite change in another object.
The angular momentum of the universe never changes.

31. Conservation of Momentum
Q. How can linear momentum be conserved if p increases?
A. Total linear momentum is zero in a rotating system!

32. Conservation of Energy
Q. What happens to kinetic energy when p increases?
A. Kinetic energy increases! DE = w. Work is done to pull rotating parts inward.

33. Rotational Kinetic Energy
Point mass KE = ½ mv2
What is this in terms of I?

34. Rotating Kinetic Energy
SCI 340 L24 Torque and angular momentum
Rotating Kinetic Energy
KE = 1/2 Iw2
I = moment of inertia (rotational analogue of mass)
units?
kg m2/s2 = I rad2/s2
I = kg m2/rad2