1. Torque
We know that Newton’s second law ( ) explains that the net force is the source of an object’s acceleration.
What is the source of a rotating object’s angular acceleration? It can’t be just a force, because it matters where on the object that force is applied.
The answer lies in the quantity called torque.

2. Torque…
Torque, t, is the tendency of a force to rotate an object about some axis
Torque is a vector
t = r F sin f = F d
F is the force
f is the angle the force makes with the horizontal
d is the moment arm (or lever arm)

3. …Torque…
The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force
d = r sin Φ

4. …Torque
The horizontal component of F (F cos f) has no tendency to produce a rotation
Torque will have direction
If the turning tendency of the force is counterclockwise, the torque will be positive
If the turning tendency is clockwise, the torque will be negative

5. Conceptest…
You are trying to open a door that is stuck by pulling on the doorknob in a direction perpendicular to the door. If you instead tie a rope to the doorknob and then pull with the same force, is the torque you exert increased?
yes no

6. …Conceptest
You are trying to open a door that is stuck by pulling on the doorknob in a direction perpendicular to the door. If you instead tie a rope to the doorknob and then pull with the same force, is the torque you exert increased?
yes no

7. Conceptest…
You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:
                                                                 

8. …Conceptest
You are using a wrench and trying to loosen a rusty nut. Which of the arrangements shown is most effective in loosening the nut? List in order of descending efficiency the following arrangements:
                                                                 

9. Conceptest…
A plumber pushes straight down on the end of a long wrench as shown. What is the magnitude of the torque he applies about the pipe at lower right?
A. (0.80 m)(900 N)sin 19°
B. (0.80 m)(900 N)cos 19°
C. (0.80 m)(900 N)tan 19°
D. none of the above

10. …Conceptest
A plumber pushes straight down on the end of a long wrench as shown. What is the magnitude of the torque he applies about the pipe at lower right?
A. (0.80 m)(900 N)sin 19°
B. (0.80 m)(900 N)cos 19°
C. (0.80 m)(900 N)tan 19°
D. none of the above

11. Net Torque
The force F1 will tend to cause a counterclockwise rotation about O
The force F2 will tend to cause a clockwise rotation about O
St = t1 + t2 = F1d1 – F2d2

12. Torque vs. Force Forces can cause a change in linear motion
Described by Newton’s Second Law
Forces can cause a change in rotational motion
The effectiveness of this change depends on the force and the moment arm
The change in rotational motion depends on the torque

13. Torque Units The SI units of torque are N.m
Although torque is a force multiplied by a distance, it is very different from work and energy
The units for torque are reported in N.m and not changed to Joules

14. Torque and Angular Acceleration, Wheel Example
The wheel is rotating and so we apply St = Ia
The tension supplies the tangential force
The mass is moving in a straight line, so apply Newton’s Second Law
SFy = may = mg - T

15. Torque and Angular Acceleration, Multi-body Ex., 1
Both masses move in linear directions, so apply Newton’s Second Law
Both pulleys rotate, so apply the torque equation

16. Torque and Angular Acceleration, Multi-body Ex., 2
The mg and n forces on each pulley act at the axis of rotation and so supply no torque
Apply the appropriate signs for clockwise and counterclockwise rotations in the torque equations

17. Problem
A model airplane with mass kg is tethered by a wire so that it flies in a circle 30.0 m in radius. The airplane engine provides a net thrust of N perpendicular to the tethering wire.
Find the torque the net thrust produces about the center of the circle.
Find the angular acceleration of the airplane when it is in level flight.
Find the linear acceleration of the airplane tangent to its flight path.

18. Answers
24.0 N-m
rad/s2
1.07 m/s2

19. Review: The Vector Product
Given two vectors, A and B
The vector (“cross”) product of A and B is defined as a third vector, C
C is read as “A cross B”
The magnitude of C is AB sin q
q is the angle between A and B

20. More About the Vector Product
The quantity AB sin q is equal to the area of the parallelogram formed by A and B
The direction of C is perpendicular to the plane formed by A and B
The best way to determine this direction is to use the right-hand rule

21. Properties of the Vector Product
The vector product is not commutative. The order in which the vectors are multiplied is important
To account for order, remember
A x B = - B x A
If A is parallel to B (q = 0o or 180o), then A x B = 0
Therefore A x A = 0

22. Vector Products of Unit Vectors
Signs are interchangeable in cross products
A x (-B) = - A x B

23. The Vector Product and Torque
The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vector
t = r x F
The torque is the vector (or cross) product of the position vector and the force vector

25. Torque Vector Example
Given the force
t = ?