Torque

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 0.750 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 0.800 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 • 0.0356 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 = ?