2. R v v Resultant force toward center F c = mv 2 R Consider TOP of circle: AT TOP: T mg T + mg + T = mv 2 R T = - mg mv 2 R

3. R v v Resultant force toward center F c = mv 2 R Consider bottom of circle: AT Bottom: T mg + T - mg = mv 2 R T = + mg mv 2 R T mg

4. R v v AT TOP: T mg + AT BOTTOM: T mg + T = - mg mv 2 R T = + mg mv 2 R

5. R v v T mg mg + T = mv 2 R At Top: T = - mg mv 2 R T = 25 N - 19.6 N T = 5.40 N A 2-kg rock swings in a vertical circle of radius 8 m. The speed of the rock as it passes its highest point is 10 m/s. What is tension T in rope?

6. R v v T mg T - mg = mv 2 R At Bottom: T = + mg mv 2 R T = 25 N + 19.6 N T = 44.6 N A 2-kg rock swings in a vertical circle of radius 8 m. The speed of the rock as it passes its lowest point is 10 m/s. What is tension T in rope?

7. Torque is defined as the tendency to produce a change in rotational motion. : Examples: What is torque ?

8. What would help tighten the lug nuts, a short handle or a longer handle?

9. How do you produce more torque for picture (a) and picture (b)?

10. Torque is proportional to the magnitude of F and to the distance r from the axis. Thus, a tentative formula might be:  = Fr Units: N  m or lb  ft 6 cm 40 N  = (40 N)(0.60 m) = 24.0 N  m = 24.0 N  m  = 24.0 N  m

11. The magnitude of the applied force. The direction of the applied force. The location of the applied force. The magnitude of the applied force. The direction of the applied force. The location of the applied force. 20 N Magnitude of force 40 N The 40-N force produces twice the torque as does the 20-N force. Each of the 20-N forces has a different torque due to the direction of force. 20 N Direction of Force 20 N   Location of force The forces nearer the end of the wrench have greater torques. 20 N

12. Torque is a vector quantity that has direction as well as magnitude. Turning the handle of a screwdriver clockwise and then counterclockwise will advance the screw first inward and then outward.

13. By convention, counterclockwise torques are positive and clockwise torques are negative. Positive torque: Counter- clockwise, out of page cw ccw Negative torque: clockwise, into page

14. Extend line of action, draw, calculate r.  = (80 N)(0.104 m) = 8.31 N m r = 12 cm sin 60 0 = 10.4 cm An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque.