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1. PA Multi-Region STEM Partnership Sept 2013 – May 2014 Monthly Meetings (Torque & Center of Mass) Eastern Region Philadelphia/Chester County

2. What we will cover tonight: • Circular Motion • Rotational Inertia • Torque • Center of Mass and Center of Gravity • Centripetal Acceleration • Centripetal Force • Angular Momentum • Conservation of Angular Momentum

3. Circular Motion Notice the direction of the velocity vector is continually changing Slide 6-23

4. Centripetal Acceleration changes the Direction a = v2/r = ω2r

5. The Effect of Radius on Centripetal Acceleration The bobsled track contains turns with radii of 33 m and 24 m. Find the centripetal acceleration at each turn for a speed of 34 m/s.

6. Centripetal Force Any force directed toward a fixed center is called a centripetal force. Centripetal means “center-seeking” or “toward the center.” Centripetal Force cause centripetal acceleration

7. Different Centripetal Forces Friction Tension Gravity

8. From Newton’s Second Law of Motion: For circular motion:

9. Torque The tendency of a force to cause rotation Torque depends upon three factors: Magnitude of the force (weight) The direction in which it acts (down) The point at which it is applied on the object Point of rotation

10. Torque How to determine Torque Torque  lever arm  force Lever Arm Force

11. Lever Arm Lever arm is the perpendicular distance from the axis of rotation to the line along which the force acts. Axis of rotation

12. Lever Arm 1st picture: Lever arm is equal to length of handle. 2nd picture: Lever arm is longer than length of handle.

13. Lever Arm What we do when the lever arm is not perpendicular distance from point of rotation to force: Line of Action: an extended line drawn colinear with the force Lever Arm: perpendicular distance between line of action and axis of rotation

14. Torque Torques: Force X Lever Arm Lever Arm: perpendicular distance between line of action and axis of rotation The amount of torque depends on where and in what direction the force is applied, as well as the location of the axis of rotation.

15. Torque Causes Rotation Figure 8.18 Axis of Rotation? Clockwise rotation is positive Counterclockwise rotation is negative

16. Torque Causes Rotation Torque = Force x Lever Arm Figure 8.18 Girl: (250 N) x (3 m) = +750 N·m Boy: (250 N) x (3 m) = -750 N·m

17. Torque Causes Rotation Figure 8.18 Equal and opposite torque results in equilibrium

18. Torque Causes Rotation

19. Knife edge under center of mass Knife edge not under center of mass, produces a torque If a mass having the same weight of the beam is placed equidistant from the pivot on the opposite side, the beam will once again be brought back to equilibrium

20. Using Torque to Mass a Paper Clip

21. The center of mass is the point where all of the mass of the object is concentrated. When an object is supported at its center of mass there is no net torque acting on the body and it will remain in static equilibrium Length of lever arm Length of lever arm weight weight Rotation Point

22. Center of Mass and Center of Gravity Center of mass is the average position of all the mass that makes up the object. Center of gravity (CG) is the average position of weight distribution. Since weight and mass are proportional, center of gravity and center of mass usually refer to the same point of an object.

23. Determine the Center of Mass of Contiguous United States

24. Determining Center of Mass Suspend the mass from several points and trace the plumb line's location. The center of mass will fall below the suspension point (to reduce any torques from the object's weight) the center of mass will be at the intersection of all of the plumb lines

25. Center of Mass Determine the Center of Mass of the Contiguous United States:

26. Geographic Center of the Contiguous United States: Lebanon, Kansas

27. The Geographical Center of the United StatesLatitude 39 degrees 50 min. Longitude 98 degrees 35 min.

28. Center of Gravity—Stability The location of the center of gravity is important for stability. If we draw a line straight down from the center of gravity and it falls inside the base of the object, it is in stable equilibrium; it will balance. If it falls outside the base, it is unstable.

29. Edgar J. Kaufmann House (Fallingwater) Frank Lloyd Wright 1935-39 Concrete cantilevered terraces

30. Stability of a Car Slide 8-19

31. The centers of gravity of three trucks parked on a hill are shown by the dots. Which truck(s) will tip over?

32. The center of gravity of Truck A is not above an area of support: The centers of gravity of Trucks B and C are above their areas of support. Therefore only Truck A will tip over

33. On the sketches shown above, explain to your partner why the same person cannot stand on tiptoes against the wall

34. Tiptoeing Why can’t you stand on tiptoes if your toes are against a wall? Center of gravity has to be over toes – the base of support – to balance. That requires shifting your body slightly forward. But you can’t shift your body forward if your toes are against the wall. Slide 8-20

35. Line of action Gravity acts at the center of gravity. This force exerts no torque about her toes. Base of support Balance For an object to balance, its center of gravity must reside over its base of support. That way gravity does not exert a torque. Slide 8-18

36. Checking Understanding The four forces shown have the same strength. Which force would be most effective in opening the door? • Force F1 • Force F2 • Force F3 • Force F4 • Either F1 or F3 Slide 7-23

37. Answer The four forces shown have the same strength. Which force would be most effective in opening the door? • Force F1 • Force F2 • Force F3 • Force F4 • Either F1 or F3 Slide 7-24

38. The pull cord of a lawnmower engine is wound around a drum of radius 6.00 cm, while the cord is pulled with a force of 75.0 N to start the engine. What magnitude torque does the cord apply to the drum? Is it positive or negative? F=75 N R=6.00 cm

39. The pull cord of a lawnmower engine is wound around a drum of radius 6.00 cm, while the cord is pulled with a force of 75.0 N to start the engine. What magnitude torque does the cord apply to the drum? F=75 N R=6.00 cm Direction of rotation is counterclockwise so the torque is positive