1. Rotational Motion Chapters 10, 11, & 12

2. Rotation vs Revolution An axis is the straight line around which rotation takes place. When an object turns about an internal axis, the motion is called rotation, or spin. When an object turns about an external axis, the motion is called revolution.

3. Rotation vs. Revolution The Ferris wheel turns about an axis. The Ferris wheel rotates, while the riders revolve about its axis. Earth undergoes both types of rotational motion.

4. Rotational Speed The turntable rotates around its axis while a ladybug sitting at its edge revolves around the same axis. Which part of the turntable moves faster—the outer part where the ladybug sits or a part near the orange center? Answer: It depends on whether you are talking about linear speed or rotational speed.

5. Rotational Speed Linear speed is the distance traveled per unit of time. The linear speed is greater on the outer edge of a rotating object than it is closer to the axis (travels a greater distance in one rotation than a point near the center). The speed of something moving along a circular path can be called tangential speed because the direction of motion is always tangent to the circle.

6. Rotational Speed Rotational speed (sometimes called angular speed) is the number of rotations per unit of time. All parts of the rigid turntable rotate about the axis in the same amount of time. All parts have the same rate of rotation, or the same number of rotations per unit of time. It is common to express rotational speed in revolutions per minute (RPM).

7. Rotational Speed All parts of the turntable rotate at the same rotational speed. a.A point farther away from the center travels a longer path in the same time and therefore has a greater tangential speed. b.A ladybug sitting twice as far from the center moves twice as fast.

8. Rotational Speed At the axis of the rotating platform, you have no tangential speed, but you do have rotational speed. You rotate in one place. As you move away from the center, your tangential speed increases while your rotational speed stays the same. Move out twice as far from the center, and you have twice the tangential speed.

9. Rotational Speed think! Q. At an amusement park, you and a friend sit on a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. What is her rotational speed? What is her linear speed?

10. Rotational Speed Why does a tapered cup roll in a curved path?

11. Rotational Speed Why does this shape remain on the track?

12. Rotational Speed Why is the tapered shape (exaggerated) of railroad train wheels essential on the curves of railroad tracks?

13. Rotational Speed think! Train wheels ride on a pair of tracks. For straight-line motion, both tracks are the same length. But which track is longer for a curve, the one on the outside or the one on the inside of the curve?

14. Centripetal Force Velocity (a vector) involves both speed and direction. When an object moves in a circle, even at constant speed, the object still undergoes acceleration because its direction is changing. This change in direction is due to a net force (otherwise the object would continue to go in a straight line). Any object moving in a circle undergoes an acceleration that is directed to the center of the circle—a centripetal acceleration.

15. Centripetal Force Centripetal means “toward the center.” The force directed toward a fixed center that causes an object to follow a circular path is called a centripetal force.

16. Centripetal Force Centripetal force holds a car in a curved path. a.For the car to go around a curve, there must be sufficient friction to provide the required centripetal force. b.If the force of friction is not great enough, skidding occurs.

17. Centripetal Force Calculating Centripetal Force Centripetal force, F c, is measured in newtons when m is expressed in kilograms, v in meters/second, and r in meters.

18. Centripetal vs. Centrifugal Force Sometimes an outward force is also attributed to circular motion. This apparent outward force on a rotating or revolving body is called centrifugal force. Centrifugal means “center-fleeing,” or “away from the center.” This force does not exist!

19. Centripetal vs. Centrifugal Force In the case of the whirling can, it is a common misconception to state that a centrifugal force pulls outward on the can. In fact, when the string breaks the can goes off in a tangential straight-line path because no force acts on it. So when you swing a tin can in a circular path, there is no force pulling the can outward (no centrifugal force). Only the force from the string acts on the can to pull the can inward.

20. Chapter 11: Rotational Equilibrium This chapter is about the factors that affect rotational equilibrium.

21. Torque Every time you open a door, turn on a water faucet, or tighten a nut with a wrench, you exert a turning force. Torque is produced by this turning force and tends to produce rotational acceleration. – Torque is different from force. Forces tend to make things accelerate. Torques produce rotation.

22. Torque A torque produces rotation.

23. Torque The distance from the turning axis to the point of contact is called the lever arm. If the force is not at right angle to the lever arm, then only the perpendicular component of the force will contribute to the torque.

24. Torque Although the magnitudes of the applied forces are the same in each case, the torques are different.

25. Torque If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?

26. Balanced Torques Children can balance a seesaw even when their weights are not equal.

27. Balanced Torques What is the weight of the block hung at the 10-cm mark?

28. Center of Mass The center of mass, is where all the mass of an object can be considered to be concentrated. – For a symmetrical object, such as a baseball, the center of mass is at the geometric center of the object. – For an irregularly shaped object, such as a hammer, the center of mass is toward the heavier end.

29. Center of Mass The center of mass of the toy is below its geometric center.

30. Center of Mass The center of mass of the rotating wrench follows a straight-line path as it slides across a smooth surface.

31. Center of Mass vs Center of Gravity Center of mass is often called center of gravity, the average position of all the particles of weight that make up an object. For almost all objects on and near Earth, these terms are interchangeable. There can be a small difference between center of gravity and center of mass when an object is large enough for gravity to vary from one part to another. For example) The center of gravity of the Sears Tower in Chicago is about 1 mm below its center of mass because the lower stories are pulled a little more strongly by Earth’s gravity than the upper stories.

32. Center of Gravity The CG is the balance point. Supporting that single point supports the whole object.

33. Center of Gravity There is no material at the CG of these objects.

34. Torque and CG The block topples when the CG extends beyond its support base.

35. Torque and CG The Rule for Toppling “If the CG extends outside the area of support, an unbalanced torque exists, and the object will topple.”

36. Torque and CG This “Londoner” double- decker bus is undergoing a tilt test. So much of the weight of the vehicle is in the lower part that the bus can be tilted beyond 28° without toppling.

37. Torque and CG The Leaning Tower of Pisa does not topple over because its CG lies above its base.

38. Center of Gravity and People When you stand, your CG is somewhere above the area bounded by your feet.

39. Center of Gravity and People You can lean over and touch your toes without toppling only if your CG is above the area bounded by your feet.

40. Center of Gravity and People Think… When you carry a heavy load—such as a pail of water—with one arm, why do you tend to hold your free arm out horizontally?