Chapter 12

1. Chapter 12 Rotation of a Rigid Body

2. Vector (or “cross”) Product Cross Product is a vector perpendicular to the plane of vectors A and B

3. Cross Product of Vectors Right hand rule: Curl your right hand around the center of rotation with the fingers going from the first vector to the second vector and the thumb will be pointing in the torque direction AP Physics C

4. Cross Product Problem Csin(110) C 110° D AP Physics C

5. Test your Understanding 1. Clockwise. 2. Counter-clockwise. 3. Not at all. 4. Not sure what will happen. Which way will it rotate once the support is removed?

6. Torque If the forces are equal, which will open the heavy door more easily? AP Physics C

7. Interpretation of torque • Measures tendency of any force to cause rotation • Torque is defined with respect to some origin – must talk about “torque of force about point X”, etc. • Torques can cause clockwise (+) or anticlockwise rotation (-) about pivot point

8. Torque con’t AP Physics C

9. Torque con’t AP Physics C

10. Definition of Torque: where is the vector from the reference point (generally either the pivot point or the center of mass) to the point of application of the force . If r and F are not perpendicular then: where q is the angle between the vectors and .

11. Definition of Torque:

12. Torque Problem Adrienne (50 kg) and Bo (90kg) are playing on a 100 kg rigid plank resting on the supports seen below. If Adrienne stands on the left end, can Bo walk all the way to the right end with out the plank tipping over? If not, How far can he get past the support on the right? AP Physics C

13. Torque Problem con’t N1 N2 2m 3m 4m x 50kg 100kg 90kg AP Physics C

14. Moments M1 M2 d1 d2 Suppose we have masses m1 and m2 on the seesaw at distances d1 and d2, respectively, from the fulcrum, when does the seesaw balance? By Archimedes’ Law of the lever, this occurs when m1d1 + m2d2 = 0 AP Physics C

15. Moments con’t M1 M2 x1 x2 If we place a coordinate system so that 0 is at the fulcrum and if we let xi be the coordinate at which is placed then: m1x1 + m2x2 = m1d1 + m2d2 = 0 AP Physics C

16. Moments con’t M2 M3 M4 M5 M1 x1 x2 x3 x5 x4 More generally, if we place masses m1, m2, …, mr at points x1, x2, …. , xr, respectively, then the see saw balances with the fulcrum at the origin, if and only if m1x1 + m2x2 + …+ mrxr = 0 AP Physics C

17. Moments con’t M2 M3 M4 M5 M1 x1 x2 x3 x5 x4 Now, suppose that we place masses m1, m2, … , mr at points x1, x2, … xr, respectively, then where should we place the fulcrum so that the seesaw balances? The answer is that we place the fulcrum at x-bar where: m1(x1 - (x-bar) ) + m2(x2 - (x-bar) ) + …+ mr(xr - (x-bar) )= 0 AP Physics C

18. Moments con’t M2 M3 M4 M5 M1 x1 x2 x3 x5 x-bar x4 m1(x1 - (x-bar) ) + m2(x2 - (x-bar) ) + …+ mr(xr - (x-bar) ) is called the moment about x-bar. Moment is from the Greek word for movement, not time. If positive, movement is counter-clockwise, negative it is clockwise. AP Physics C

19. Moments con’t M2 M3 M4 M5 M1 x1 x2 x3 x5 x-bar x4 Suppose that We want to solve for AP Physics C

20. Center of Mass M2 M3 M4 M5 M1 x1 x2 x3 x5 x-bar x4 AP Physics C

21. Center of Mass con’t Suppose m1, m2, … , mr are masses located at points (x1, y1), (x2, y2), … , (xr, yr). M1 M4 The moment about the y-axis is: M2 The moment about the x-axis is: M3 Center of Mass is AP Physics C

22. Center of Mass con’t Now lets find the center of mass of a thin plate with uniform density, ρ. First we need the mass of the plate: AP Physics C

23. Center of Mass con’t Next we need the moments of the region: f(x) g(x) ∆x To find the center of mass we divide by mass: AP Physics C

24. Center of Mass Problem Determine the center of mass of the region bounded by y = 2 sin (2x) and y = 0 on the interval, [0, π/2] Given the symmetry of the curve it is obvious that x-bar is at π/4. First find the area. Using the table of integrals: AP Physics C

25. Moment of Inertia M r CR F AP Physics C

26. Moment of Inertia con’t AP Physics C

27. Calculating Moment of Inertia AP Physics C

28. Calculating Moment of Inertia AP Physics C

29. Moment of Inertia con’t AP Physics C

30. Parallel Axis Theorem AP Physics C

31. ||-axis Theorem Proof AP Physics C

32. Rotational Kinetic Energy AP Physics C

33. Rotational Dynamics AP Physics C

34. Rotation About a Fixed Axis AP Physics C

35. Bucket Problem A 2.0 kg bucket is attached to a mass-less string that is wrapped around a 1.0 kg, 4.0 cm diameter cylinder, as shown. The cylinder rotates on an axel through the center. The bucket is released from rest 1.0 m above the floor, How long does it take to reach the floor AP Physics C

36. Bucket Problem con’t AP Physics C

37. Bucket Problem Con’t AP Physics C

38. Static Equilibrium AP Physics C

39. Statics Problem A 3.0 m ladder leans against a frictionless wall at an angle of 60°. What is the minimum value of μs, that prevents the ladder from slipping? AP Physics C

40. Statics Problem AP Physics C

41. Balance and Stability θc h t/2 θc h t/2 AP Physics C

42. Rolling Motion AP Physics C

43. Rolling Motion con’t AP Physics C

44. Rolling Motion con’t AP Physics C

45. Rolling Motion con’t AP Physics C

46. Rolling Kinetic Energy AP Physics C

47. Great Downhill Race AP Physics C

48. Downhill Race con’t AP Physics C

49. Angular Momentum AP Physics C

50. Angular Momentum AP Physics C