Banked Curves Ch 7 and 8

1. Banked Curves Ch 7 and 8

2. Banked Curves If the curve is banked at an angle, then the normal force can provide the centripetal force needed to make the turn. What happens to mass?

3. Coin Drop, 30 sec video • http://www.youtube.com/watch?v=3zhjXvJSib8&safety_mode=true&persist_safety_mode=1

4. Banked Curves Example The Daytona 500 is the major event of the NASCAR season. It is held at the Daytona International Speedway in Daytona, Florida. The turns in this oval track have a maximum radius (at the top) of r = 316 m and are banked steeply, with  = 31o. Suppose these maximum radius turns were frictionless. At what speed would the cars have to travel around them?

5. Banked Curves Example The Daytona 500 is the major event of the NASCAR season. It is held at the Daytona International Speedway in Daytona, Florida. The turns in this oval track have a maximum radius (at the top) of r = 316 m and are banked steeply, with  = 31o. Suppose these maximum radius turns were frictionless. At what speed would the cars have to travel around them? tan = v2 rg v = 43 m/s Since no friction is available to provide the centripetal force, the horizontal component of the normal force must provide it.

6. Draw this table into your notes. Allow a few spare lines at the bottom. Everyone stand and give a potential answer.

7. Draw this table into your notes. Allow a few spare lines at the bottom.

8. Question • If I have an object moving in a circle, what is the kinetic energy? (List Eqn.)

9. Question • If I have an object moving in a circle, what is the kinetic energy? • It's ½ mv2

10. Question • If I have an object moving in a circle, what is the kinetic energy? • It's ½ mv2 • Since v = rω KE=1/2 mr2ω2

11. Inertia, I, New Symbol • Moment of Inertia: I, is an objects resistance to change in rotational motion. I = mr2 in kg m2. (from Newton’s 1st law.) For a particle some distance from the pivot point. • It is the rotational equivalent of mass. • An object rotating tends to stay rotating and an object not rotating tends to stay not rotating until acted upon by an outside torque.

12. Inertia, I, New Symbol • Moment of Inertia: I, is an objects resistance to change in rotational motion. I = mr2 in kg m2. Mass is a resistance to change in motion (comapre earth globe to classroom wall.) • An object rotating tends to stay rotating and an object not rotating tends to stay not rotating until acted upon by an outside torque. • An object in motion tends to stay in motion and an object at rest tends to stay at rest until acted upon by an outside force.

13. Remember KE = ½ mv2, • “ω” replaces v • “I” replaces m

15. Calculating Moment of Inertia • For a system of more than one mass: • Inet = I1 + I2 + I3 … • Inet =m1r12 + m2r22 + m3r32 ...

16. Calculating Moment of Inertia • For a solid object it's more complicated. • You have to break the object up into tiny little bits, calculate the mass and radius of each and add up the moments of inertia of each one.

17. Calculating Moment of Inertia • For a solid object it's more complicated. • You have to break the object up into tiny little bits, calculate the mass and radius of each and add up the moments of inertia of each one. • Calculus: volume integral

18. Calculating Moment of Inertia • For a solid object it's more complicated. • You have to break the object up into tiny little bits, calculate the mass and radius of each and add up the moments of inertia of each one. (Calculus: volume integral) • Or just look it up in a table.

19. Your text book, Bing, and wikipedia all have good tables of moments of inertia. Do not take the time to memorize any of them

20. Question What is the moment of inertia of an ordinary dice cube? Mass = 1.5 g Side length = 1 cm

21. Wikipedia sez:

22. Question What is the moment of inertia of an ordinary dice cube? Mass = 1.5 g = .0015 kg Side length = 1 cm = .01 m I = ms2/6

23. Question What is the moment of inertia of an ordinary dice cube? Mass = 1.5 g = .0015 kg Side length = 1 cm = .01 m I = ms2/6 I = .0015 kg (.01m)2 / 6 I = 2.5 * 10 -8 kgm2

24. What causes acceleration? Linear acceleration is caused by force. Angular acceleration is caused by “Torque”. This mean force and torque are analagous.

25. New Symbol “” Greek Lowercase “Tau” Stands for Torque (just like F = ma) Torque is like force: , F Moment of Inertia (I) is like mass: , m Angular acceleration (is like linear acceleration: , a Torque = the amount of angular acceleration a force causes. (how much a force makes something rotate.)

26. Calculating Torque The equation relating torque and force is: r x F It is the “cross product” of force and radius. Cross Product uses sine and right hand rule. a x b = |a| |b| sinθ where a and b are vectors. Radius is the vector from the pivot point “line of action” Line of action is the line along which the force acts. http://en.wikipedia.org/wiki/File:Right_hand_rule_cross_product.svg

27. Two variables that affect Torque • First is angle force is applied. • 2nd we will discuss is distance force is applied. • The question, how do angle and distance affect torque?

28. How does angle affect torque? The angle you apply the force at changes the distance to the “line of action.” Which force will be best for opening the door? (Draw theta angles and force causing torque.)

29. How does angle affect torque? Which force will be best for opening the door? ANS: 90Deg has most torque.

31. From Mr. Burkholder • or F d sin θ • F = force, D = distance force is applied, • θ is the angle between the Force and direction of Torque Motion. θ d = distance

32. Examples of Theta Theta’s for the different angles of force. What is the torque for each drawing? (Draw theta angles and force causing torque.)

33. Torque with a knife,Chopping Nuts Hand pushing down Fixed hand, Creates axis of rotation Hand pushing down Which is easiest to cut? Hand pushing down (Use door stop at different distances as demo.)

34. Torque with a knife Hand pushing down Hand pushing down Which is easiest to cut? Hand pushing down

35. How does Distance affect torque? A door is a device that works when you apply torque to it. Which force will get the door opened the fastest?

36. Torque is the cross product of force and radius. It is a vector. What is the direction of the vector??? Please Note: Torque is not Work, even though they have the same units. Work is the dot product of force and displacement. It is a scalar

37. Torque as a vector. 2nd right hand rule: 1) Line your fingers up with the radius vector. 2) Curl your fingers along the α the force would cause. Your thumb points in the same direction as the torque. The direction of the Torque matches the Direction of  from The 1st right hand rule

38. 2 equations What does ΣF equal?

39. 2 equations What does ΣF equal? Newton’s 2nd F = ma

40. Solve problems with 2 sums. • Remember, • Σ F = ma • So, Σ τ = Iα From Newton’s laws, there can only be an angular acceleration when there is a net tau. • And when STATIC (No accelerations) • Σ F = 0 (Linear Motion) • So, Σ τ = 0 (Rotation)

41. Solve problems with 2 sums. • How many axis do rocket scientist work with? • So how many problems do they solve?

42. Solve problems with 2 sums. • How many axis do rocket scientist work with? • x, y, z • So how many problems do they solve? • x2 for force and torque = 6 problems

43. Solve problems with 2 sums. • How many axis do rocket scientist work with? • x, y, z • So how many problems do they solve? • x2 for force and torque = 6 problems • Apollo astronauts did this in their heads!!!!! • No room sized computers in the capsule. • And they did them very fast. VERY SMART!

44. Example A Diving Board A woman whose weight is 530 N is poised at the right end of a diving board with length 3.90 m. The board has negligible weight and is supported by a fulcrum 1.40 m away from the left end. Find the forces that the bolt and the fulcrum exert on the board. Add to WOD torque: Direction of torque. Positive = CCW Negative = CW

45. Example A Diving Board A woman whose weight is 530 N is poised at the right end of a diving board with length 3.90 m. The board has negligible weight and is supported by a fulcrum 1.40 m away from the left end. Find the forces that the bolt and the fulcrum exert on the board. Axis of rotation is your frame of reference And can be selected anywhere. Pick it So that one radius will be zero for easier Math work.

46. Example A Diving Board Axis of rotation is your frame of reference And can be selected anywhere. Pick it So that one radius will be zero for easier Math work. How many forces do we have? How many torques do we have?

47. Example A Diving Board Axis of rotation is your frame of reference And can be selected anywhere. Pick it So that one radius will be zero for easier Math work. How many forces do we have? 3 How many torques do we have? 3 If we pick the axis of rotation on a force, Then 1 torque will go away because r = 0.

48. Fb = Force from bolt, Ww = Force from Woman’s weight, r = radius (note: book moved axis To middle, but not needed.)

49. Make sure the signs of the torque and forces are correct.

50. Question Will there be a net torque on this object? What is the net force? Where is the line of action?