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# Rotational Dynamics

Rotational Dynamics. Radian measure compared to degree measure. 1 radian = angle of rotation where the arc length of rotation = radius of the circle. 1 rad = about 57 ˚ 2 π rad = 360˚. Rotational displacement. Change in the angle of rotation Symbol is θ (theta), unit is rads

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## Rotational Dynamics

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1. Rotational Dynamics

2. Radian measure compared to degree measure • 1 radian = angle of rotation where the arc length of rotation = radius of the circle 1 rad = about 57˚ 2π rad = 360˚

3. Rotational displacement • Change in the angle of rotation • Symbol is θ (theta), unit is rads Counterclockwise motion is + Clockwise motion is -

4. Rotational velocity • Change in angular displacement over time • Measures rate of angular motion • ω = ∆θ / t • Symbol is ω (omega), unit is rads / sec

5. Rotational Acceleration • Change in the rate of angular velocity over time • α =∆ω / t • Symbol is α (alpha), unit is rads / sec2

6. Relationship between linear and rotational motion values • Linear values for a rotating object depends on its distance from the pivot point • Relationships between linear and angular values depend on the radius • d = θr v = ωr a = αr

7. Angular frequency • Cycles through rotation • ƒ = ω / 2π • Remember v = 2πr / T and v = ωr?

8. Question • What is the angular displacement of each of the following hands of a clock in 1 hour? • The second hand • The hour hand

10. Next question The rotational velocity of a merry-go-round in increased from 1.5 rads/s to 3.5 rads/s over 9.5 seconds. What is the rotational acceleration of the merry-go-round?

11. Homework on pg 200 Question 1-10

12. DEFINITION OF TORQUE

13. Rotating a door • Where is the best place to apply force to open the door? • How does the direction that force is applied relate to the pivot point?

14. Torque and pipe wrench, extenders

15. Big ideas for this section After this section you should be able to: define what torque is identify examples of torque in real life identify lever arm and applications points calculate torque

16. Torque Is a measure of how effectively a force causes rotation. Generated by an application of force on an object in a direction that does not go through the pivot point

17. Result of Torque • Application of Torque can result in object changing the rate of spin

18. Torque Symbol is T (Tau), Units are Nm Not Joules (no displacement) Direction (clockwise-, counter clockwise +)

19. How to calculate the torque Need: Size of force applied = F Distance between application of force and pivot point (lever arm length = r ) Angle between force and lever arm (use the smaller of the two angles = Θ)

20. Torque equation • T = rFsinA, • Where • r means______________ • F means _____________ • A means ______________

21. Torque is larger when: A larger force is applied The length of the lever arm is increased The angle between force and arm is 90°

22. Questions on torque • A bolt of a car engine needs to be tightened with a torque of 35 Nm. You use a 25 cm long wrench and pull on the end of the wrench at an angle of 120˚from body of the wrench. • How much force do you exert on the wrench?

23. Homework • 12-15 on pg 203

24. Answers to questions on pg 200 • a: -120π or -377 rad b: -2π or -6.28 rad c) –π/6 rad 2) Diameter = 0.707 m 3) a) Linear acc is same as truck b) α = 7.71 rad/s2 4) Angular velocity decreases # revolution decreases

25. Answers to pg 200 cont… 5) a: -π/3 or -1.05 rad/s b) -4π or -12.6 rad 6) a: 2358720 sec = T b: 4.24x10-8 cycles/s or 2.66x10-6 rad/s c: V = .46 m/s on moon d: v = 463 m/s on earth (1000 times larger)

26. Answers for 200 7) 3.8π or 12 radians 8) Yes to same angular displacement No, to traveling the same linear distance 9) α = - 8.3 rad/s2 10) Α = -0.0059 rad/s2

27. See-saws and torque

28. Net torque • Sum of all torque exerted on an object • (Not the sum of all forces) • (Not all forces exert torque) • Tnet = 0 means that the clockwise torque is balanced by the counterclockwise torque

29. Net torque question • 2 kids, the 1st is 65 kg , 2nd is 45 kg want to balance on a 3 m long seesaw. • If the 45 kg kid wants to be at the end of the see-saw on the left, where would you place the other kid? • Can they remain balanced only if the seesaw is horizontal?

30. Homework • Pg 205 16-19

31. Answers 16-19 on pg 205 16) About 1.49 meters from center 17) About 2.7 Nm in the counterclockwise (or +) direction 18) About 0.056 kg 19) About .042 kg

32. What happens to an object if Tnet ≠ 0?

33. Torque and Angular Acceleration • When a rigid object is subject to a net torque (≠0), it undergoes an angular acceleration • The angular acceleration is directly proportional to the net torque • The relationship is analogous to ∑F = ma • Newton’s Second Law

34. Moment of Inertia • The angular acceleration is inversely proportional • to the object’s mass • its position of mass in a rotating system • This mass component is called the moment of inertia, I, of the object • Inertia of rotation

35. Moment of Inertia I = mr2 (if a single point mass) where I = moment of inertia m = mass r =distance mass is from pivot SI units are kg m2

36. Which moves done the ramp faster (greater α) ?

37. Moment inertia depends on the shape and size of mass The farther and larger the mass is from the pivot, the greater tits moment of inertia

38. More About Moment of Inertia • There is a major difference between moment of inertia and mass: the moment of inertia depends on the quantity of matter and its distribution in the rigid object. • The moment of inertia also depends upon the location of the axis of rotation

39. Homework on Moment of inertia and rotational acceleration • Pg 208 21-24

40. Answers to homework 21) Done in class 22) Hollow ball has greater moment of inertia, mass is farther away 23) A has greater moment of inertia (5 mr2 compared to 2 mr2) 24) .02 kg m2 compared to 0.008 kg m2 Challenge: least A (0), D (5), C (6) , B (14) most

41. Newton’s Second Law for a Rotating Object • The angular acceleration is directly proportional to the net torque • The angular acceleration is inversely proportional to the moment of inertia of the object

42. Remember: • Torque is the application of force, not force • Moment of inertia is based on the mass shape and position from the pivot point

43. Example problem • A solid steel wheel has a mass of 15 kg and a diameter of 0.44m. It starts from rest. You want to make it rotate at 8.0 rev/s in 15 s. • What torque must be applied to the wheel? • If you apply the torque by wrapping a strap around the wheel, how much force should you exert on the strap?

44. Moment of Inertia of a Uniform Ring • Image the hoop is divided into a number of small segments, m1 … • These segments are equidistant from the axis

45. Other Moments of Inertia

46. Homework • Pg 210 25-35 • To be turned in on Thursday

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