Rotational Motion

1. Rotational Motion Chapter 7

2. Rotational Motion • Motion about an axis of rotation. • A record turntable rotates; • A bug sitting on the record revolves around the axis and is said to undergo circular motion.

3. Particles in the rings of Saturn rotate using circular motion.

4. Spin cycle of washer • The spin cycle of your washer works on the principle that your clothing is forced to follow a circular path, but the water in the clothing escapes through holes in the side of the drum, not following a circular path.

5. Measuring rotational motion: • You have probably already encountered the radian, the measure of angular displacement: • Angle whose arc length = its radius • ΔΘ = Δs/r • Θ is anglular in radians, s is arc length, r is radius • Converting to degrees: • 2 π(rad) = 360 (deg)

6. Just like linear displacement, the direction matters. • Conventionally, rotation is….. • Positive when Counterclockwise • Negative when Clockwise • Lets do the problem on P246…. • H/W P247 Q1-4

7. P247 answers • 1. 1.7 rad • 2. Pi rad, 1.2 m • 3. 0.34 rad • 4. 2.5 rad, 6.4m, - 320 ° , 1.1m

8. Angular velocity: • Think of it as how quickly something is turning • A unit that is often used is revolutions per minute (RPM) • Old records spun at 33.3, 45 or 76 RPM • Car engines often run most efficiently at about 2500 RPM and produce the maximum power about 4500rpm • Most electric motors spin at a multiple or sub multiple of 3600RPM or 60 revolutions/sec

9. Angular velocity (speed) • Just like motion in a straight line, after displacement comes speed …. • Angular speed is the rate of change of angular displacement • ω = ΔΘ/Δt • Units are rad/s • Lets do problem on P248 • H/W P248 Q1-4

10. P248 answers 1. 29 rad/s • 2.2 rad/s • 7.3 X 10-5rad/s • A) 0.23 rad/s b) 0.24 rad c) -6.3 rad/s d) 0.75s

11. Angular acceleration: • Think of it as how quickly a rotating object speeds up or slows down. • The angular acceleration of the earth is High:Low: Zero • The angular acceleration of a bicycle wheel is pulling away from a stop High:Low:Zero • The angular acceleration of a motorcycle doing a constant 150mphis High:Low:Zero • The angular acceleration of a motorbike wheel pulling away from a race start is High:Low: Zero

12. Angular acceleration • Rate of change of angular velocity, α α = ωf – ωi Δt • Units are rad/s2 • Let’s do Problem on P249 • H/W P. 250 Q1,2,3

13. P250 Answers • 4.3rad/s2 • 1.3rad/s2 • a) 17rad/s2 b) 0.038rad/s c) -6.3 rad/s2

14. Angular kinematic equations: ωf = ωi + α tθ = ωit + 1/2 α t2ωf2 = ωi2 + 2 α (θf-θi) θ =1/2 (ωi + ωf) t

15. Answers to P 252 • 9.0 rad/s • 25 rad/s2 • 15 rad/s • 31 rad/s • 0.89 rad/s

16. Section review: • 1. 0.44rad, 0.61 rad, 2.23 rad, 4.7 rad • 2. -1.0 rad • 3. 0.314 rad/s • 4. 0.20 rad/s2 • 5. 0.70 rad/s • Page 269 Q10: 0.042rad/s , Q11a) 821rad/s2 , b) 4.2 X103 rad

17. Remember the strategy: • Write down the givens and unknown. • Find the equation that has all the givens and unknown and nothing else. • If necessary, rearrange the equation to find the unknown • and then substitute to solve.

18. Tangential Speed (7.2) • Speed of an object (m/s) traveling in a circle is called Tangential Speed because the direction of motion is always in a tangent to the circle.

19. Tangential speed: Tangential speed would be important to find out how fast a point on the earth is travelling in a given time etc vt (m/s)= r ω

20. Tangential Acceleration: • The rate of change of tangential speed. It is the linear acceleration of a point undergoing angular acceleration: at (m/s2) = r α

21. Centripetal acceleration: • Acceleration directed toward the center of a circle that an object undergoing circular motion must experience. (Note spinning cup with water in it) ac= vt2 / r ac= r ω2

22. H/W : P255 1-4, P256 1-3, P258 1-5 • P250 • 1.8m/s • 6.9 m/s • 9.2 m/s • 3.6 m/s, 15 rad/s, 29m/s, 1.3m • P256: • 2.11 m/s2, 0.18m/s2, 1.0m/s2

23. P258 answers: • 3.0m/s2 • 250m/s2 • 1.5m/s, 1.0rad/s • 12.6m/s2 • 84m/s2

24. Centripetal force: • In order for an object to travel in a circle, something must provide a force that is directed at all times toward the center of the circle. This force is called CENTRIPETAL FORCE. • For a car going around the corner, the force is provided by the ______. • For a stone being twirled in a slingshot it is provided by the _______. • For clothes in the spin cycle it is provided by______

25. For the moon traveling around the earth it is provided by _______ . • For the earth traveling around the sun it is provided by _______ . • Can you think of any other objects that undergo circular motion and identify what provides the centripetal force?

26. Demonstration • The object on the left travels with inertia, while the object on the right is caused to travel in a circle by the wooden block. Centripetal force is applied.

27. Calculating Centripetal Force

28. Inertia should cause the car to continue in the direction in which it was traveling. What causes it to travel in a circular direction? What applies the centripetal force?

29. If you let go, you’ll be like Mary Poppins and fly off the Merry-go-Round.

30. You do not fly straight outward. • Instead you follow tangential motion, and continue in a straight line from the point where the circular motion ends.

31. As usual, there is a formula: (From F=ma) Fc= mvt2 / r Fc= mrω2 Homework: P261 Q1-5

32. Newton’s universal law of gravitation • “There is an attractive force between any two masses or particles in the universe” F = - G m1m2 r2 Where G is the universal gravitational constant, m is each mass in kg, and r is the distance separating their centers of mass G = 6.67 X 10 -11 N m2 / kg2

33. P265 1-3 top of page • And Section review • Keep in mind that, for an orbiting body, centripetal force = gravitational force.

34. Speed of an orbiting satellite: • Vs = (G Mc /r)1/2 • Where Mcis central mass, r is the total distance from center of rotation.

35. Escape Velocity: • There is a speed at which an object shot straight up from a planet will have enough energy to escape the gravitational field of the planet. • Vesc = ( 2MG/R)1/2 • M is the mass of the planet.

36. g, the acceleration due to gravity on any planet surface : • g = G Mp/ rp2

37. Homework: • Find your gravitational force on the earth’s surface using universal G formula. (1lb = 0.45kg) • Compare with the weight formula result. • Compare with your gravitational force in orbit 300km above the earth’s surface. • Find g for each planet and the moon. • Find the escape velocity for each planet and the moon.

38. Rotational Speed (angular speed) • The number of rotations/unit of time. • RPM = rotations/min

39. Centripetal Force • Centripetal force is a force that causes an object to travel in a circle.

40. How does mass impact Centripetal Force

41. Centrifugal Force • Centrifugal means “Center-fleeing” and it is a force that seems to push you outward. • Think playground “Merry-go-Round”

42. What it really is is inertia. • Newton’s First Law applies always.

43. Inertia, Centrifugal Force • In a car.

44. Kids, Don’t try this at home… • Experts state that you can swing a bucket of water over your head and it won’t fall out because of centrifugal force (INERTIA). • What they don’t say is that when you stop swinging, it will drench you!

45. The breaking string revisited… • What kind of tension would be in that string?

46. In action…

47. Rotational Mechanics • Torque – Rotational analog of Force; • Produces rotation • More “leverage” = More Torque

48. Torque changes the rotational motion of an object.

49. What is Torque?? • Used when you use a hammer claw to remove a nail • Used when you use a long-handled wrench to loosen a bolt • The longer the handle, the greater the torque

50. Important facts to increase Torque • The force must be applied perpendicular to the plane. • The Longer the Lever, the greater the force.