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
rad
2. Pi rad, 1.2 m
rad
rad, 6.4m, ° , 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 10-5 rad/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. Δt Units are rad/s2 Let’s do Problem on P249 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. 4.3rad/s2 1.3rad/s2 a) 17rad/s2 b) 0.038rad/s c) -6.3 rad/s2
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 t θ = ωi t + 1/2 α t2 ωf2 = 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:
rad, 0.61 rad, 2.23 rad, 4.7 rad
rad
rad/s
rad/s2
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. vt (m/s) = r ω 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 : P , P , P
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= mr ω2
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 N m2 / kg2

33. P 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 Mc is 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 / rp 2

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. Used when you use a hammer claw to remove a nail
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.

51. Formula
Torque = force (perpendicular) x Lever Arm

52. Look at the pictures on page 151.
How does having the doorknob in the center of the door impact the torque?

53. See-Saw Torque
When a large child and a small child play on the same see-saw, how do they balance the torques?

54. The same mass moved farther down the arm produces more torque.
Triple Beam Balances
Triple Beam Balances work the same way as the see-saw. You slide the weights on the arms to balance the torques.
The same mass moved farther down the arm produces more torque.

55. Rotational motion and torque in an auto engine.

56. Torque Diagram

57. Torque measurement
Torque = Force x lever arm

58. Torque & Center of Gravity
Stand with your back and heels against the wall.
Then try to lean forward to touch your toes.
What happens to you??

59. You now have your center of gravity located somewhere other than over your
feet, so you …

60. See the sketches on page 154.
If you kick a football at its center of gravity, what happens?
If you kick a football above or below its center of gravity, what happens?

61. Rotational Inertia Just like in the inertia as learned before…
An object that is rotating about its axis will continue to rotate about its axis.
Rotational inertia depends on the mass of the object and the distribution of mass relative to the axis of rotation.
The more mass and the further it is on average from the axis, the higher the moment of inertia.

62. See P285 (P262 Hons)

63. Moment of inertia for rotating objects is analogous to mass for objects in linear motion.
Which will roll down a hill first, a basketball or a bowling ball?
A tire or a wheel/tire assembly?
A solid golf ball or a hollow lead sphere?
A tennis ball or a ping-pong ball?

64. Just like momentum (P = m v) for objects moving in a straight line
Angular momentum:
Just like momentum (P = m v) for objects moving in a straight line
Angular momentum, L = I ω , where L is the angular momentum, I is the moment of inertia and ω is angular speed.
Angular momentum is conserved if there is no net external torque.

65. Examples:
Skater
Diver
Aerialist skier / snowboarder
Planets in elliptical orbits.
Tornado
Hurricane

68. Newtons second law for rotation Ch 9:4 hons.
Before we had F = ma
Can you think of what the rotational equivalent might be?
Net external Τorque = I X α
Where I is the moment of inertia and α is angular acceleration.

69. Rotation and energy, Ch 9:5 hons.
Before we had work = F x d
What do you think the rotational equivalent of work might be?
Kinetic energy = ½ I x ω2
Δ Work (Joules) = Net external Torque x Θ
Homework : P156: Q13, 15, 18. P 281:28, 29, 31 and P282 Q43, 44, 45, 46

70. Baseball and torque…
Why does a batter choke up on the bat?
Page 155

71. Formulas for rotational inertia
See page 157 of the text book.

72. Which will roll down the slope faster?
See page 158.
The hoop will roll faster that has the least inertia.
Why?

73. Gymnastics and Inertia
What are the three principal axes of rotation of the human body? (page 159)
Each axis has a different rotational inertia.

74. Angular Momentum Angular momentum is a vector quantity
And the momentum is conserved.
A gyroscope swivels around, but the spin stays the same.
See page 161

75. How does angular momentum impact the balance of a bicycle rider?
See page 162
How does angular momentum impact the balance of a bicycle rider?

76. Conservation of Angular Momentum
Law of Conservation of Angular Momentum states:
If no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant.

77. Read page 163.
How does that apply to figure skating?

78. Why does a cat land on its feet (usually) when it falls?
Page 163

79. Space and Angular Momentum
Read the box on page 163.
How does angular momentum relate to the shape and speed of
rotation of a galaxy?