1. Circular Motion, Gravitation, Rotation, Bodies in Equilibrium
Chapter 6,9,10
Circular Motion, Gravitation, Rotation, Bodies in Equilibrium

2. Circular Motion Ball at the end of a string revolving
Planets around Sun
Moon around Earth

3. The Radian The radian is a unit of angular measure
The radian can be defined as the arc length s along a circle divided by the radius r
57.3°

4. More About Radians Comparing degrees and radians
Converting from degrees to radians

5. Angular Displacement Axis of rotation is the center of the disk
Need a fixed reference line
During time t, the reference line moves through angle θ

6. Angular Displacement, cont.
The angular displacement is defined as the angle the object rotates through during some time interval
The unit of angular displacement is the radian
Each point on the object undergoes the same angular displacement

7. Average Angular Speed
The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval

8. Angular Speed, cont. The instantaneous angular speed
Units of angular speed are radians/sec
rad/s
Speed will be positive if θ is increasing (counterclockwise)
Speed will be negative if θ is decreasing (clockwise)

9. Average Angular Acceleration
The average angular acceleration of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:

10. Angular Acceleration, cont
Units of angular acceleration are rad/s²
Positive angular accelerations are in the counterclockwise direction and negative accelerations are in the clockwise direction
When a rigid object rotates about a fixed axis, every portion of the object has the same angular speed and the same angular acceleration

11. Angular Acceleration, final
The sign of the acceleration does not have to be the same as the sign of the angular speed
The instantaneous angular acceleration

12. Analogies Between Linear and Rotational Motion
Linear Motion with constant acc.
(x,v,a)
Rotational Motion with fixed axis
and constant a
(q,,a)

13. Examples 78 rev/min=? A fan turns at a rate of 900 rpm
Tangential speed of tips of 20cm long blades?
Now the fan is uniformly accelerated to 1200 rpm in 20 s

14. Relationship Between Angular and Linear Quantities
Displacements
Speeds
Accelerations
Every point on the rotating object has the same angular motion
Every point on the rotating object does not have the same linear motion

15. Centripetal Acceleration
An object traveling in a circle, even though it moves with a constant speed, will have an acceleration
The centripetal acceleration is due to the change in the direction of the velocity

16. Centripetal Acceleration, cont.
Centripetal refers to “center-seeking”
The direction of the velocity changes
The acceleration is directed toward the center of the circle of motion

17. Centripetal Acceleration, final
The magnitude of the centripetal acceleration is given by
This direction is toward the center of the circle

18. Centripetal Acceleration and Angular Velocity
The angular velocity and the linear velocity are related (v = ωR)
The centripetal acceleration can also be related to the angular velocity

19. Forces Causing Centripetal Acceleration
Newton’s Second Law says that the centripetal acceleration is accompanied by a force
F = ma 
F stands for any force that keeps an object following a circular path
Tension in a string
Gravity
Force of friction

20. Examples
Ball at the end of revolving string
Fast car rounding a curve

21. More on circular Motion
Length of circumference = 2R
Period T (time for one complete circle)

22. Example
200 grams mass revolving in uniform circular motion on an horizontal frictionless surface at 2 revolutions/s. What is the force on the mass by the string (R=20cm)?

23. Newton’s Law of Universal Gravitation
Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.

24. Universal Gravitation, 2
G is the constant of universal gravitational
G = x N m² /kg²
This is an example of an inverse square law

25. Universal Gravitation, 3
The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1
The forces form a Newton’s third law action-reaction

26. Universal Gravitation, 4
The gravitational force exerted by a uniform sphere on a particle outside the sphere is the same as the force exerted if the entire mass of the sphere were concentrated on its center

27. Gravitation Constant Determined experimentally Henry Cavendish
1798
The light beam and mirror serve to amplify the motion

29. Applications of Universal Gravitation
Weighing the Earth

31. Applications of Universal Gravitation
“g” will vary with altitude

32. Escape Speed
The escape speed is the speed needed for an object to soar off into space and not return
For the earth, vesc is about 11.2 km/s
Note, v is independent of the mass of the object

33. Various Escape Speeds
The escape speeds for various members of the solar system
Escape speed is one factor that determines a planet’s atmosphere

34. Motion of Satellites Consider only circular orbit Radius of orbit r:
Gravitational force is the centripetal force.

35. Motion of Satellites
Period 
Kepler’s 3rd Law

36. Communications Satellite
A geosynchronous orbit
Remains above the same place on the earth
The period of the satellite will be 24 hr
r = h + RE
Still independent of the mass of the satellite

37. Satellites and Weightlessness
weighting an object in an elevator
Elevator at rest: mg
Elevator accelerates upward: m(g+a)
Elevator accelerates downward: m(g+a) with a<0
Satellite: a=-g!!

40. Force vs. Torque Forces cause accelerations
Torques cause angular accelerations
Force and torque are related

41. Torque The door is free to rotate about an axis through O
There are three factors that determine the effectiveness of the force in opening the door:
The magnitude of the force
The position of the application of the force
The angle at which the force is applied

42. Torque, cont
Torque, t, is the tendency of a force to rotate an object about some axis
t is the torque
F is the force
symbol is the Greek tau
l is the length of lever arm
SI unit is N.m
Work done by torque W=

43. Direction of Torque
If the turning tendency of the force is counterclockwise, the torque will be positive
If the turning tendency is clockwise, the torque will be negative

44. Multiple Torques
When two or more torques are acting on an object, the torques are added
If the net torque is zero, the object’s rate of rotation doesn’t change

45. Torque and Equilibrium
First Condition of Equilibrium
The net external force must be zero
This is a necessary, but not sufficient, condition to ensure that an object is in complete mechanical equilibrium
This is a statement of translational equilibrium

46. Torque and Equilibrium, cont
To ensure mechanical equilibrium, you need to ensure rotational equilibrium as well as translational
The Second Condition of Equilibrium states
The net external torque must be zero

47. Equilibrium Example
The woman, mass m, sits on the left end of the see-saw
The man, mass M, sits where the see-saw will be balanced
Apply the Second Condition of Equilibrium and solve for the unknown distance, x

48. Moment of Inertia
The angular acceleration is inversely proportional to the analogy of the mass in a rotating system
This mass analog is called the moment of inertia, I, of the object
SI units are kg m2

49. 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

50. 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

51. 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

52. Other Moments of Inertia

53. Example
Wheel of radius R=20 cm and I=30kg·m². Force F=40N acts along the edge of the wheel.
Angular acceleration?
Angular speed 4s after starting from rest?
Number of revolutions for the 4s?
Work done on the wheel?

54. Rotational Kinetic Energy
An object rotating about some axis with an angular speed, ω, has rotational kinetic energy KEr=½Iω2
Energy concepts can be useful for simplifying the analysis of rotational motion
Units (rad/s)!!

55. Total Energy of a System
Conservation of Mechanical Energy
Remember, this is for conservative forces, no dissipative forces such as friction can be present
Potential energies of any other conservative forces could be added

56. Rolling down incline Energy conservation
Linear velocity and angular speed are related v=R
Smaller I, bigger v, faster!!

57. Work-Energy in a Rotating System
In the case where there are dissipative forces such as friction, use the generalized Work-Energy Theorem instead of Conservation of Energy
(KEt+KER+PE)i+W=(KEt+KER+PE)f

58. Angular Momentum
Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum
Angular momentum is defined as
L = I ω
and

59. Angular Momentum, cont
If the net torque is zero, the angular momentum remains constant
Conservation of Angular Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero.
That is, when

60. Conservation Rules, Summary
In an isolated system, the following quantities are conserved:
Mechanical energy
Linear momentum
Angular momentum

61. Conservation of Angular Momentum, Example
With hands and feet drawn closer to the body, the skater’s angular speed increases
L is conserved, I decreases, w increases

64. Example
A 500 grams uniform sphere of 7.0 cm radius spins at 30 rev/s on an axis through its center.
Moment of inertia
Rotational kinetic energy
Angular momentum

65. Example
Find work done to open 30 a 1m wide door with a steady force of 0.9N at right angle to the surface of the door.

66. Example
A turntable is a uniform disk of metal of mass 1.5 kg and radius 13 cm. What torque is required to drive the turntable so that it accelerates at a constant rate from 0 to 33.3 rpm in 2 seconds?

67. Center of Gravity
The force of gravity acting on an object must be considered
In finding the torque produced by the force of gravity, all of the weight of the object can be considered to be concentrated at a single point

68. Calculating the Center of Gravity
The object is divided up into a large number of very small particles of weight (mg)
Each particle will have a set of coordinates indicating its location (x,y)

69. Calculating the Center of Gravity, cont.
We wish to locate the point of application of the single force whose magnitude is equal to the weight of the object, and whose effect on the rotation is the same as all the individual particles.
This point is called the center of gravity of the object

70. Coordinates of the Center of Gravity
The coordinates of the center of gravity can be found

71. Center of Gravity of a Uniform Object
The center of gravity of a homogenous, symmetric body must lie on the axis of symmetry.
Often, the center of gravity of such an object is the geometric center of the object.

72. Example
Find the center of mass (gravity) of these masses: 3kg (0,1), 2kg (0,0)
And 1kg (2,0)

73. Example
Find the center of mass (gravity) of the dumbbell, 4 kg and 2 kg with a 4m long 3kg rod.

74. Torque, review SI unit is N.m t is the torque F is the force
symbol is the Greek tau
l is the length of lever arm
SI unit is N.m

75. Direction of Torque
If the turning tendency of the force is counterclockwise, the torque will be positive
If the turning tendency is clockwise, the torque will be negative

76. Multiple Torques
When two or more torques are acting on an object, the torques are added
If the net torque is zero, the object’s rate of rotation doesn’t change

77. Example
A 2 m by 2 m square metal plate rotates about its center. Calculate the torque of all five forces each with magnitude 50N.

79. Torque and Equilibrium
First Condition of Equilibrium
The net external force must be zero
The Second Condition of Equilibrium states
The net external torque must be zero

80. Example
The system is in equilibrium. Calculate W and find the tension in the rope (T).

81. Example
A 160 N boy stands on a 600 N concrete beam in equilibrium with two end supports. If he stands one quarter the length from one support, what are the forces exerted on the beam by the two supports?