# Circular Motion, Gravitation, Rotation, Bodies in Equilibrium - PowerPoint PPT Presentation

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## Circular Motion, Gravitation, Rotation, Bodies in Equilibrium

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### Chapter 6,9,10 Circular Motion, Gravitation, Rotation, Bodies in Equilibrium Example A turntable is a uniform disk of metal of mass 1.5 kg and radius 13 cm. – PowerPoint PPT presentation

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Title: Circular Motion, Gravitation, Rotation, Bodies in Equilibrium

1
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 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
• 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 ?

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, ?, 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
• Speed will be positive if ? is increasing
(counterclockwise)
• Speed will be negative if ? 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 ?R)
• The centripetal acceleration can also be related
to the angular velocity

19
Forces Causing Centripetal Acceleration
• Newtons 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 (R20cm)?

23
Newtons 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 6.673 x 10-11 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 Newtons 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

28
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29
Applications of Universal Gravitation
• Weighing the Earth

30
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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
planets atmosphere

34
Motion of Satellites
• Consider only circular orbit
• Gravitational force is the centripetal force.

35
Motion of Satellites
• Period ?

Keplers 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(ga)
• Elevator accelerates downward m(ga) with alt0
• Satellite a-g!!

38
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39
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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
• 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,
• If the net torque is zero, the objects rate of
rotation doesnt 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
Newtons 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
• 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 R20 cm and I30kgm². Force
F40N 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, ?, has rotational kinetic energy
KEr½I?2
• Energy concepts can be useful for simplifying the
analysis of rotational motion

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

56
Rolling down incline
• Energy conservation
• Linear velocity and angular speed are related
vR?
• 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
• (KEtKERPE)iW(KEtKERPE)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 ?
• 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
skaters angular speed increases
• L is conserved, I decreases, w increases

62
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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
• 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,
• If the net torque is zero, the objects rate of
rotation doesnt 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.

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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?

82
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