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Circular Motion; Gravitation

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Chapter 5 Circular Motion; Gravitation 5-8 Satellites and Weightlessness The satellite is kept in orbit by its speed it is continually falling, but the ... – PowerPoint PPT presentation

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Title: Circular Motion; Gravitation


1
Chapter 5 Circular Motion Gravitation
2
Units of Chapter 5
  • Kinematics of Uniform Circular Motion
  • Dynamics of Uniform Circular Motion
  • Highway Curves, Banked and Unbanked
  • Nonuniform Circular Motion
  • Centrifugation
  • Newtons Law of Universal Gravitation

3
Units of Chapter 5
  • Gravity Near the Earths Surface Geophysical
    Applications
  • Satellites and Weightlessness
  • Keplers Laws and Newtons Synthesis
  • Types of Forces in Nature

4
5-1 Kinematics of Uniform Circular Motion
Uniform circular motion motion in a circle of
constant radius at constant speed Instantaneous
velocity is always tangent to circle.
5
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

6
Centripetal Acceleration, cont.
  • a ?v (eq. I)
  • ?t
  • By similar triangles
  • ?v ?s
  • v r
  • Therefore
  • ?v ?s v
  • r
  • Sub into eq. I
  • a ?s v v2
  • r ?t r
  • Since ?s v
  • ?t

7
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
  • ac v2 (r?)2 r?2
  • r r

8
5-1 Kinematics of Uniform Circular Motion
This acceleration is called the centripetal, or
radial, acceleration, and it points towards the
center of the circle.
9
5-2 Dynamics of Uniform Circular Motion
We can see that the force must be inward by
thinking about a ball on a string
10
5-2 Dynamics of Uniform Circular Motion
For an object to be in uniform circular motion,
there must be a net force acting on it.
We already know the acceleration, so can
immediately write the force
SFr Fc mac mv2
r
11
5-2 Dynamics of Uniform Circular Motion
There is no centrifugal force pointing outward
what happens is that the natural tendency of the
object to move in a straight line must be
overcome. If the centripetal force vanishes, the
object flies off tangent to the circle.
12
5-3 Highway Curves, Banked and Unbanked
When a car goes around a curve, there must be a
net force towards the center of the circle of
which the curve is an arc. If the road is flat,
that force is supplied by friction.
FcFfriction
13
5-3 Highway Curves, Banked and Unbanked
If the frictional force is insufficient, the car
will tend to move more nearly in a straight line,
as the skid marks show.
14
4-8 Applications Involving Friction, Inclines
The static frictional force increases as the
applied force increases, until it reaches its
maximum. Then the object starts to move, and the
kinetic frictional force takes over.
15
5-3 Highway Curves, Banked and Unbanked
Banking the curve can help keep cars from
skidding. In fact, for every banked curve, there
is one speed where the entire centripetal force
is supplied by the
horizontal component of the normal force, and no
friction is required. This occurs when
16
5.2 Centripetal Acceleration
Example 3 The Effect of Radius on Centripetal
Acceleration The bobsled track contains turns
with radii of 33 m and 24 m. Find the
centripetal acceleration at each turn for a
speed of 34 m/s. Express answers as multiples
of
17
5.2 Centripetal Acceleration
18
5.3 Centripetal Force
Recall Newtons Second Law When a net external
force acts on an object of mass m, the
acceleration that results is directly
proportional to the net force and has a magnitude
that is inversely proportional to the mass. The
direction of the acceleration is the same as the
direction of the net force.
19
5.3 Centripetal Force
Thus, in uniform circular motion there must be a
net force to produce the centripetal
acceleration. The centripetal force is the name
given to the net force required to keep an
object moving on a circular path. The
direction of the centripetal force always points
toward the center of the circle and continually
changes direction as the object
moves. Centripetal force can be caused by,
tension, friction, or gravitational attraction.
In which case Fc T Fc Ffr Fc Fg
20
5.3 Centripetal Force
Example 5 The Effect of Speed on Centripetal
Force The model airplane has a mass of 0.90 kg
and moves at constant speed on a circle that is
parallel to the ground. The path of the airplane
and the guideline lie in the same horizontal
plane because the weight of the plane is
balanced by the lift generated by its wings.
Find the tension in the 17 m guideline for a
speed of 19 m/s.
21
5.4 Banked Curves
  • On an unbanked curve, the static frictional force
  • provides the centripetal force.
  • A car rounds a curve having a 100m radius
  • Travelling at 20m/s. What is the minimum
  • Coefficient of friction between the tires and
  • the road required?
  • Fc Ffr ?Fn
  • mv2 ?mg
  • r
  • ? v2 (20m/s)2
  • gr (9.8m/s2)(100m)
  • 0.41

Fn Ffr W mg
22
5.4 Banked Curves
On a frictionless banked curve, the centripetal
force is the horizontal component of the normal
force. The vertical component of the normal
force balances the cars weight.
23
5.4 Banked Curves
24
5.4 Banked Curves
25
5.4 Banked Curves
Example 8 The Daytona 500 The turns at the
Daytona International Speedway have a maximum
radius of 316 m and are steeply banked at
31 degrees. Suppose these turns were
frictionless. At what speed would the cars have
to travel around them?
26
5-6 Newtons Law of Universal Gravitation
If the force of gravity is being exerted on
objects on Earth, what is the origin of that
force?
Newtons realization was that the force must come
from the Earth. He further realized that this
force must be what keeps the Moon in its orbit.
27
5-6 Newtons Law of Universal Gravitation
The gravitational force on you is one-half of a
Third Law pair the Earth exerts a downward force
on you, and you exert an upward force on the
Earth. When there is such a disparity in masses,
the reaction force is undetectable, but for
bodies more equal in mass it can be significant.
28
5-6 Newtons Law of Universal Gravitation
Therefore, the gravitational force must be
proportional to both masses. By observing
planetary orbits, Newton also concluded that the
gravitational force must decrease as the inverse
of the square of the distance between the
masses. In its final form, the Law of Universal
Gravitation reads where
(5-4)
29
5-6 Newtons Law of Universal Gravitation
The magnitude of the gravitational constant G can
be measured in the laboratory.
This is the Cavendish experiment.
30
5-7 Gravity Near the Earths Surface Geophysical
Applications
Now we can relate the gravitational constant to
the local acceleration of gravity. We know that,
on the surface of the Earth Solving for g
gives Now, knowing g and the radius of the
Earth, the mass of the Earth can be calculated
(5-5)
31
Example
A 10kg mass and a 15kg mass are separated by
1.5m. Find the force of attraction between the
two masses.
32
Gravitational Force and Satellites
  • Orbiting objects are in free fall.
  • To see how this idea is true, we can use a
    thought experiment that Newton developed.
    Consider a cannon sitting on a high mountaintop.

Each successive cannonball has a greater
initial speed, so the horizontal distance that
the ball travels increases. If the initial speed
is great enough, the curvature of Earth will
cause the cannonball to continue falling without
ever landing.
33
5-8 Satellites and Weightlessness
Satellites are routinely put into orbit around
the Earth. The tangential speed must be high
enough so that the satellite does not return to
Earth, but not so high that it escapes Earths
gravity altogether.
34
5-8 Satellites and Weightlessness
The satellite is kept in orbit by its speed it
is continually falling, but the Earth curves from
underneath it.
35
5.5 Satellites in Circular Orbits
There is only one speed that a satellite can have
if the satellite is to remain in an orbit with a
fixed radius.
36
5.5 Satellites in Circular Orbits
Fg Fc
37
5.5 Satellites in Circular Orbits
Example 9 Orbital Speed of the Hubble Space
Telescope Determine the speed of the Hubble
Space Telescope orbiting at a height of 598 km
above the earths surface.
38
5.5 Satellites in Circular Orbits
4?2r3 GM
39
5-9 Keplers Laws and Newton's Synthesis
  • Keplers laws describe planetary motion.
  • The orbit of each planet is an ellipse, with the
    Sun at one focus.

40
5-9 Keplers Laws and Newton's Synthesis
2. An imaginary line drawn from each planet to
the Sun sweeps out equal areas in equal times.
41
Keplers Third Law
  • The square of the orbital period of any planet is
    proportional to cube of the average distance from
    the Sun to the planet.
  • T circumference of orbit
  • orbital speed
  • For orbit around the Sun, KS 2.97x10-19 s2/m3
  • K is independent of the mass of the planet
  • K 4?2
  • GM
  • Example A planet is in orbit 109 meters from the
    center of the sun. Calculate its orbital period
    and velocity.
  • Ms1.991 x 1030 kg

42
5-9 Keplers Laws and Newton's Synthesis
The ratio of the square of a planets orbital
period is proportional to the cube of its mean
distance from the Sun.
43
5.5 Satellites in Circular Orbits
44
5.5 Satellites in Circular Orbits
Global Positioning System
4?2r3 GM
T (24 hours)(3600s/hour) 86400s
T2GMe 4?2
r
3
r (86400s)2(6.67 x 10-11 Nm2/kg2)(5.98 x
1024kg) 4?2
3
r 42250474m distance from center of the
earth to GPS r Re h ? h r Re
42250474m 6380000m 35870474m

22,300 mi
45
5.6 Apparent Weightlessness and Artificial Gravity
Example 13 Artificial Gravity At what speed
must the surface of the space station move so
that the astronaut experiences a push on his feet
equal to his weight on earth? The radius is
1700 m.
130 m/s
46
5.7 Vertical Circular Motion
47
Example
A satellite orbits the earth at an altitude of
1000km. What must the velocity of the satellite
be in order for it to maintain a circular orbit.
Once in circular orbit, what happens if something
causes the satellite to speed up or slow down.
48
5-8 Satellites and Weightlessness
Objects in orbit are said to experience
weightlessness. They do have a gravitational
force acting on them, though! The satellite and
all its contents are in free fall, so there is no
normal force. This is what leads to the
experience of weightlessness.
49
5-10 Types of Forces in Nature
  • Modern physics now recognizes four fundamental
    forces
  • Gravity
  • Electromagnetism
  • Weak nuclear force (responsible for some types
    of radioactive decay)
  • Strong nuclear force (binds protons and neutrons
    together in the nucleus)

50
5-10 Types of Forces in Nature
So, what about friction, the normal force,
tension, and so on? Except for gravity, the
forces we experience every day are due to
electromagnetic forces acting at the atomic level.
51
Summary of Chapter 5
  • An object moving in a circle at constant speed
    is in uniform circular motion.
  • It has a centripetal acceleration
  • There is a centripetal force given by
  • The centripetal force may be provided by
    friction, gravity, tension, the normal force, or
    others.

52
Summary of Chapter 5
  • Newtons law of universal gravitation
  • Satellites are able to stay in Earth orbit
    because of their large tangential speed.

53
5-4 Nonuniform Circular Motion
If an object is moving in a circular path but at
varying speeds, it must have a tangential
component to its acceleration as well as the
radial one.
54
5-4 Nonuniform Circular Motion
This concept can be used for an object moving
along any curved path, as a small segment of the
path will be approximately circular.
55
5-5 Centrifugation
A centrifuge works by spinning very fast. This
means there must be a very large centripetal
force. The object at A would go in a straight
line but for this force as it is, it winds up at
B.
56
5-7 Gravity Near the Earths Surface Geophysical
Applications
The acceleration due to gravity varies over the
Earths surface due to altitude, local geology,
and the shape of the Earth, which is not quite
spherical.
57
5-8 Satellites and Weightlessness
More properly, this effect is called apparent
weightlessness, because the gravitational force
still exists. It can be experienced on Earth as
well, but only briefly
58
5-9 Keplers Laws and Newton's Synthesis
Keplers laws can be derived from Newtons laws.
Irregularities in planetary motion led to the
discovery of Neptune, and irregularities in
stellar motion have led to the discovery of many
planets outside our Solar System.
59
5.6 Apparent Weightlessness and Artificial Gravity
Conceptual Example 12 Apparent Weightlessness
and Free Fall In each case, what is the weight
recorded by the scale?
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