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Nicholas J. Giordano
www.cengage.com/physics/giordano
Circular Motion and Gravitation
Introduction
• Circular motion
• Acceleration is not constant
• Cannot be reduced to a one-dimensional problem
• Examples
• Car traveling around a turn
• Parts of the motion of a roller coaster
• Centrifuge
• The Earth orbiting the Sun
• Gravitation
• Explore gravitational force in more detail
• Look at Kepler’s Laws of Motion
• Further details about g
Introduction
Uniform Circular Motion – Overview
• Circular motion is an example of accelerated motion
• It can be analyzed in terms of acceleration and
forces
• A problem-solving strategy can be applied in a
manner similar to one- and two-dimensional motion
approaches
Section 5.1
Uniform Circular Motion
• Assume constant speed
• The direction of the
velocity is continually
changing
• The vector is always
tangent to the circle
• Uniform circular motion
is circular motion at
constant speed
Section 5.1
Uniform Circular Motion, cont.
• Examine one trip around the track
• The distance traveled is the circumference of the track
• The period of the motion, T is
• r is the radius of the track
• v is the speed of the motion
• The period does not depend on the location on the object
• The speed depends on the radius of the circle of motion
and so depends on location
• Speeds are fastest at the outside edge
Section 5.1
Centripetal Acceleration
• Although the speed is
constant, the velocity is
not constant
• Direction of acceleration
• Always directed toward
the center of the circle
• This is called the
centripetal acceleration
•
Centripetal means “centerseeking”
• Magnitude of acceleration
Section 5.1
Circular Motion and Forces
• Newton’s Second Law can be applied to circular
motion:
• The force must be directed toward the center of the
circle
• The centripetal force can be supplied by a variety of
physical objects or forces
• The “circle” does not need to be a complete circle
Section 5.1
Centripetal Force Example
• The centripetal
acceleration is
produced by the tension
in the string
• If the string breaks, the
object would move in a
direction tangent to the
circle at a constant
speed
Section 5.1
Problem Solving Strategy –
Circular Motion
• Recognize the principle
• If the object moves in a circle, then there is a
centripetal force acting on it
• Sketch the problem
• Show the path the object travels
• Identify the circular part of the path
• Include the radius of the circle
• Show the center of the circle
• Selecting a coordinate system that assigns the
positive direction toward the center of the circle is
often convenient
Section 5.1
Problem Solving Strategy, cont.
• Identify the principles
• Find all the forces acting on the object
•
A free body diagram is generally useful
• Find the components of the forces that are directed
toward the center of the circle
• Find the components of the forces perpendicular to the
center
• Apply Newton’s Second Law for both directions
•
The acceleration directed toward the center of the circle is a
centripetal acceleration
Section 5.1
Problem Solving Strategy, final
• Solve for the quantities of interest
• Check your answer
• Consider what the answer means
• Does the answer make sense
Section 5.1
Centripetal Acceleration Example: Car
• A car rounding a curve
travels in an approximate
circle
• The radius of this circle is
called the radius of
curvature
• Forces in the y-direction
• Gravity and the normal
force
• Forces in the x-direction
• Friction is directed toward
the center of the circle
Section 5.1
Car Example, cont.
• Since friction is the only force acting in the x-
direction, it supplies the centripetal force
• Solving for the maximum velocity at which the car
can safely round the curve gives
Section 5.1
Example: Car on Banked Curve
• The maximum speed can
be increased by banking
the curve
• Assume no friction
between the tires and the
road
• The car travels in a circle,
so the net force is a
centripetal force
• There are forces due to
gravity and the normal
force acting on the car
Section 5.1
Banked Curve, cont.
• There is a horizontal component of the normal force
• Letting the horizontal be the x-direction
• The speed at which the car will just be able to
negotiate the turn without sliding up or down the
banked road is
• When θ = 0, v = 0 and you cannot turn on a very icy
unbanked road without slipping
Section 5.1
Examples of Circular Motion
• When the motion is uniform, the total acceleration is
the centripetal acceleration
• Remember, this means that the speed is constant
• The motion does not need to be uniform
• Then there will be a tangential acceleration included
• Many examples can be analyzed by looking at the
two components
Section 5.2
Non-Uniform Circular Motion
• If the speed is also
changing, there are two
components to the
acceleration
• One component is
tangent to the circle, at
• The other component is
directed toward the
center of the circle, ac
Section 5.2
Circular Motion Example:
Vertical Circle
• The speed of the rock
varies with time
• At the bottom of the
circle:
• Tension and gravity are
in opposite directions
•
•
The tension supports the
rock (mg) and supplies
the centripetal force
Section 5.2
Vertical Circle Example, cont.
• At the top of the circle:
• Tension and gravity are
in the same direction
•
Pointing toward the
center of the circle
•
Section 5.2
Vertical Circle Example, Final
• There is a minimum value of v needed to keep the
string taut at the top
• Let Ttop = 0
•
•
If the speed is smaller than this, the string will become slack
and circular motion is no longer possible
Circular Motion Example: Roller Coaster
• The roller coaster’s path
is nearly circular at the
minimum or maximum
points on the track
• There is a maximum
speed at which the
coaster will not leave the
top of the track:
•
• If the speed is greater
than this, N would have
to be negative
• This is impossible, so the
coaster would leave the
track
Section 5.2
Circular Motion Example:
Artificial Gravity
• Circular motion can be
used to create “artificial
gravity”
• The normal force acting
on the passengers due to
the floor would be
• If N = mg it would feel like
the passengers are
experiencing normal Earth
gravity
Section 5.2
Circular Motion Example: Centrifuge
• A centrifuge is a device
used in many laboratories
• It can be used to separate
particles or molecules
• Or remove them
• The effective force causes
the particle to move to the
bottom of the test tube
• Similar to artificial gravity
• To someone outside of the
test tube, the particle
appears to spiral
Section 5.2
Frames of Reference and the Centrifuge
• The stationary observer is in an inertial reference
frame
• He can apply Newton’s Second Law
• His interpretation is correct
• There is no actual force acting on the particle
• The force FAG is a fictitious force
• The observer moving with the particle is in an
accelerated frame
• He cannot apply Newton’s Second Law
• He thinks there is a force acting on the particle
• They agree on the particle’s motion
Section 5.2
Newton’s Law of Gravitation
• In many cases, the orbits of planets and moons are
approximately circular
• The Law of Gravitation plays a key role in physics
• It allows us to calculate and understand the motion of
a wide variety of objects
• Newton’s application of his law of gravitation to
motion of planets and moons was the first time physics
was successfully applied to describe the motion of the
solar system
•
•
Showed that the laws of physics apply to all objects
Had an effect on how people viewed the universe
Section 5.3
Newton’s Law of Gravitation, Equation
• Law states: There is a
gravitational attraction
between any two
objects. If the objects
are point masses m1
and m2, separated by a
distance r the
magnitude of the force
is
Section 5.3
Law of Gravitation, cont.
• Note that r is the distance between the objects
• G is the Universal Gravitational Constant
• G = 6.67 x 10-11 N . m2/ kg2
• The gravitational force is always attractive
• Every mass attracts every other mass
• The gravitational force is symmetric
• The magnitude of the gravitational force exerted by
mass 1 on mass 2 is equal in magnitude to the force
exerted by mass 2 on mass 1
• The two forces form an action-reaction pair
Section 5.3
Gravitation and the Moon’s Orbit
• The Moon follows an
approximately circular
orbit around the Earth
• There is a force
required for this motion
• Gravity supplies the
force
Section 5.3
Notes on the Moon’s Motion
• We assumed the Moon orbits a “fixed” Earth
• It is a good approximation
• It ignores the Earth’s motion around the Sun
• The Earth and Moon actually both orbit their center
of mass
• We can think of the Earth as orbiting the Moon
• The circle of the Earth’s motion is very small
compared to the Moon’s orbit
Section 5.3
Gravitation and g
• Assuming a spherical
Earth, we can consider
all the mass of the Earth
to be concentrated at its
center
• The value of r in the
Law of Gravitation is
just the radius of the
Earth
Section 5.3
Gravitation and g, cont.
• Since the weight of a person is also the gravitational
force between the person and the Earth, we can find
the value of g:
• The value of g is a function only of the Earth’s mass
and radius, and the value of G
Section 5.3
Gravitation Force From The Earth –
Assumptions
• We assumed
• A spherical Earth
• That the gravitational force could be calculated as if all
the mass of the Earth was located at its center
• Assumptions are true as long as the density of the
object is spherically symmetric
• The object has a constant density
• The object’s density varies with depth as long as the
density depends only on the distance from the center
Section 5.3
Measuring G
• Henry Cavendish
measured the force of
gravity between two
large lead spheres
• By an experimental setup similar to the picture,
he was able to
determine the value of
G
Section 5.3
More About Gravity – Newton’s Apple
• G is a constant of nature
• Gravity is a weak force
• Much smaller than typical normal or tension forces
• Gravity is considered the weakest of the
fundamental forces of nature
• Newton showed the motion of celestial bodies and
the motion of terrestrial motion are caused by the
same force and governed by the same laws of
motion
Section 5.3a
Johannes Kepler
• Kepler studied results of other astronomer’s
measurements of portions of the Moon, planets, etc.
• He found the motion of the Moon and planets could be
described by a series of laws
• Now called Kepler’s Laws of Planetary Motion
• Kepler’s Laws are mathematical rules inferred from the
available information about the motion in the solar
system
• Kepler could not give a scientific explanation or derivation
of his laws
• Newton’s Laws of Motion and Gravitation give the
explanation
Section 5.4
Kepler’s First Law of Planetary Motion
• Planets move in
elliptical orbits
• The Sun is at one focus
• This was very different
from the previous idea
that the planets moved
in perfect circles with
the Sun at the center
Section 5.4
Kepler’s First Law of Planetary
Motion, Planetary Orbits
Section 5.4
Kepler’s Second Law of Planetary Motion
• A line connecting a planet
to its sun sweeps out
equal areas in equal times
as the planet moves
around its orbit
• If the time required for the
planet to sweep out area
A1 is equal to the time to
sweep out A2, the areas
will be equal
• The planet’s speed will
be slowest when it is
farthest from its sun and
fastest when it is closest
Section 5.4
Kepler’s Third Law of Planetary Motion
• The square of the period of an orbit is proportional to
the cube of the orbital radius
• For simplicity, apply to a circular orbit
• Period is T and the gravitational force supplies a
centripetal force
• Also applies to satellites with MSun replaced by Mplanet
Section 5.4
Orbit Examples
• Low Earth orbit
• International Space Station, for example
• T ~ 90 minutes
• r ~ 6.66 x 106 m
• Geosynchronous Orbit
• T = 1 day
•
Always above the same position above the Earth
• r = 4.2 x 107 m
Section 5.4
Kepler’s Laws and Orbits, Summary
• Kepler’s Three Laws of planetary motion apply to all
types of gravitationally produced orbital motion
• Different motions result from different ways of initially
setting objects into motion
• For example, launching a satellite to the east into an
equatorial orbit takes advantage of the speed of the
Earth’s rotation
Section 5.4
Tides
• Tides are the fluctuations of the level of the Earth’s
oceans
• Tides are due to the Moon’s gravitational force on
the oceans
• Also exerts a force on the solid Earth
• The bulge in the ocean is due to the slightly greater
force acting on the side of the Earth facing the Moon
• Due to the dependence of gravity on distance
• The bulge is a high tide
Section 5.5
Tides, cont.
• One high tide occurs
when the Moon is directly
overhead
• The acceleration of the
water closest to the Moon
is more than that of the
rest of the Earth
• Twelve hours later the
acceleration of the solid
Earth is more than the
ocean water on the farther
side
• There are generally two
high tides per day, 12
hours apart
Section 5.5
Tides, final
• The Sun also affects tides
• Its effects are smaller than the Moon’s
• When the Sun and the Moon are aligned and on the
same of the Earth, the tide is higher than when
produced by the Moon alone
Section 5.5
Inverse Square Law
• Newton’s Law of Gravitation is an example of what
is called in mathematics an inverse square law
• A number of other forces in nature are also inverse
square laws
• Force between two electric charges, for example
• How can you explain why so many forces follow this
pattern?
Section 5.6
Field Lines
• Imagine that an object
possesses gravitational
field lines that emanate
from it
• Also imagine that the
number of lines is
proportional to the mass
• When the lines intersect
another object, there is a
force on that object
directed parallel to the
lines
Section 5.6
Field Lines, cont.
• Since the force lines emanate in three-dimensional
space, the number of lines that intercept the second
object falls off with distance as 1/r2
• The result implies that gravity follows an inverse
square law because we live in three-dimensional
space
• It also means we should expect other forces
described by the field line picture to have the same
inverse square dependence
Section 5.6
Field Lines, final
• No one has devised a way to see the line
• You can observe the resulting force that the lines are
presumed to cause
• Gravitational force is also an example of “action at a
distance”
• Newton’s Law of Gravitation tells us that action at a
distance does occur
• The Law doesn’t tell us how it occurs
• The field line picture was invented to answer this
“how” question
Section 5.6
Mass
• The mass in the gravitational force is sometimes
called gravitational mass
• The mass in Newton’s Second Law of Motion is
called inertial mass
• The gravitational mass of an object is exactly equal
to its inertial mass
• Needed the theories of relativity to explain why they
were equal
• Also tells us that a photon of light will be accelerated
by gravity
•
Even though it is a massless particle
Section 5.6