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![Page 1: Nicholas J. Giordano - University of Massachusetts Lowellfaculty.uml.edu/arthur_mittler/Teaching/chapter5.pdf · Uniform Circular Motion, ... Examples of Circular Motion ... • It](https://reader031.fdocuments.net/reader031/viewer/2022030502/5aaec7877f8b9a6b308c804d/html5/thumbnails/1.jpg)
Nicholas J. Giordano
www.cengage.com/physics/giordano
Circular Motion and Gravitation
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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
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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
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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
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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
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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 “center-
seeking” • Magnitude of acceleration
Section 5.1
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Measuring G
• Henry Cavendish measured the force of gravity between two large lead spheres
• By an experimental set-up similar to the picture, he was able to determine the value of G
Section 5.3
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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
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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
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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
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Section 5.4
Kepler’s First Law of Planetary Motion, Planetary Orbits
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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
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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
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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
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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
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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
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• 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, cont.
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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
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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
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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
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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
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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
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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