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PreClass Notes: Chapter 8

• From Essential University Physics 3rd Edition

• by Richard Wolfson, Middlebury College

• ©2016 by Pearson Education, Inc.

• Narration and extra little notes by Jason Harlow,

University of Toronto

• This video is meant for University of Toronto

students taking PHY131.

© 2012 Pearson Education, Inc. Slide 1-2

Toward a Law of Gravity

• Newton was not the first to

discover gravity. Newton

discovered that gravity is

universal.

• Legend: Newton, sitting

under an apple tree, realizes

that the Earth’s pull on an

apple extends also to pull on

the Moon.

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Outline

“Newton’s genius was to

recognize that the motion of the

apple and the motion of the

Moon were the same, that both

were ‘falling’ toward Earth under

the influence of the same force.”

– R.Wolfson

Image of the Moon from http://www.salon.com/2014/07/18/nasa_believes_caves_on_the_moon_could_shelter_astronauts/ ]

• 8.1,8.2 Newton’s Law of

Universal Gravitation

• 8.3 Orbital Motion

• 8.4 Gravitational Potential

Energy

• 8.5 The Gravitational Field

© 2012 Pearson Education, Inc. Slide 1-4

Toward a Law of Gravity

• In Aristotle’s time, motion of planets and stars in

the heavens was not expected to be governed by

the same laws as objects on Earth.

• Newton recognized that a force directed toward

the Sun must act on planets

– This is similar to force that Earth exerts on an apple

that falls toward it.

• Newtonian synthesis: The same set of laws

apply to both celestial and terrestrial objects.

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© 2012 Pearson Education, Inc. Slide 1-5

Universal Gravitation

Law of universal gravitation:

• Everything pulls on everything else.

• Every body attracts every other body with a

force that is directly proportional to the product

of their masses and inversely proportional to the

square of the distance separating them.

© 2012 Pearson Education, Inc. Slide 1-6

Universal Gravitation

F

Gm1m

2

r 2

• Here G = 6.6710–11 N·m2/kg2 is the constant of universal

gravitation.

• Newton invented calculus to show that this law applies to

spherical masses using the centre-to-centre distance for r.

• Introduced by Isaac Newton, the Law of

Universal Gravitation states that any two

masses m1 and m2 attract with a force F that is

proportional to the product of their distances

and inversely proportional to the distance r

between them.

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© 2012 Pearson Education, Inc. Slide 1-7

Inverse Square Law

1 16N

If the masses of two planets are each somehow

doubled, the force of gravity between them

A. doubles.

B. quadruples.

C. reduces by half.

D. reduces by one-quarter.

Got it?

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Fast moving projectiles: Satellites!

• Satellite motion is an example of

a high-speed projectile.

• A satellite is simply a projectile

that falls around Earth rather

than into it.

Images from https://en.wikipedia.org/wiki/GPS_%28satellite%29#/media/File:Navstar-2F.jpg and https://upload.wikimedia.org/wikipedia/commons/8/86/GPS-IIRM.jpg

• Sufficient tangential

velocity is needed for orbit.

• With no air drag to reduce

speed, a satellite goes

around Earth indefinitely.

Orbits

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Orbits

• The downward acceleration due to gravity is g ≈ 10 m/s2.

• t = 1 second after a ball is thrown horizontally, it has fallen a

distance

• No matter how fast the girl throws ball sideways, 1 second

later it has fallen 5 m below the horizontal line

Orbits

Orbits

Curvature of Earth

• Earth surface drops a vertical distance of 5

meters for every 8000 meters tangent to the

surface

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Orbits

What speed will allow the ball to clear the gap?

8000 m per second: 8 km/s!

Kepler’s Laws of Planetary Motion

1st Law: The path of each planet

around the Sun is an ellipse

with the Sun at one focus.

2nd Law: The line from the Sun to

any planet sweeps out equal

areas of space in equal time

intervals.

3rd Law: The square of the orbital

period of a planet is directly

proportional to the cube of the

average distance of the planet

from the Sun (for all planets).

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Orbits

Ellipse

• specific curve, an oval path

Example: A circle is a special case of an

ellipse when its two foci coincide.

Projectile Motion and Orbits

• The “parabolic” trajectories of

projectiles near Earth’s surface are

actually sections of elliptical orbits

that intersect Earth.

• The trajectories are parabolic only

in the approximation that we can

neglect Earth’s curvature and the

variation in gravity with distance

from Earth’s center.

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• In a circular orbit, gravity provides the force of

magnitude mv2/r needed to keep an object of mass m in

its circular path about a much more massive object of

mass M. Therefore,

• Orbital speed:

• Orbital period:

– Kepler’s third law:

– For satellites in low-Earth orbit, the period is about 90

minutes.

Circular Orbits

• Because the gravitational force changes with distance, it’s

necessary to integrate to calculate potential energy

changes over large distances.

• This integration gives

Gravitational Potential Energy

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• It’s convenient to take the zero of gravitational potential

energy at infinity. Then the gravitational potential energy

becomes

Gravitational Potential Energy

( ) =GMm

U rr

• This result holds regardless of whether the two points are

on the same radial line.

Gravitational Potential Energy

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• The total energy E = K + U determines the type of

orbit an object follows:

• E < 0: The object is in a bound, elliptical orbit.

– Special cases include circular orbits and the

straight-line paths of falling objects.

• E > 0: The orbit is unbound and hyperbolic.

• E = 0: The borderline case gives a parabolic orbit.

Energy and Orbits

Energy and Orbits

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Got it?

• Suppose the paths in the figure are the paths of four

projectiles. All four projectiles were launched from a

common point at the top of the figure. Which projectile

had the second-highest initial speed?

A. The projectile with the

closed path.

B. The projectile with the

hyperbolic path.

C. The projectile with the

parabolic path.

D. The projectile with the

elliptical path.

Escape Speed

• An object with total energy E less than zero is in a bound

orbit and can’t escape from the gravitating center.

• With energy E greater than zero, the object is in an unbound

orbit and can escape to infinitely far from the gravitating

center.

• The minimum speed required to escape is given by

• Solving for v gives the escape speed:

– Escape speed from Earth’s surface is about 11 km/s.

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Energy in Circular Orbits

• In the special case of a circular orbit, kinetic energy and

potential energy are precisely related:

U 2K• Thus in a circular orbit the total energy is

E K U K 1

2U

GMm

2r

– This negative energy shows that the orbit is bound.

– The lower the orbit, the lower the total energy—but the

faster the orbital speed.

• This means an orbiting spacecraft needs to lose energy to

gain speed.

Got it?

• A moon is orbiting around Planet X. Which of the

following statements is always true about its kinetic

energy (K), and its gravitational potential energy (U)?

A. K < 0 and U < 0

B. K < 0 and U > 0

C. K > 0 and U < 0

D. K > 0 and U > 0

E. K < 0 and U = 0

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The Gravitational Field

Fields are represented by field

lines radiating into the object

(Earth).

• The inward direction of arrows

indicates that the force is always

attractive to Earth.

• The crowding of arrows closer to

Earth indicates that the

magnitude of the force is larger

closer to Earth.

The Gravitational Field

• Inside a planet, it decreases to zero at the center

– because pull from the mass of Earth below you

is partly balanced by what is above you.

• Outside a planet, it decreases to zero at infinity

– because you are farther away from planet.