Please pick up problem set #1. Median score = 63 (B-).
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Transcript of Please pick up problem set #1. Median score = 63 (B-).
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Please pick up problem set #1. Median score = 63 (B-).
Recitation session: Every Monday before a problem set is due. 4:30 – 5:30 pm. 4054 McPherson (“The Coffee Room”).
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Gravity for BeginnersGravity for Beginners
Monday, October 13 Next Planetarium Show: Wed, Oct 15
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Flashback:
Kepler’s 1st Law of Planetary Motion:
Orbits of planets around the Sun are ellipsesellipses with
the Sun at one focusfocus.
Kepler could describedescribe orbits, but not provide a motivation.
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Isaac Newton (1642/3-1727), English
Discovered 3 Laws of Motion, 3 Laws of Motion, Law of Gravity: Law of Gravity: explained Kepler.explained Kepler.
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“Mathematical Principles of Natural Philosophy”
Newton’s laws: mathematicalmathematical in form, universaluniversal in scope.
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FirstFirst Law of Motion: An object remains at rest, or moves in
a straight line at constant speed, unless acted on by an outside force.
Precise mathematical laws require precise definitions of terms.
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SPEED = rate at which an object changes its position.
VELOCITY = speed plus direction of travel
Example: 65 miles per hour.
Example: 65 miles per hour to the north.
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ACCELERATION = rate at which an object changes its velocity.
Acceleration can involve:
1) increase in speed
2) decrease in speed
3) change in direction.
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Example of acceleration: an apple falls from a tree.
Acceleration = 9.8 meters/second/second.
After 1 second, speed = 9.8 meters/second,
After 2 seconds, speed = 19.6 m/sec, etc…
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FORCE = a push or pull acting to accelerate an object.
Examples:
Gravity = pull
Electrostatic attraction = pull
Electrostatic repulsion = push
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Restatement of First Law: In the absence of outside forces, velocity is constantconstant.
after one
second
after two
seconds
after three
seconds
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SecondSecond Law of Motion: The acceleration of an object is
directly proportional to the force acting on it, and inversely proportional to its mass.
mFa /
amF or
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Example: a package of cookies has a mass m = 0.454 kilograms.
It experiences the gravitational acceleration a = 9.8 meters/second2.
How large is the force acting on the cookies?
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amF F = (0.454 kg) (9.8 m/sec2)
F = 4.4 kg m / s2
F = 4.4 NewtonsNewtons
F = 1 poundpound
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ThirdThird Law of Motion: For every action, there is an equal
and opposite reaction.
If A exerts a force on B, then B exerts a force on A that’s equalequal in magnitude and oppositeopposite in direction.
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after 1 sec
after 2 sec
after 3 sec
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Example: I balance a package of cookies on my hand.
Cookies push on hand: F = 1 pound, downward.
Hand pushes on cookies: F = 1 pound, upward.
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Example: I remove my hand.
Earth pulls on cookies: F = 1 pound, downward.
Cookies pull on Earth:Cookies pull on Earth: F = 1 pound, upward.
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Third LawThird Law states: force on Earth = force on cookies.
Second LawSecond Law states: acceleration = force divided by mass.
Mass of Earth = 10Mass of Earth = 102525 ×× mass of cookies. mass of cookies. Therefore, acceleration of cookies = 1025 times acceleration of Earth.
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Newton’s Law of Gravity
Gravity is an attractiveattractive force between allall pairs of massive objects.
How bigbig is the force? That’s given by a (fairly) simple formula.
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Newton’s Law of Gravity
2r
M mG F
F = force
m = mass of one object
M = mass of other object
r = distance between centers of objects
G = “universal constant of gravitation”
(G = 6.7 × 10-11 Newton meter2 / kg2)
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What is gravitational force between Earth and cookies?
2r
M mG F
M = mass of Earth = 6.0 × 1024 kg
m = mass of cookies = 0.454 kg
r = radius of Earth = 6.4 × 106 meters
G = 6.7 × 10-11 Newton meter2 / kg2
F = 4.4 Newtons = 1 pound
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What is acceleration of cookies?
2r
M mG F
Newton’s 2nd law of motion:
Newton’s law of gravity:
m / F a
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Combining the two equations:
22 r
MG
m
1
r
M mG a
For the Earth,
8.9r
MG a
2 meters/sec2
INDEPENDENT OF MASS OF THE COOKIES!INDEPENDENT OF MASS OF THE COOKIES!
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Gravitational acceleration decreases with distance from the Earth’s center.
Base of CN Tower: weight = 180 pounds.
Top of CN Tower: weight = 180 pounds minus ½ ounce.
560 m
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Gravity makes apples fall; it also keeps the Moon on its orbit around the Earth, & the Earth on its orbit around the Sun.
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Artificial satellites as envisaged by Newton:
To put an object into orbit, launch it sideways with a large enough speed.
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Newton: shapeshape of orbit depends on speedspeed of satellite at launch.
Low speed = closedclosed orbit (circle, ellipse).
High speed = openopen orbit (parabola, hyperbola).
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A satellite will have a circular orbit if its initial speed = circular speedcircular speed ( vcirc )
r
GMvcirc
Presented without proof (life is too short).
r = radius of circular orbit M = mass of object being orbited
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Wednesday’s Lecture:
Stars & Galaxies in Motion
Reading:
Chapter 4