Lecture 5: Gravity and Motion
Describing Motion and Forces
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speed, velocity and acceleration
momentum and force
mass and weight
Newton’s Laws of Motion
conservation of momentum
analogs for rotational motion
Torque and Angular Momentum
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A torque is a twisting
force
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Torque = force x
length of lever arm
Angular momentum is
torque times velocity
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For circular motion,
L=mxvxr
Laws for Rotational Motion
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Analogs of all of Newton’s Laws exist for
rotational motion
For example, in the absence of a net
torque, the total angular momentum of a
system remains constant
There is also a Law of Conservation of
Angular Momentum
Conservation of Angular Momentum
during star formation
Newton’s Universal Law
of Gravitation
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Every mass attracts every other mass
through a force called gravity
The force is proportional to the product of
the two objects’ masses
The force is inversely proportional to the
square of the distance between the
objects’ centers
Universal Law of Gravitation
The Gravitational Constant G
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The value of the constant G in Newton’s
formula has been measured to be
G = 6.67 x 10 –11 m3/(kg s2)
This constant is believed to have the same
value everywhere in the Universe
Remember Kepler’s Laws?
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Orbits of planets are ellipses, with the Sun
at one focus
Planets sweep out equal areas in equal
amounts of time
Period-distance relation:
(orbital period)2 = (average distance)3
Kepler’s Laws are just a special
case of Newton’s Laws!
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Newton explained Kepler’s Laws by
solving the law of Universal Gravitation
and the law of Motion
Ellipses are one possible solution, but
there are others (parabolas and
hyperbolas)
Conic Sections
Bound and Unbound Orbits
Unbound (comet)
Unbound (galaxy-galaxy)
Bound
(planets,
binary stars)
Understanding Kepler’s Laws:
conservation of angular momentum
L = mv x r = constant
r
larger distance
smaller v
planet moves slower
smaller distance
smaller r
bigger v
planet moves faster
Understanding Kepler’s Third Law
Newton’s generalization of Kepler’s Third Law is given by:
4p2 a3
p2 =
G(M1 + M2)
Mplanet << Msun, so 
4p2 a3
p2 =
GMsun
This has two amazing implications:
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The orbital period of a planet depends
only on its distance from the sun, and this
is true whenever M1 << M2
An Astronaut and the Space Shuttle
have the same orbit!
Second Amazing Implication:
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If we know the period p and the average
distance of the orbit a, we can calculate
the mass of the sun!
Example:
Io is one of the large Galilean moons orbiting Jupiter.
It orbits at a distance of 421,600 km from the center of
Jupiter and has an orbital period of 1.77 days.
How can we use this information
to find the mass of the Sun?
Tides
The Moon’s Tidal Forces on the Earth
Tidal Friction
Synchronous Rotation
Galactic Tidal Forces