Universal Gravitation

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Universal Gravitation

• Unit 8

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Lesson 1 Newtons Law of Universal Gravitation
In 1687 Newton published Mathematical Principles
of Natural Philosophy
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G m1 m2
Fg
r2
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Measuring the Gravitational Constant (G)
Henry Cavendish (1798)
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Gravitational attraction forms an
action-reaction pair.
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The gravitational force exerted by a finite-size,
spherically symmetric mass distribution on a
particle outside the distribution is the same as
if the entire mass of the distribution were
concentrated at the center.
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Newtons Test of the Inverse Square Law
The acceleration of the apple has the same cause
as the centripetal acceleration of the moon.
2.75 x 10-4
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Centripetal acceleration of the Moon
Newton calculated ac from its mean distance and
orbital period
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Example 1
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Free-Fall Acceleration and Gravitational Force
Equating mg and Fg
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Example 2
Find the mass of the Earth and the average
density of the Earth.
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Lesson 2 Keplers Laws of Planetary Motion
Keplers First Law
F1 , F2 are foci located c from center
r1 r2 constant
2a major axis
a semimajor axis
2b minor axis
b semiminor axis
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e 0.25
0 lt e lt 1
For circles, c 0, e 0.
e 0.97
Earths orbit e 0.017
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Keplers Second Law
Since t r x F 0, the angular momentum L is
constant.
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In the time interval dt radius vector r sweeps
out the area dA, which equals half the area (r x
dr) of the parallelogram formed by vectors r and
dr.
The radius vector (r) from the Sun to any planet
sweeps out equal areas in equal times.
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Keplers Third Law
Since gravitational force provides centripetal
force,
Since orbital speed 2pr / T,
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Solving for T2 ,
Replacing r with a,
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Example 1
Calculate the mass of the Sun using the fact that
the period of the Earths orbit around the Sun is
3.156 x 107 s and its distance from the Sun is
1.496 x 1011 m.
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Example 2
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a) Determine the speed of the satellite in terms
of G, h, RE (radius of Earth), and ME (mass of
Earth).
b) If the satellite is to be geosynchronous (that
is, appearing to remain over a fixed position on
the Earth), how fast is it moving through space
?
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Lesson 3 Gravitational Potential Energy
What is the general form of the gravitational
potential energy function ?
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Gravitational Force is Conservative
Work done by gravitational force is independent
of path taken by an object.
As particle moves from point A to B, it is acted
upon by a central force F, which is a radial
force.
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Work done by force F is always perpendicular to
displacement.
So, work done along any path between points A and
B 0
The total work done by force F is the sum of the
contributions along the radial segments.
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dr
1
rf
Uf Ui GMEm
GMEm -
r2
r
ri
Taking Ui 0 at ri infinity,
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In General (for any two particles)
Gravitational potential energy varies as 1/r,
whereas gravitational force varies as 1/r2.
U becomes less negative as r increases.
U becomes zero when r is infinite.
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Example 1
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Lesson 4 Energy in Planetary and Satellite
Motion
E KE U
Newtons Second Law applied to mass m
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Multiplying both sides by r and dividing by 2,
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For elliptical orbits, we replace r with
semimajor axis a,
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In all isolated gravitationally bound two-object
systems
Total energy is constant.
Total angular momentum is constant.
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Example 1
The space shuttle releases a 470 kg
communications satellite while in an orbit 280 km
above the surface of the Earth. A rocket engine
on the satellite boosts it into a geosynchronous
orbit, which is an orbit in which the satellite
stays directly over a single location on the
Earth. How much energy does the engine have to
provide ?
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Escape Speed
Minimum value of the initial speed needed to
allow the object to move infinitely far away from
the Earth.
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If we let rmax ? infinity,
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Example 2
Calculate the escape speed from the Earth for a
5000 kg spacecraft, and determine the kinetic
energy it must have at the Earths surface in
order to move infinitely far away from the Earth.
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Example 3 (AP 1984 2)
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a) Calculate the orbital speed vo of the
satellites in terms of G, Me, and Re.
b) Assume that the satellites collide head-on and
stick together. In terms of vo, find the speed
v of the combination immediately after the
collision.
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c) Calculate the total mechanical energy of the
system immediately after the collision in terms
of G, m, Me, and Re. Assume that the
gravitational potential energy of an object is
defined to be zero at an infinite distance from
the Earth.
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Example 4 (AP 1992 3)
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Determine each of the following for the
spacecraft when it is at point A.
a) The total mechanical energy of the spacecraft,
assuming that the gravitational potential
energy is zero at an infinite distance from the
Earth.
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b) The magnitude of the angular momentum of the
spacecraft about the center of the Earth.
Later the spacecraft is at point B on the exact
opposite side of the orbit at a distance rB 3.6
x 107 m from the center of the Earth.
c) Determine the speed vB of the spacecraft at
point B.
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Suppose that a different spacecraft is at point
A, a distance rA 1.2 x 107 m from the center of
the Earth. Determine each of the following.
d) The speed of the spacecraft if it is in a
circular orbit around the Earth.
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e) The minimum speed of the spacecraft at point A
if it is to escape completely from the Earth.