Newton’s Law of Universal Gravitation
The Moon’s nearly circular
orbit about the earth has a
radius of about 384,000 km
and a period T of 27.3 days.
Determine the acceleration
of the Moon toward the
Earth.
Newton’s Law of Universal Gravitation
„
Every particle in the universe attracts every other
particle in the universe with a force that is
proportional to the product of their two masses
and inversely proportional to the square of the
distance between them. The force acts along the
line joining the two particles.
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Newton’s Law of Universal Gravitation
„
Scalar form
„
Vector form
‰
Where
„
„
„
„
„
„
‰
F =G
m1m2
r212
F12 = − G
m1m2
r̂21
r212
G = universal gravitational constant = 6.67 x 10-11 N⋅m2/kg
m1 = mass of body 1
m2 = mass of body 2
r21 = position vector from body 2 to body 1
r21 = magnitude of the position vector from body 2 to body 1
r̂21 = position vector from body 2 to body 1
How does this relate to g?
Effect of a Planet’s Rotation on g
The plant Jupiter is about 320 as massive as the
Earth. It is claimed that a person would be crushed
by the force of gravity on Jupiter since people can’t
survive more than a few gE’s. Calculate the number
of gE’s a person would experience if she could stand
a) at one of the poles and b) at the equator. MJ =
1.90 x 1027 kg, rJ = 7.14 x 104 km, and TJ = 9 hrs 55
min.
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Weightlessness and Satellite Motion
„
„
Since space shuttle astronauts are close to Earth, why
do they experience weightlessness?
Also, what keeps the space shuttle “up in space?”
How to Experience “Weightlessness”
„
Go to deep space
‰
„
far away from everything
Go into “free fall”
‰
Example: Elevator
Other Ways to Experience Free Fall
„
“Topping a Hill”
300. ft
A car “tops” a hill
with a radius of
curvature of 300. ft.
At what speed will
the car become
airborne?
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Other Ways to Experience Free Fall
„
NASA’s “Vomit Comet”
mg = m
v2
r
The Vomit Comet’s speed is 550. knots at the
2
apex. What is the radius of curvature?
⎧
⎛ 1.15MPH ⎞⎛ 1.4667 ft / s ⎞⎫
⎟⎜
⎟⎬
⎨550kt ⎜
kt
v2 ⎩
⎝
⎠⎝ MPH ⎠⎭
r=
=
g
32.2 ft / s 2
r = 26,700 ft = 5.06 mi
1 knot (kt) = 1.15 MPH
Newton’s Cannon
„
„
So what does this have to do
with the space shuttle?
Newton’s thought experiment
‰
‰
Throw a projectile parallel to the
Earth’s surface. See where it lands.
Double the initial velocity. Repeat
„
Double the velocity again. Repeat
‰
‰
Double the velocity again. Repeat
ƒ Double the velocity again. Repeat
Eventually, it will never land!!!
„
It will remain in continuous free fall!!!
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Satellite Motion
„
The objective is to “balance” the
force of gravity with the normal
(as in direction) acceleration.
With what horizontal velocity would a satellite have to
be launched from the top of Mt. Everest (h = 8848 m)
to be placed in a circular orbit around the earth?
r
rE = 6.38 x 106 m
ME = 5.97 x 1024 kg
v = 7890 m / s
Geosynchronous Satellite Motion
„
The objective is to “circle” the
equator in exactly 24.0 hours.
r
rE = 6.38 x 106 m
ME = 5.97 x 1024 kg
r = 4.22 ×107 m
v =
⇒ h = r − rE = 3.58 ×107 m = 3.58 × 10 4 km
2π r
= 3070 m / s
T
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