Download Lecture-17-10-31 - University of Virginia

Gravity
Newton’s Law of Universal Gravitation
Newton’s insight: The force
accelerating an apple downward is the
same force that keeps the Moon in its
orbit.
Universal Gravitation
m1m2
 F12  G 2 rˆ12  force on m1 due to m2
r12
Nature is nice!
m1m2
2
12
r
1.00000001
1
m
2.0000001
12
r
1.00000001
2
m
The gravitational force is always attractive, and
points along the line connecting the two masses:
The two forces shown are
an action-reaction pair.
G is a very small number; this means that the force of
gravity is negligible unless there is a very large mass
involved (such as the Earth).
Ex. How strong is gravitational attraction between you and the
person next to you?
Solution:
Estimate: m1  m2  60 kg ; r12
 F   6.67  10
11
1m

 60  60 
2
1
3  10 7 N
If an object is being acted upon by several different
gravitational forces, the net force on it is the vector
sum of the individual forces.
This is called the principle of superposition.
Gravitational Attraction of Spherical Bodies
Gravitational force between a point mass and a sphere*:
the force is the same as if all the mass of the sphere
were concentrated at its center.
a consequence of 1/r2
(inverse square law)
*Density of sphere must be radial symmetric
Gravitational Force at the Earth’s Surface
The center of the Earth is one Earth radius away, so
this is the distance we use:
g
The acceleration of gravity
decreases slowly with altitude...
...until altitude becomes comparable to the
radius of the Earth. Then the decrease in the
acceleration of gravity is much larger:
In the Space Shuttle
a) they are so far from Earth that Earth’s gravity
doesn’t act any more
Astronauts in the b) gravity’s force pulling them inward is cancelled
by the centripetal force pushing them outward
space shuttle
c) while gravity is trying to pull them inward, they
are trying to continue on a straight-line path
float because:
d) their weight is reduced in space so the force of
gravity is much weaker
In the Space Shuttle
a) they are so far from Earth that Earth’s gravity
doesn’t act any more
Astronauts in the b) gravity’s force pulling them inward is cancelled
by the centripetal force pushing them outward
space shuttle
c) while gravity is trying to pull them inward, they
are trying to continue on a straight-line path
float because:
d) their weight is reduced in space so the force of
gravity is much weaker
Astronauts in the space shuttle float because
they are in “free fall” around Earth, just like a
satellite or the Moon. Again, it is gravity that
provides the centripetal force that keeps them in
circular motion.
Follow-up: How weak is the value of g at an altitude of 300 km?
Satellite Motion: FG and acp
Consider a satellite in circular motion*:
Gravitational Attraction:
Necessary centripetal acceleration:
• Does not depend on mass of the satellite!
• larger radius = smaller velocity
smaller radius = larger velocity
Relationship between FG and acp will be important
for many gravitational orbit problems
*
not all satellite orbits are circular!
A geosynchronous satellite is one whose orbital period is equal to
one day. If such a satellite is orbiting above the equator, it will be in
a fixed position with respect to the ground.
These satellites are used for communications and weather
forecasting.
How high are they?
RE = 6378 km
ME = 5.87 x 1024 kg
Averting Disaster
a) it’s in Earth’s gravitational field
b) the net force on it is zero
The Moon does not
c) it is beyond the main pull of Earth’s gravity
crash into Earth
d) it’s being pulled by the Sun as well as by
Earth
because:
e) none of the above
Averting Disaster
a) it’s in Earth’s gravitational field
b) the net force on it is zero
The Moon does not
c) it is beyond the main pull of Earth’s gravity
crash into Earth
d) it’s being pulled by the Sun as well as by
Earth
because:
e) none of the above
The Moon does not crash into Earth because of its high
speed. If it stopped moving, it would, of course, fall
directly into Earth. With its high speed, the Moon would
fly off into space if it weren’t for gravity providing the
centripetal force.
Follow-up: What happens to a satellite orbiting Earth as it slows?
Two Satellites
Two satellites A and B of the same mass
are going around Earth in concentric
orbits. The distance of satellite B from
Earth’s center is twice that of satellite A.
What is the ratio of the centripetal force
acting on B compared to that acting on A?
a) 1/8
b) ¼
c) ½
d) it’s the same
e) 2
Two Satellites
Two satellites A and B of the same mass
are going around Earth in concentric
orbits. The distance of satellite B from
Earth’s center is twice that of satellite A.
What is the ratio of the centripetal force
acting on B compared to that acting on A?
Using the Law of Gravitation:
we find that the ratio is .
a) 1/8
b) ¼
c) ½
d) it’s the same
e) 2
Note the
1/R2 factor
Gravitational Potential Energy
Gravitational potential energy of an object of mass
m a distance r from the Earth’s center:
(U =0 at r -> infinity)
Very close to the Earth’s
surface, the gravitational
potential increases linearly
with altitude:
Gravitational potential energy, just like all
other forms of energy, is a scalar. It
therefore has no components; just a sign.
Energy Conservation
Total mechanical energy of an
object of mass m a distance r from
the center of the Earth:
This confirms what we already know – as an object
approaches the Earth, it moves faster and faster.
Escape Speed
Escape speed: the initial upward speed a
projectile must have in order to escape from
the Earth’s gravity
from total energy:
If initial velocity = ve, then velocity at large distance goes to zero. If
initial velocity is larger than ve, then there is non-zero total energy, and
the kinetic energy is non-zero when the body has left the potential well
Maximum height vs. Launch speed
Speed of a projectile as it leaves the Earth,
for various launch speeds
Kepler’s Laws of Orbital Motion
Johannes Kepler made detailed studies of the apparent motions of the
planets over many years, and was able to formulate three empirical laws
1. Planets follow elliptical orbits, with the Sun at one
focus of the ellipse.
Elliptical orbits are stable under inverse-square force law.
You already know about circular
motion... circular motion is just a
special case of elliptical motion
Only force is central gravitational attraction - but for elliptical
orbits this has both radial and tangential components
Kepler’s Laws of Orbital Motion
2. As a planet moves in its orbit, it sweeps out an
equal amount of area in an equal amount of time.
Apogee
r
v Δt
Perigee
This is equivalent to
conservation of angular
momentum
L  mrp v p  mra va
 rp v p t  ra va t  Ap  Aa
Kepler’s Laws of Orbital Motion
3. The period, T, of a planet increases as its mean
distance from the Sun, r, raised to the 3/2 power.
This can be shown to be a consequence of the
inverse square form of the gravitational force.
Orbital Maneuvers
Which stable circular orbit has
the higher speed?
How does one move from the
larger orbit to the smaller orbit?
Binary systems
If neither body is “infinite” mass, one should consider
the center of mass of the orbital motion
2 r1 2 r2
T1  T2 

v1
v2
CM : m1r1  m2 r2
Four equations in four
unknowns
Guess My Weight
If you weigh yourself at the equator of
Earth, would you get a bigger, smaller,
or similar value than if you weigh
yourself at one of the poles?
a) bigger value
b) smaller value
c) same value
Guess My Weight
If you weigh yourself at the equator of
Earth, would you get a bigger, smaller,
or similar value than if you weigh
yourself at one of the poles?
a) bigger value
b) smaller value
c) same value
The weight that a scale reads is the normal force exerted by the
floor (or the scale). At the equator, you are in circular motion, so
there must be a net inward force toward Earth’s center. This
means that the normal force must be slightly less than mg. So the
scale would register something less than your actual weight.
Earth and Moon I
a) the Earth pulls harder on the Moon
Which is stronger,
b) the Moon pulls harder on the Earth
Earth’s pull on the
c) they pull on each other equally
Moon, or the
d) there is no force between the Earth and
the Moon
Moon’s pull on
Earth?
e) it depends upon where the Moon is in its
orbit at that time
Earth and Moon I
a) the Earth pulls harder on the Moon
Which is stronger,
b) the Moon pulls harder on the Earth
Earth’s pull on the
c) they pull on each other equally
Moon, or the
d) there is no force between the Earth and
the Moon
Moon’s pull on
Earth?
e) it depends upon where the Moon is in its
orbit at that time
By Newton’s Third Law, the forces
are equal and opposite.
Principle of Equivalence
You’re standing at rest on a scale. The display
shows that it is exerting a force on you of
F = mg = 60 * 9.81 = 589 N
You’re now at rest in outer space, far from any star
or planet. You fire your thruster which exerts a
force of 150 N on you (m = 60 kg). At what rate
do you accelerate?
F = ma so a = F/m = 150/60 = 2.5 m/s2
m and m represent two different concepts. Why
can we treat them interchangeably?
General Relativity
More complete theory of gravity.
Replaces “spooky” action-at-a-distance with
curvature of space, an idea that is just about as
“spooky” as action-at-a-distance.
Required to make GPS work!
Black holes
If an object is sufficiently massive and
sufficiently small, the escape speed
will equal or exceed the speed of light –
light itself will not be able to escape the
surface.
This is a black hole.
The light itself has mass (in the
mass/energy relationship of Einstein), or
spacetime itself is curved
Gravity and light
Light will be bent by any
gravitational field; this can be
seen when we view a distant
galaxy beyond a closer galaxy
cluster. This is called gravitational
lensing, and many examples have
been found.