Download gravitational fields

Question
If we build a very tall tower with a height of one Earth radius. What
would be your weight when you make a measurement with our regular
spring scale on the top of the tower?
 It would be only one-forth of your weight on the ground.
F
Fground 
Ftower 
Gmearthm you
Gmearthm you
(2rearth ) 2
2
rearth
Gmearthm you
r2
 100kg  9.8m / s 2  98km  m  s  2
1 Gmearthm you 100kg  9.8m / s 2
2



27
.
5
km

m

s
2
4
rearth
4
Question
If you take your spring scale with you to a space tour, orbiting the
Earth at a distance of 1 Earth radius above the ground. What would
be your weight when you make a measurement with you spring scale
in orbit?
 It would read zero!
F
Fground 
Ftower 
Gmearthm you
Gmearthm you
(2rearth ) 2
2
rearth
Gmearthm you
r2
 100kg  9.8m / s 2  98km  m  s  2
1 Gmearthm you 100kg  9.8m / s 2
2



27
.
5
km

m

s
2
4
rearth
4
This calculation is still valid. That is, you still experience the force. But
when you are in orbit, you and the spring scale are both in constant free
fall!
Centrifugal Force?
If you take your spring scale with you to a space tour, orbiting the Earth
at a distance of 1 Earth radius above the ground. What would be your
weight when you make a measurement with you spring scale in orbit?
 It would read zero!
A different answer is that the gravitational forces are balanced by the
centrifugal forces.
Centrifugal force is a fictitious force that is “experienced” by an object in
circular motion. When you are in circular motion, there is a force that
acts on you all the time to change your direction. Therefore, you feel
there is a force that’s throwing you out.
Orbital Motion
Newton’s theory of gravitation explains why planets move in elliptical orbits
(Kepler’s First Law of Planet Motion), Additionally, it also tells us that there are
other type of orbits an object can have in gravitational fields:
• Bound Orbit: Ellipses
Objects in bound orbits circle around the Sun in elliptical orbits.
• Unbound Orbit: Parabola, Hyperbola
Objects in unbound orbits pass
through the Sun only once, and never return.
Capture of a comet
The encounter with
Jupiter may put the
originally unbound
comet in the right
place with the right
speed of a bound
orbit.
Escape Velocity
Escape Velocity is the minimum velocity that an object is required to attain
in order to escape from gravitational field of another object.
• Escape velocity does not depends on the mass of the escaping object.
• However, heavier object will require more energy to escape!
• Vescape= 40,000 km/hr = 11 km/sec from the surface of the Earth.
Click image to start movie
An object with speed exceeding
the escape velocity can escape
from the gravitational field of
Earth
An object with the right velocity
will stay in orbit
Tides
Because the gravitational force decrease quadratically with distance, the side of the
Earth facing the Moon experiences stronger gravitational pull then the side facing
away from the Moon. The net effect is a stretching of the entire Earth into elongated
shape in the direction toward the Moon.
The gravitational
pull by the moon on
this side is weaker
The gravitational
force is stronger on
this side
Spring and Neap Tides
The Sun also causes tides on Earth.
The tidal effect due to the Sun is
about 3 times smaller than the tidal
effect due to the Moon, because the
Sun is too far away.
Spring Tides
When the Sun, the Earth, and the
Moon as (more or less) aligned on a
straight line, the tidal effect of the
Sun and the Moon works together
to cause much more pronounced
tidal effect.
Neap Tides
When the tidal forces of the Sun
and the Moon oppose each other,
we get relatively small tides.
Why do we have two tides per day?
Click image to start animation
Tidal Friction
Gravitational pull of the Moon on the tidal bulges of the Earth generates a non-zero
net torque opposite to the rotation of the Earth, causing the rotation of the Earth to
slow down.=> Friction.
Gravitational Pull of
the Moon on Earth’s
bulge
Effects of Tidal Friction
1. Slowing down of Earth’s Rotation (previous page)
2. Increasing distance between the Earth and the Moon (this page)
3. Synchronous Rotation (next page)
Because the tidal bulge of the Earth
is always ahead of the Earth-Moon
line due to Earth’s rotation, the
Earth is pulling the Moon ahead of
its orbit, making it rotates faster
around the earth, thus moving it
farther away from the Earth.
This effect can also be explained
by the conservation of angular
momentum. The reduced angular
momentum of the Earth (slower
rotation) is transferred to the
Moon, causing it to rotate faster
around the Earth.
Net force
Gravitational pull due to the
tidal bulge of the Earth on the
Moon (Newton’s Third Law)
Synchronous Rotation
•
•
•
Tidal friction also applies to the
Moon’s rotation. The Moon may
have being rotating much faster
before than it does today, but the
tidal friction effect due to the tidal
force of the Earth had slowed the
Moon’s rotation to the point where
its rotational period is the same as
its orbital period => Synchronous
Rotation.
Most of the moons of the jovian
planets rotate synchronously.
Pluto and its companion (not its
moon any more!) Charon both
rotate synchronously.
Why do we always see the same
face of the Moon?
• The tidal force of the Earth stretches the Moon, just like the tidal
force of the Moon causes the tide on Earth.
• If the Moon is trying to rotate faster or slower, the gravitational pull of
the Earth on the bulge A is stronger than on bulge B (because of
shorter distance, Newton’s law of gravity), it will be pulled back.
A
Moon
Earth
B
Momentum and Energy
Momentum
Momentum
•
A quantity describing the motion of an object that depends on both the mass and the
velocity of the object
•
An object with mass m moving with a velocity v has momentum P defined by
P=mv
Example of Momentum
•
You can stop a rolling shopping cart in a slopped parking lot, but you cannot easily
stop a rolling car (with the same speed) in the same parking lot…
 The momentum of the heavy car is much larger than that of the shopping cart
moving at the same speed, and much larger force is needed to stop the car.
•
Consider a baseball (heavy object) and a bullet (light object)…
•
The baseball thrown by a person cannot easily break a wooden board.
•
The bullet fired by a gun can easily penetrate the wooden board.
 The momentum of the fast moving light object is much higher than that of the
slow moving heavy object.

Momentum is the product of mass and velocity!
2m
v
P = 2mv
m
2v P = 2mv
Force and Momentum
• Force and momentum are related by
Force = rate of change in momentum
or
F = dP/dt
• This means that to change the momentum of a moving object, we
need to apply a force to it (assuming that the mass of the object
remains constant)…
Linear and Angular Motions
Correspondence Between Linear and Angular Motions
Linear Motion
Angular Motion
Linear Velocity
Angular Velocity
Force
Torque
Linear Momentum = m  v
Angular Momentum = m  r 
v
Examples of Angular Momentum
Orbital motions are more easily
described as angular motion. For
example, an object in circular
motion has constant angular
momentum. But its linear
momentum is constantly
changing. The 24-hour rotation of
the Earth possesses angular
momentum also.
v
r
Basic Types of Energy
• Energy of Motion, or kinetic energy
– Energy associated with motion, E = ½
mv2.
– Thermal Energy is associated with the
collective kinetic energy of a system of
many particles.
• Energy carried by light, or radiative
energy
• Stored energy, or potential energy
– Gravitational Potential Energy
– Chemical potential energy is stored in
gasoline and battery
– A person on the top floor of a tall
building has more gravitational potential
energy than one that sits at the ground
floor of the building.
– A compressed spring has energy
stored in it.
– Mass energy: Matter can be converted
into energy (Einstein’s famous
equation).
Thermal Energy
• Thermal energy is the total
kinetic energy of a system of
many particles in random
motion…
• Temperature is a measure of
how much thermal energy a
system has.
• Thermal energy does not
include the kinetic energy of
the whole system moving as
a whole.
v
Gravitational Potential Energy
The gravitational potential energy
between two bodies with mass m1 and
m2 separated by a distance r is given by
Gm1m2
Eg  
r
m2
r
m1
Mass Energy
• Mass can be converted into
energy (Einstein)…
E = mc2
• In nuclear fission and fusion
reactions, a small amount of
the mass is converted into
energy according to
Einstein’s formula, generating
a very large amount of
energy.
Energy Comparison
• Table 4.1 of Textbook.
Conservation Laws
Conservation law states that certain properties of a physical system
remain the same unless something is done to change it.
Conservation of Momentum
– The momentum of a moving object will remain unchanged unless a
force is acted upon it. This is true regardless of how far the object has
moved.
– the total momentum of all interacting objects always stays the same.
Conservation of energy means that
– the total energy of a system remains constant unless more energy is
added into the system, or some energy is removed from the system.
– Within a closed system, the energy can change from one form into
another, but the total energy is always the same.
Conservation of momentum and conservation of energy are the two
types of conservation laws that we encounter most frequently in
astronomy.
Conservation of Linear Momentum
Consider two balls each with mass m, initially at rest placed on the two ends of a compressed
spring, as depicted in (a). Then, the spring is released, pushing the two balls moving with
speed –v and v in opposite direction, as depicted in (b)…
• The total momentum of the two balls in (a) is zero.
• The total momentum of the two balls after the spring is released (b) is still zero, although
the two balls are now moving.
 The total linear momentum is conserved. It is the same before and after the compressed
spring is released.
Spring is compressed
(a)
v=0
v=0
m
P=0
m
Spring is released
(b)
-v
v
m
P = -mv + mv = 0
m
Conservation of Angular
Momentum
• The angular momentum of a rotating
body is constant unless an torque is
applied to it.
• When a net torque is applied to an
object, the object will change its
rotational speed.
• Like mass, which determines how fast
an object can react to applied force,
the moment of inertia determines how
fast an object can respond to an
applied torque.
• The ice skater is in fact changing her
moment of inertia with respect to her
rotation axis. It is larger when she
extends her arms, thus in order to
satisfy angular momentum
Conservation of Angular Momentum in
Astronomy—Orbital Motion
• Although the orbital speed of Earth around the Sun is
changing according to its distance to the Sun, its angular
momentum is constant regardless of its orbital speed…
Conservation of Angular
Momentum—Earth-Moon System
• The total angular momentum of the Earth-Moon system is also
conserved—Earth probably use to rotate much faster, with the Moon
much closer to Earth. Because of the tidal effect, the rotation of the
Earth slowed while the Moon move farther away from Earth (and
thus rotate around Earth faster, Kepler’s third law). In terms of
angular momentum conservation, the angular momentum of the
Earth decreases while the angular momentum of the Moon increase,
but the total angular momentum remains constant.
Conservation of Angular
Momentum—The Solar System
• All the planets of the solar
system rotate in the same
direction around the Sun,
and their orbital planes are
pretty much the
same…Why?
– Angular Momentum
Conservation:
The orbital rotation follows
the original rotation of the
planetary nebular that
forms the planets.
Examples of Angular Momentun
Conservation in Daily Life
Riding Bicycle:
• When you ride a bicycle, you don’t fall off the bike easily. But when
you stop, is it very difficult to maintain your balance. Why?
• When you make a turn, you tip the bicycle to the side. Why doesn’t
the bicycle just fall to the ground?
• If you are riding a motorcycle real fast, then you can tip over much
more when you make the turn. Why?
 These are consequences of Angular
Momentum Conservation!
Conservation of Energy in the Ball-Spring Sytem
Consider two balls each with mass m, initially at rest placed on the two ends of a compressed spring, as
depicted in (a). Then, the spring is released, pushing the two balls moving with speed –v and v in
opposite direction, as depicted in (b)…
• The total kinetic energy E of the two balls is zero in (a)
• There are potential energy V stored in the compressed spring in (a)
• The potential energy stored in the compressed spring is released in (b), and transfomed into the
kinetic energy of the two balls.
 The total energy U of the balls and spring system is conserved. The potential energy of the spring is
converted into kinetic energy of the balls.
Spring is compressed
(a)
v=0
v=0
m
U = V, E=0
m
Spring is released
(b)
-v
v
m
U=E = ½
m(-v)2
m
+ ½ mv2 = mv2, V=0
Conservation of Energy of a Falling Ball
When a ball is thrown up and then
falls down to the ground, the total
energy of the ball is conserved…
– Chemical potential energy stored in
our muscle is converted in kinetic
energy of the ball going up.
– The kinetic energy is converted into
gravitational energy as the ball gains
height but loss speed.
– The gravitational potential energy is
converted back into kinetic energy
again as the ball falls and gains
speed.
– When the ball falls to a level lower
than where it started, its original
gravitational energy is converted into
kinetic energy, making it falls at a
higher speed.
Conversion of Gravitation Energy
into Thermal Energy
• A cloud of interstellar gas has
more gravitational potential
energy when it is more spread
out.
• As the cloud collapse under its
own gravitational pull, the
gravitational potential energy is
converted into thermal energy
of the system.
• If a star is formed in this
process, then some of the
mass energy will be converted
into radiative energy.
Weighing the Earth
We can use Newton’s Theory of Gravity to derive
2
4

p2 
a3
G (m1  m2 )
Where m1 and m2 are the masses of two objects (in kg) orbiting each
other, p is the orbital period (in second), and a is the average radius of
the orbit (in m). (Note that this is exactly Kepler’s Third Law of Planet
Motion).
Therefore, we can use it to determine the mass of the Sun, the planets,
or even black holes as long as we can measure
1. The period p of the orbital motion,
2. The average radius a of the orbit,
And that one of the object is much more massive than the other…
We know that
• The average orbital period of the Moon around the Earth is about 27.3
days
 p = 2.35  106 sec
• The average distance between the Earth and the Moon is 384,000 km.
 a = 3.84  108 m
If we assume that mearth >> mmoon, so that we can write
2
4

p2 ~
a3 ,
Gmearth
or
4 2 3
4 2  (2.35 106 )3
24
mearth ~
a

[
kg
]

6

10
[kg]
2
11
8 2
Gp
6.67 10  (3.84 10 )
Using mearth derived from the observing Moon’s orbital motion around Earth, we
can calculate the gravitational acceleration on the surface of Earth:
Gmearth 6.67 1011  6 1024
2
a


9
.
8
m
/
sec
r2
(6.4 106 ) 2