Question

1
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.

2
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!

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!
3
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
  thats throwing you out.

4
Orbital Motion

• Newtons theory of gravitation explains why
  planets move in elliptical orbits (Keplers 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.
5
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
6
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 force is stronger on this side
The gravitational pull by the moon on this side
is weaker
7
Spring and Neap Tides
Click image to start animation
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?
8
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 Earths bulge
9
Effects of Tidal Friction

• Slowing down of Earths Rotation (previous page)
• Increasing distance between the Earth and the
  Moon (this page)
• Synchronous Rotation (next page)

Because the tidal bulge of the Earth is always
ahead of the Earth-Moon line due to Earths
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 (Newtons Third Law)
10
Synchronous Rotation

• Tidal friction also applies to the Moons
  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 Moons 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.

11
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, Newtons law of gravity), it
  will be pulled back.

A
Moon
Earth
B
12
Momentum and Energy
13
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!

14
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)

15
Linear and Angular Motions
Correspondence Between Linear and Angular Motions
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
16
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
  (Einsteins famous equation).

17
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
18
Gravitational Potential Energy
The gravitational potential energy between two
bodies with mass m1 and m2 separated by a
distance r is given by
m2
r
m1
19
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 Einsteins formula, generating a
  very large amount of energy.

20
Energy Comparison

• Table 4.1 of Textbook.

21
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.

22
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
P 0
Spring is released
23
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 law, she
  rotates slower.

24
Conservation of Angular Momentum in
AstronomyOrbital 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

25
Conservation of Angular MomentumEarth-Moon System

• The total angular momentum of the Earth-Moon
  system is also conservedEarth 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, Keplers 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.

26
Conservation of Angular MomentumThe 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 sameWhy?
• Angular Momentum Conservation
• The orbital rotation follows the original
  rotation of the planetary nebular that forms the
  planets.

27
Examples of Angular Momentun Conservation in
Daily Life

• Riding Bicycle
• When you ride a bicycle, you dont 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 doesnt 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!
28
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
U V, E0
Spring is released
29
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.

30
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.

31
Weighing the Earth

• We can use Newtons Theory of Gravity to derive
• 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
  Keplers 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
• The period p of the orbital motion,
• The average radius a of the orbit,
• And that one of the object is much more massive
  than the other

32

• 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

Using mearth derived from the observing Moons
orbital motion around Earth, we can calculate the
gravitational acceleration on the surface of
Earth