1. Fundamentals of Physics School of Physical Science and Technology
Mechanics
(Bilingual Teaching)
张昆实
School of Physical Science and Technology
Yangtze University

2. Chapter 14 Gravitation 14-1 The World and the Gravitational Force
14-2 Newton's Law of Gravitation
14-3 Gravitation and the Principle of
Superposition
14-4 Gravitation Near Earth's Surface
14-5 Gravitation Inside Earth
14-6 Gravitational Potential Energy
14-7 Planets and Satellites: Kepler's Laws
14-8 Satellites: Orbits and Energy

3. 14-1 The World and the Gravitational Force
Have you ever
imaged how vast is
the universe?
The sun is one of
millions of stars
that form the Milky
Way Galaxy.
We are near the
edge of the disk of
the galaxy, about
26000 light-years
from its center.
Milky Way galaxy

4. 14-1 The World and the Gravitational Force
The universe is made
up of many galaxies,
each one containing
millions of stars.
One of the galaxies is
the Andromeda galaxy.
The great galaxy M31
in the Constellation
Andromeda is more than
light-years across.
Andromeda galaxy

5. 14-1 The World and the Gravitational Force
The most distant galaxies are known to be over
10 billion light years away !
What force binds together these progressively
larger structures, from star to galaxy to
supercluster ?
It is the gravitational force that not only holds
you on Earth but also reaches out across
intergalactic space.

6. 14-1 The World and the Gravitational Force
The great steps
of China
toward the space
Yang Liwei
and
ShenZhou 5

7. 14-1 The World and the Gravitational Force
China CE-1 project
Exploring the Moon
Orbit around
the Moon :
Lauching:
shifting:
Moon orbit

8. 14-2 Newton's Law of Gravitation
Nowton published the law of gravitation
In It may be stated as follows:
Every particle in the universe attracts every
other particle with a force that is directely
proportional to the product of the masses
of the particles and inversely proportinal
to the square of the distance between them.
( Nowton’s law of gravitation )
(14-1)
Translating this into an equation

9. 14-2 Newton's Law of Gravitation
( Nowton’s law of gravitation )
(14-1)
is the gravitational constant with a value of
(14-2)
Fig.14-2
Particle 2 attracts particle 1
with
Particle 1 attracts particle 2
and are equal in magnitude
but opposite in direction.
These forces are not changed even if there are bodies lie between them

10. 14-2 Newton's Law of Gravitation
( Nowton’s law of gravitation )
(14-1)
is the gravitational constant with a value of
(14-2)
Fig.14-2
Particle 2 attracts particle 1
with
Particle 1 attracts particle 2
and are equal in magnitude
but opposite in direction.
These forces are not changed even if there are bodies lie between them

11. 14-2 Newton's Law of Gravitation
Nowton’s law of gravitation applies
strictly to particles; also applies to
real objects as long as their sizes
are small compared to the distance
between them (Earth and Moon).
What about an apple and Earth?
Shell theorem:
A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

12. 14-2 Newton's Law of Gravitation
Nowton’s law of gravitation applies
strictly to particles; also applies to
real objects as long as their sizes
are small compared to the distance
between them (Earth and Moon).
What about an apple and Earth?
Shell theorem:
A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its center.

13. 14-3 Gravitation and the Principle of Superposition
Given a group of n particles, there are gravitational forces between any pair of particles. 1 1
extented body 3 5
Finding the net force acting on particle 1 from the others 2 i n
First, compute the gravitational force that acts on particle 1 due to each of the other particles, in turn. 4
a group of n particles
Then, add these forces vectorialy.
(14-3)
(14-4)
(14-5)
extented body
For particle-
the Principle of Superposition

14. 14-4 Gravitation Near Earth's Surface
A particle (m) locates outside Earth
a distance r from Earth’s center. The magnitude of the gravitational force from Earth (M) acting on it equals
(14-8)
If the particle is releaced, it will fall
towards the center of Earth with the
gravitatonal acceleration :
Gravitation Near Earth's Surface
(14-9)
the gravitatonal acceleration
(14-10)

15. 14-4 Gravitation Near Earth's Surface
the gravitatonal acceleration
(14-10)
Gravitation Near Earth's Surface

16. 14-4 Gravitation Near Earth's Surface
We have assumed that Earth is an
inertial frame (negnecting its actual
rotation). This allowed us to assume
the free-fall acceleration is the same
as the gravitational acceleration m
However differs from
(14-10)
Weight differs from
(14-8)
Because:
(1) Earth is not uniform,
(2) Earth is not a perfect sphere,
(3) Earth rotates.

17. 14-4 Gravitation Near Earth's Surface
Crust
Inner core
Oute core
Mantle
(1) Earth is not uniform
The density of Earth varies
radially:
Inner core (103 kg/m3)
Outer core (103 kg/m3)
Mantle (103 kg/m3)
and the density of the crust (outer section) of Earth
varies from region to region
over Earth’s surface.
Thus, varies from region to region over the surface.

18. 14-4 Gravitation Near Earth's Surface
(2) Earth is not a perfect sphere
Earth is approximately an ellipsoid, flattened at the poles and bulging at the equattor. Its equatorial radius is greater than its polar radius by 21km.
equator
Thus, a point at the poles is closer to the dense core of Earth than is a point on the equator.
This is one reason the free-fall acceleration increases as one
proceeds, at sea level, from the equator toward either pole.

19. 14-4 Gravitation Near Earth's Surface
(3) Earth is rotating.
An object located on Earth’s surface anywhere
(except at two poles) must rotate in a circle about
the Earth’s rotation axis and thus have a centripital
acceleration ( requiring a centripital net force )
directed toward the center of the ciecle.
equator
How Earth’s rotation causes to differ from ? Put a crate of mass on a scale at the equator and analyze it.
Free-body diagram
Normal force (outward in direction )
Gravitational force (inward in direction )
Centripital acceleration (inward in direction )
Newton’s secend law for the axis
(14-11)

20. 14-4 Gravitation Near Earth's Surface
(3) Earth is rotating.
Newton’s secend law for the axis
(14-11)
Reading on the scale
(14-12)
mearsure
weight =
magnitude of
gravitation force
mass times
centripetal acceleration -
(14-13)
Relation between and =
Free-fall
acceleration
gravitation
centripetal -

21. Gravitation Inside Earth
14-4
Newton’s shell theorem can also be applied
to a particle located Inside a uniform shell:
A uniform spherical shell of matter exerts no net gravitational force on a particle located inside it.
If a particle were to move into Earth, the
gavitational Force would change :
It would tend to increase because the
particle would be moving closer to the
center of Earth.
(2) It would tend to decrease because the
thickening shell of material lying outside
the particle’s radial position would not
exert any net force on the particle.

22. 14-6 Gravitational Potential Energy
In section 8-3 (P144) the gravitational potential energy of a
particle-Earth system is studied. Usually we chose the
reference configuration ( the particle was on Earth’s surface)
as having a gravitational potential energy of zero.
(8-9)
Here, considerthe gravitational potential
energy of two particles, of masses
and , seperated by a distance
However, we now choose a referance
configuration with equal to zero as
the seperation distance is large
enough to be approximated as infinite.
gravitational potential energy
At finite

23. 14-6 Gravitational Potential Energy
(14-20)
For any finite value of , the
value of is negative.
The gravitational potential energy is a
property of the system of the two
particles rather than of either particle along
However, for Earth and a apple,
We often speak of “potential energy of the
apple”, because when a apple moves in
the vicinity of Earth, (apple)

24. 14-6 Gravitational Potential Energy
(14-20)
For a system of three particles, the
gravitational potential energy of
the system is the sum of the
gravitational potential energies of
all three pairs of particles.
( calculating as if the other
particle were not there )
(14-21)

25. 14-6 Gravitational Potential Energy
Find the gravitational
potential energy of a ball at point P, at
radial distance R from Earth’s center.
Proof of (14-20):
Differential displacement
The work done on the ball by the gravita- tional force as the ball travels from point P to a great (infinite) distance from Earth is
(14-22)
(14-23)
(14-24)

26. 14-6 Gravitational Potential Energy
Differential displacement
(14-24)
From Eq. 8-1
(14-20)

27. 14-6 Gravitational Potential Energy
Path Independence
Earth
Moving a ball from A to G along a path :
consisting of three radial lengths and
three circular arcs (cented on Earth).
The work done by the gravitational force on the ball as it moves along ABCDEFG:
The work done along each circular arc is zero, because at every point.
the gravitational force is a conservative force, the work done by it on a particle is independent of the actual path taken between points A and G.

28. 14-6 Gravitational Potential Energy
Earth
From Eq. 8-1:
(8-1)
(14-25)
Since the work done by a conservative force is independent of the actual path taken.
The change in gravitational potential energy is also independent of the actual path taken.

29. 14-6 Gravitational Potential Energy
We derived the potential energy function
from the force function
potential energy and force
Now let’s go the other way: derive the force function from the potential energy function
(14-26)
radially
inward
This is Newton’s law of gravitation (14-1) .
( Derivation is the inverse operation of integration )

30. 14-6 Gravitational Potential Energy
Escape Speed
The initial speed that will cause a projectile to move up forever is called the (Earth) escape speed.
Consider a projectile ( ) leaving the surface of a planet with escape speed
Its potential energy
Its kinetic energy
When the projectile reches infinity, it stops.
Its kinetic energy
Its potential energy
From the principle of conservation of energy
(14-27)
Escape Speed:

31. 14-6 Gravitational Potential Energy
Escape Speed
(14-27)
From Earth:
The escape speed does not depend on the direction in which a projectile is fired from a planet.
eastward
However, attaining that speed is easier if the projectile is fired in the direction the launch site is moving as the planet rotates about its axis .
For example, rockets are launched eastward at XiChang to take the advantage of the eastward speed of 1500km/h due to Earth’s rotation.

32. 14-1 The World and the Gravitational Force
China CE-1 project
Exploring the Moon
Orbit around
the Moon :
launched eastward
Lauching:
shifting:
Moon orbit

33. 14-7 Planets and Satellites: Kepler's Laws
The motion of the planets have been a puzzle since the dawn of history.
Johannes Kepler ( ) worked out the empirical laws that governed these motions based on the data from the observations by Tycho Brahe ( ).
1 THE LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus.
is the semimajor axis of the orbit
is the eccentricity of the orbit
is the distance from the center of the ellipse to either focus
the eccentricity of Earth’s orbit is only

34. 14-7 Planets and Satellites: Kepler's Laws
2 THE LAW OF AREAS: A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal times; that is, the rate dA/dt at which it sweeps out area A is constant.
This second law tell us that the planet will move most slowly when it is farthest from the Sun and most rapidly when it is nearest to the Sun.

35. 14-7 Planets and Satellites: Kepler's Laws
Proof of Kepler’s second law is totally equivalent to the law of conservation of angular momentum.
The area of the wedge
The instantaneous rate at which area is been sweept out is
(14-29)
The magnitude of the angular momen- tum of the planet about the Sun is
(14-30)
constant
constant

36. 14-7 Planets and Satellites: Kepler's Laws
3 THE LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.
Applying Newton’s second law to the orbiting planet :
(14-32)
From Eq
(14-33)
The quantity in parentheses is a constant that depends only the mass M of the central body about which the planet orbits.

37. 14-7 Planets and Satellites: Kepler's Laws
(14-33)
3 THE LAW OF PERIODS:
（水星）
（金星）
（地球）
（火星）
（木星）
（土星）
（天王星）
（海王星）
（冥王星）

38. 14-8 Satellites: Orbits and Energy
As a satellite orbits Earth on its elliptical path, its speed and the distance from the center of Earth fluctuate with fixed periods. However, the mechanical energy E of the satellite remains constant.
The potential energy of the
system (or the satellite) is
To find the kinetic energy of the satellite, use Newton’s second law
Compare U and K
(14-41)
(14-43)
( Circular orbit )
(14-42)

39. 14-8 Satellites: Orbits and Energy
The total mechanical energy E of the satellite is
(14-44)
( circular orbit )
Compare E and K
(14-45)
( circular orbit )
For a satellite in an elliptical orbit of semimajor axis
(14-46)
( elliptical orbit )

40. 14-8 Satellites: Orbits and Energy