Gravitation

Halliday, Resnick & Walker Chapter 13
Gravitation
Physics 1A – PHYS1121
Professor Michael Burton
II_A2: Planetary Orbits in the Solar System
13-1 Newton's Law of Gravitation
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+ Galaxy Interactions (You Tube)
21 seconds
The gravitational force
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Holds us to the Earth
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Holds Earth in orbit around the Sun
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Holds the Sun together with the stars in our Galaxy
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Holds together the Local Group of galaxies
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Holds together the Local Supercluster of galaxies
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Attempts to slow the expansion of the Universe
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Is responsible for black holes
Gravity is far-reaching and very important!
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13-1 Newton's Law of Gravitation
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Gravitational attraction depends on mass of an object
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Between an apple & the Earth: ~0.8 N
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Between 2 people: < 1 µN
Where G is the gravitational constant:
Eq. (13-2)
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13-1 Newton's Law of Gravitation
The force always points from one particle to the other,
so this equation can be written in vector form:
Eq. (13-3)
13-1 Newton's Law of Gravitation
M
M
M
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The shell theorem describes
gravitational attraction for objects
Earth is a nesting of shells, so we
feel Earth's mass as if it were all
located at its centre
Gravitational force forms third-law
force pairs (i.e. N3L)
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e.g. Earth-apple and apple-Earth
forces are both 0.8 N
F
The magnitude of the force is given by:
Newton's Law of Gravitation defines the strength of this
attractive force between particles
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F
Eq. (13-1)
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Bodies attract each other through gravitational
attraction
Newton realized this attraction was responsible for
maintaining the orbits of celestial bodies
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m1
Earth has a large mass and produces a large attraction
The force is always attractive, never repulsive
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13-1 Newton's Law of Gravitation
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Earth-apple and apple-Earth
forces are both ~0.8 N
The difference in mass causes
the difference in the apple:Earth
accelerations:
~10 m/s2 vs. ~ 10-25 m/s2
m2
Consider the objects of various masses indicated below. The objects are each separated
from another object by the distance indicated. In which of these situations is the
gravitational force exerted on the two objects the largest?
a) #1
b) #2
13-2 Gravitation and the Principle of Superposition
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The principle of superposition applies.
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i.e. Add the individual forces as vectors:
c) #3
Eq. (13-5)
d) #2 and #3
e) #1, #2, and #3
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For a real (extended) object, this becomes an integral:
Eq. (13-6)
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If the object is a uniform sphere or shell we can treat
its mass as being at its centre instead
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13-2 Gravitation and the Principle of Superposition
b) 2
c) 4
d) 1 and 2 equally large
e) 2 and 4 are equally large
Figure 13-4
M
Consider a system of particles, each of mass m. In which one of the following
configurations is the net gravitational force on Particle A the largest? The horizontal or
vertical spacing between particles is the same in each case.
a) 1
Example Summing two forces:
M
13-3 Gravitation Near the Earth's Surface
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13-3 Gravitation Near the Earth's Surface
Combine F = GMm/r2 and F = mag:
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Eq. (13-11)
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Gives magnitude of gravitational
acceleration at a given distance
from the centre of the Earth
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Table 13-1 shows the value for ag
for various altitudes above the
Earth’s surface
13-3 Gravitation Near Earth's Surface
The calculated ag will differ slightly from the measured
g at any location on the Earth’s surface
Three reasons. The Earth….
1. 
mass is not uniformly distributed
2. 
is not a perfect sphere
3. 
rotates
13-5 Gravitational Potential Energy
Example Difference in gravitational force and weight due to
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rotation at the equator:
V2
, the centripetal acceleration.
R
Here V = #R with # the angular velocity, and FN = mg.
Gravitational potential energy for a two-particle system
is written:
N2L : FN " mag = "m
Eq. (13-21)
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Thus g = ag " # 2 R.
Exercise for the student. Show this is about 0.034 m/s2.
Use R=6,400km and !=2" radians/day.
Question: do you weigh more or less at the equator than the Poles?
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Note this value is negative and approaches 0 for r ! "
The gravitational potential energy of a system is the
sum of potential energies for all pairs of particles
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Proof comes from integrating the force to obtain the work done.
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GMm
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i.e. U = "W = " # F • dr and using F =
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r2
Figure 13-6
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13-5 Gravitational Potential Energy
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13-5 Gravitational Potential Energy
The gravitational force is conservative.
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The work done by this force does not
depend on the path followed by the
particles, only the difference in the
initial and final positions of the
particles.
For a projectile to escape the gravitational pull of a body,
it must come to rest only at infinity (if at all).
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At rest at infinity: K = 0 and U = 0 (because r ! ")
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So K + U must be # 0 at surface of the body to escape:
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This is the escape speed. The minimum value to escape.
Since the work done is independent of
path, so is the gravitational potential
energy change
Eq. (13-26)
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Figure 13-10
Rockets launch eastward to take advantage of Earth's
rotational speed, to reach vescape more easily
13-5 Gravitational Potential Energy
You move a ball of mass m away from a sphere of mass M.
1.  Does the gravitational potential energy of the system of the ball and sphere
a)  Increase, or
b)  Decrease.
2. Is the Work done by the gravitational force between the ball and the sphere
a)  Positive work, or
b)  Negative work
r
m
M
In a distant solar system where several planets are orbiting a single star of mass M, a
large asteroid collides with a planet of mass m orbiting the star at a distance r. As a
result, the planet is ejected from its solar system. What is minimum amount of energy
that the planet must receive in the collision to be removed from the solar system?
2_A3: Retrograde Motion of the Planets
a)
r
b)
c)
m
M
d)
e)
13-6 Planets and Satellites: Kepler's Laws
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13-6 Planets and Satellites: Kepler's Laws
The motion of planets in the solar system was a
puzzle for astronomers, especially curious motions
such as “retrograde motion”.
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Johannes Kepler (1571-1630) derived laws of motion
using Tycho Brahe's (1546-1601) measurements
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An eccentricity of zero corresponds to a circle
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Eccentricity of Earth's orbit is 0.0167
Figure 13-11
Figure 13-12
The orbit is defined by its semi-major axis a and its
eccentricity e
Kepler2L: Kepler’s 2nd Law
Equal Areas in Equal Times
13-6 Planets and Satellites: Kepler's Laws
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Equivalent to Conservation of Angular Momentum
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13-6 Planets and Satellites: Kepler's Laws
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The law of periods can be written mathematically as:
Holds for elliptical orbits if we replace r with a, the
semi-major axis.
See later in this course.
13-6 Planets and Satellites: Kepler's Laws
A spacecraft is in low orbit of the Earth with a period of approximately 90 minutes.
By which of the following methods could the spacecraft stay in the same orbit and
reduce the period of the orbit?
13-7 Satellites: Orbits and Energy
a) Before launch, increase the mass of the spacecraft to increase the centripetal
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force on it.
Relating the centripetal acceleration of a satellite to
the gravitational force, we can rewrite as energies:
b) Remove any unnecessary equipment, cargo, and supplies to reduce the mass
and decrease its angular momentum.
Eq. (13-38)
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Meaning that:
c) Fire rockets to increase the tangential velocity of the ship.
Eq. (13-39)
d) None of the above methods will achieve the desired effect.
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Therefore the total mechanical energy is:
Eq. (13-40)
13-7 Satellites: Orbits and Energy
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Total energy E is the negative of the kinetic energy
For an ellipse, we substitute a for r
Therefore the energy of an orbit depends only on its
semi-major axis, not its eccentricity
!  All orbits in Figure 13-15 have the same energy
Figure 13-15
13-7 Satellites: Circular Orbit
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Graph of variation in Energy for a circular orbit, radius r
Note that:
!  E(r) and U(r) are negative
!  E(r) = -K(r)
!  E(r), U(r), K(r) all ! 0 as r!!
13-7 Satellites: Orbits and Energy
13
Summary
The Law of Gravitation
Eq. (13-1)
Gravitational Behavior of
Uniform Spherical Shells
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Eq. (13-2)
Superposition
The net force on an external
object: calculate as if all the
mass were concentrated at the
centre of the shell
Gravitational Acceleration
Eq. (13-11)
Eq. (13-5)
a)  Which orbit (1, 2 or 3?) will the shuttle take when it fires a forward-pointing
thruster so as to reduce its kinetic energy?
b)  Is the orbital period T then (i) greater than, (ii) less than or (iii) the same as, that
of the circular orbit?
13
Summary
13
Free-Fall Acceleration and
Weight
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Earth's mass is not uniformly
distributed, the planet is not
spherical, and it rotates: the
calculated and measured values
of acceleration differ
Gravitational Potential
Energy
Gravitation within a
Spherical Shell
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Summary
Escape Speed
Kepler's Laws
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A uniform shell exerts no net
force on a particle inside
Eq. (13-28)
Inside a solid sphere:
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The law of orbits: ellipses
The law of areas: equal areas
in equal times
The law of periods:
Eq. (13-19)
Potential Energy of a
System
Eq. (13-34)
Energy in Planetary Motion
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Eq. (13-42)
Eq. (13-21)
Eq. (13-22)
Kepler's Laws
Gravitation and acceleration
are equivalent