Download G = 6.67 10 -11 m 3 s -2 kg -1

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Apples and Planets
PTYS206-2
28 Feb 2008
List of Symbols
• F, force
• a, acceleration (not semi-major axis in this
lecture)
• v, velocity
• M, mass of Sun
• m, mass of planet
• d, general distance
• r,radius of circle, semi-major axis of orbit
• R, radius of Earth
Newton’s Laws
Newton devised a uniform and systematic method for
describing motion, which we today refer to as the
Science of Mechanics. It remains the basic description
of motion, requiring correction only at very high
velocities and very small distances.
Newton summarized his theory in 3 laws:
1. An object remains at rest or continues in uniform motion
unless acted upon by a force.
2. Force is equal to mass x acceleration (F=ma)
3. For every action there is an equal and opposite reaction.
Newton and Gravity
Cambridge was closed because
of the Plague. As the story goes,
Newton was sitting under the
apple tree outside his farmhouse
(shown right) and while watching
the apples fall he realized that the
force that made the apples fall
also made the planets orbit the
sun. Using his newly invented
Calculus, Newton was able to
show that Kepler’s 3 laws of
planetary motion followed directly
from this hypothesis.
Link for animation
Falling Apples and Orbiting Planets
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Splat
What do these have in common?
Newton’s cannonball
From Principia
Apples and Planets
We will know analyze the motion of terrestrial falling
bodies and orbiting planets in more detail. We will
analyze both phenomenon in the same way and show
that Newton’s theory explains both. The plan is to
combine Newton’s second law with Newton’s law of
gravitation to determine the acceleration.
The interesting thing here is that we are applying laws
determined for motion on Earth to the motion of
heavenly bodies. What an audacious idea!
Gravitational Force: Units
According to Newton’s 2nd law, Force=mass x acceleration
The units must also match.
Units of mass = kilograms
Units of acceleration = meters/sec2
Unit of force must be kilograms-meters/sec2 = kg m s-2 (shorthand)
We define a new unit to make notation more simple. Let’s call it a
Newton. From the definition we can see that
1 Newton = 1 kg m s-2
From now on we measure force in Newtons.
What are the units of G?
Newton’s law of gravitation
F = GMm/d2
Let’s solve for G (multiply by d2, divide by Mm)
G = Fd2/Mm
Examine the units
Fd2/Mm has units of N m2/kg2 or N m2 kg-2
Or, expressing Newtons in kg, m, and s (1 N = 1 kg m s-2)
Fd2/Mm has units of N m2 kg-2 = (kg ms-2)m2 kg-2= m3 s-2 kg-1
G has units of m3 s-2 kg-1
Numerically, G = 6.6710-11 m3 s-2 kg-1
Newton’s Second Law
Force = mass x acceleration
F = ma
Newton’s Law of Gravity
• All bodies exert a gravitational force on each other.
• The force is proportional to the product of their masses
and inversely proportional to the square of their
separation.
F = GMm/d2
where m is mass of one object, M is the mass of the other,
and d is their separation.
• G is known as the constant of universal gravitation.
Falling Apples: Gravity on Earth
F = m a = G m M/
R2
Quic kT i me™ and a
dec om pres s or
are needed t o s ee thi s pi c ture.
F = m a = G m M / R2 (cancel the m’s)
a = G M / R2
where: G = 6.67x10-11 m3kg-1s-2
M = 5.97x1024 kg
On Earth’s surface:
R = 6371 km
Thus:
a = G M / R2 = 9.82 m s-210 m s-2
a on Earth is sometimes called g.
•
The separation, d, is
the distance between
the centers of the
objects.
Newton Explains Galileo
Newton’s 2nd Law:
F = ma
Newton’s law of gravity:
F = GMm/d2
The separation d is the
distance between the
falling body and the center
of the Earth d=R
F = GMm/R2
Set forces equal
ma = GMm/R2
Cancel m on both sides of
the equation
a = GM/R2
The acceleration does not depend on m!
Bodies fall at the same rate regardless of mass.
Planetary motion is more
complicated, but governed by
the same laws.
First, we need to consider the
acceleration of orbiting bodies
Circular Acceleration
Acceleration is any change in speed or direction
of motion.
Circular motion is
accelerated motion
because direction is
changing. For circular
motion:
a =
2
v /r
Real Life Example
A Circular Race Track
Acceleration
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Orbiting Planets Continued
So, orbiting planets are
accelerating. This
must be caused by a
force. Let’s assume
that the force is gravity.
We should be able to
calculate the force and
acceleration using
Newton’s second law
and Newton’s law of
gravity.
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Orbits come in a variety
of shapes (eccentricities).
In order to keep the math
simple, we will consider
in this lecture only circular
orbits. All of our results
also apply to elliptical
orbits, but we will not
derive them that way.
Step 1: Calculate the Velocity
We take as given that acceleration and velocity in circular motion are
related by
a = v2/r
According to Newton’s 2nd law
F = ma = mv2/r
According to Newton’s law of gravity
F = GMm/r2
Equating the expressions for force we have
mv2/r = GMm/r2
Solving for v2 gives
v2 = GM/r
Step 2: The Velocity is related to the
semi-major axis and period
The velocity is related to the semi-major axis and the
period in a simple way: velocity = distance/time
distance = 2r,
where r=semi-major axis, radius of circle
time = Period, P
v = 2r/P = distance/time
Step 3: Relate the Period to the
Orbital Radius
We have
v2 =GM/r
And
v = 2r/P
So it follows that
(2r/P)2 = GM/r
Or
42r2/P2 = GM/r
How Does This Relate to Kepler’s
Third Law?
We have
42r2/P2 = GM/r
Multiply both sides by r
42r3/P2 = GM
Multiply both sides by P2
42r3 = GM P2
Divide both sides by 42
r3 = (GM/42) P2
Newton’s form of Kepler’s Third Law
We have
r3 = (GM/42) P2
Kepler’s third law was a3=P2, where a=semi-major axis
(not acceleration). Since today we are using r=semimajor axis, this equation is the same as Kepler’s 3rd if
(GM/42) = 1 AU3/year2
Let’s check
Do Newton and Kepler Agree?
We want to know if
(GM/42) = 1 AU3/year2
Plug in G = 6.710-11 m3 s-2 kg-1, M=2.01030 kg
(GM/42) = 3.41018 m3 s-2
Recall 1 AU = 1.51011 m and 1 year = 3.1107 s
So
1 AU3/year2 = (1.51011 m)3/(3.1107 s)2
1 AU3/year2 = 3.41018 m3 s-2
Wow!!!
Using Newton’s Form of Kepler’s
Third Law: Example 1
Planet Gabrielle orbits
star Xena. The semi
major axis of Gabrielle's
orbit is 1 AU. The
period of its orbit is 6
months. What is the
mass of Xena relative to
the Sun?
Using Newton’s Form of Kepler’s
Third Law: Example 2
Planet Linus orbits star
Lucy. The mass of Lucy is
twice the mass of the Sun.
The semi-major axis of
Linus' orbit is 8 AU. How
long is 1 year on Linus?
Using Newton’s Form of Kepler’s
Third Law: Example 3
Jupiter's satellite (moon)
Io has an orbital period of
1.8 days and a semi-major
axis of 421,700 km. What
is the mass of Jupiter?
Using Newton’s Form of Kepler’s
Third Law: Example 4
The moon has an orbit with
a semi-major axis of
384,400 km and a period of
27.32 days. What is the
mass of the Earth?