Kepler's third law
• A planet is found to orbit a star
with a sidereal period of 8 years.
What will be the length of the
semimajor axis of the ellipse that it
orbits along if it obeys Kepler's
third law?
Kepler’s third law
P2
=
=a
a3
P = planet’s sidereal period, in years = time to go around
ellipse as view by someone above solar system.
a = planet’s semimajor axis, in AU Hint – just try cubing all four
P2 =
82
answers if you don’t have a
calculator that does cube roots.
P=8 so
= 64
a = 4 so a3 = 43 = 4x4x4 = 64
Kepler's second law
• In the following image, the amount of time
it takes the planet to go from A to B is the
same as the time it takes to go from C to
D. The amount of ink that would be
required to paint in each of the brown
triangles is
about equal because planet sweeps
out equal area in equal time.
Recall the discussion in lecture
where I covered triangles with little
squares to measure area.
Kepler's second law 2
• Suppose that an object was found to orbit
the Sun, and that its path was an ellipse
with eccentricity of 0.5. If the planet always
moved at the same speed, would it satisfy
Kepler's second law? Choose the best
answer of the four below.
No, because when an object
sweeps out equal areas in
equal time, it must travel
faster when it is closer to the
planet.
Kepler's second law 3
• Suppose that the time between A
and B was 1 month. What would
the time be for the planet to go
from C to D? Less than one
month.
As the planet moves closer
to the sun, it travels faster.
The distance between A
and B is about the same as
the distance between C and
D, so it will take less time to
go from C to D. (The area
from Sun-A-B is more than
Sun-C-D.)
d
d
Mass and Weight
• If you measured your weight with very high
precision scale, would you see a difference
between your weight at the top of a mountain
versus at the bottom of the mountain? Select the
best answer.
• Yes, because the force that Earth exerts on me
would be slightly less because I am slightly
farther from it. So the scale spring will compress
less and so a scale will register my weight as
less. See the next slide.
Mass and Weight continued
Mass m1
 m1m2 
Force  G 2 
 r 
A spring
Mass m2
Weight is a number that
tells you about how
much this spring will
compress. It depends
on m1 and r.
Mass and Weight 2
• If you measured your mass with very high
precision scale, would you see a difference
between your mass at the top of a mountain
versus at sea level? Select the best answer.
• No. My mass would not change because the
number of atoms and molecules in me does not
change. Note that the periodic table, which lists
intrinsic properties of the elements such as
number of electrons and protons, uses “atomic
mass”.
Newton's Law of Gravity
• Newton's Law of Gravity is
 m1m2 
Force  G 2 
 r 
• This equation states that if the distance between
two objects increases by a factor of three, the
force they exert on each other decreases by a
factor of nine. For example, if r goes from 2 to
2x3, the force equation changes from
 m1m2 
 m1m2 
G 2   G

 2 
 4 
to
 m1m2 
 m1m2 
 m1m2 


G
 G 2   G

2 
 6 
 36 
 (2x3) 
Newton's Law of Gravity 2
• Newton's Law of Gravity is
 m1m2 
Force  G 2 
 r 
• This equation states that if the mass of
each object increased by a factor of two,
the force that object 1 exerts on object 2
increases by a factor of four.
 m1m2 
 2m1 2m2 
 m1m2 
Force  G 2   G
  4G 2 
2
 r 
 r

 r 
Newton and Kepler
• In class, we came up with an equation that
related a planet's orbital speed, v, with its
distance from the sun, r given the sun had mass
m1
Gm1
v 
r
2
• Why is this equation consistent with Kepler's
second law? According to Kepler's second law,
the farther the planet is from the sun, the slower
it moves. According to this equation, the larger r
is, the smaller v is.
Tides
• In the following figure we are assuming
that Earth has two moons. If both moons
are the same distance from Earth, which
moon has a larger mass?
Moon 1. If both moons are the
same distance, the one with the
larger mass will exert greater
tidal forces. Although the sun
has a very large mass, it is very
far away, which is the reason it
has a smaller tidal force than
the moon.
Gravity
• In the following figure, at which position would
the force of gravity due to Earth be about the
same, but in opposite direction, as the force of
gravity due to Mars? Note that Mars has a
smaller mass than Earth.
• At 2, the distance between Mars and ship is equal to distance
between Earth and ship. Earth has a larger mass, so it will pull
harder to the left. If you move to the right (near 3), (a) the distance
from Earth to the ship will increase and so the force of Earth will
decrease and (b) the distance from Mars to the ship will decrease
and so the force of Mars on the ship will increase. At some point
near 3, the forces of Earth to the left will balance the force of Mars to
the right.
 mEarthmship 

FEarth  G 2
 r Earth to ship 
 mMarsmship 

FMars  G 2
 r Mars to ship 
Gravity cont.
• In the following figure, at which position would
the force of gravity due to Earth be about the
same, but in opposite direction, as the force of
gravity due to Mars? Note that Mars has a
smaller mass than Earth.
• At 2, the distance between Mars and ship is equal to distance
between Earth and ship. Earth has a larger mass, so it will pull
harder to the left. If you move to the right (near 3), (a) the distance
from Earth to the ship will increase and so the force of Earth will
decrease and (b) the distance from Mars to the ship will decrease
and so the force of Mars on the ship will increase. At some point
near 3, the forces of Earth to the left will balance the force of Mars to
the right.
 mEarthmship 

FEarth  G 2
 r Earth to ship 
 mMarsmship 

FMars  G 2
 r Mars to ship 
Debating students
•
Given that Earth is much larger and more massive than the Moon, how does the strength
of the gravitational force that the Moon exerts on Earth compare to the gravitational force
that Earth exerts on the Moon?
Student 1: I thought that whenever one object exerts a force on the second object, the
second object also exerts a force that is equal in strength, but in the other direction. So
even if Earth is bigger and more massive than the Moon, they still pull on each other with a
gravitational force of the same strength, just in different directions.
Student 2: I disagree. I said that Earth exerts the stronger force because it is way bigger
than the Moon. Because its mass is bigger, the gravitational force Earth exerts has to be
bigger too. I think that you are confusing Newton’s third law with the law of gravity.
Which student is correct? Student 1. Newton’s third law says that when one object exerts
a force on another object the other object exerts a force that is equal but in the opposite
direction. Also, Newton’s law of gravity says that the force between two objects depends
on the mass of both objects. If I push on someone that is the same size as me, he will
react by falling backward, and I will fall backward as well. We both experienced the same
force, so we both react by falling back in the same way. If I push on someone much
larger, I fall back much more and he barely moves. I experienced the same force as he
did, but because he has more mass, he reacted much less.
m m

F  G 2Earth Moon 
 r Earth to Moon 
Newton’s law of gravity