Download Mass and Weight are not the same

Lecture 6
ASTR 111 – Section 002
Ptolmey
Copernicus
Kepler
Galileo (Galilei)
Brahe
Newton
Outline
• Exam Results
• Finish Chapter 4
– Kepler’s Laws
– Newton’s Laws
Exam Results - Average = 89%
Percent error in student’s guess of their
score out of 100 – Average = -0.88.
Kepler proposed elliptical paths for the
planets about the Sun
• Using data collected by
Brahe, Kepler deduced
three laws of planetary
motion:
1. the orbits are ellipses
2. a planet’s speed varies
as it moves around its
elliptical orbit
3. the orbital period of a
planet is related to the
size of its orbit
Text these numbers
Abbreviation
Circle with radius 1. x
from -1.0 to 1.0 in steps
of 0.1. Compute y using
2
2
x
y


1
2
2
r
r
Equation for a circle
2
2
x
y


1
2
2
r
r
Equation for an ellipse
2
2
x
y


1
2
2
a
b
Kepler’s First Law
Planets orbit the Sun in an ellipse
b
=a
Kepler’s Second Law
Kepler’s Third Law
Kepler’s Laws
• Planet orbit is ellipse
• Equal area in equal time
• Farther away planets orbit slower
Is this an ellipse?
•
Suppose that you are
looking down on a solar
system with one planet
#9
orbiting a star. You take
a picture every 10 days.
1. Does this planet obey
Kepler’s laws? How do
#10
you know?
2. How would the speed of
this planet change?
How would you measure #11
the change in speed?
#8
#7
#6
#5
#4
#3
#12
#2
#1
Based on Lecture-Tutorials for Introductory Astronomy
2nd
ed., Prather et. al, page 21
• The following planet obeys Kepler’s second law.
3. Draw two lines: one connecting the planet at Position A to the star and a
second line connecting the planet at Position B to the star. Shade in the
triangular area swept out by the planet when traveling from A to B.
4. Which other two planet positions, out of C-I, could be used together to
construct a second swept-out triangular area that would have
approximately the same area as the one you shaded in for Question 3?
Shade in the second swept-out area using the planet positions that you
chose. Note: Your triangular area needs to be only roughly the same size;
no calculations are required.
5. How would the time it takes the planet to travel from A to B compare to the
time it takes to travel between the two positions you selected in the
previous questions? Explain your reasoning!
6. During which of the two time intervals for which you sketched the triangular
areas in questions 3 and 4 is the distance traveled by the planet greater?
7. During which of the tow time intervals for which you sketched the triangular
areas in Questions 3 and 4 would the planet be traveling faster? Explain
your reasoning!
C
D
B
E
A
F
G
H
I
8. The drawing on the following slide shows another
planet. In this case, the twelve positions are exactly
one month apart. As before, the plane obeys Kepler’s
second law.
9. Does the planet appear to be traveling the same
distance each month?
10. At which position would the planet have been traveling
the fastest? The slowest? Explain your reasoning.
11. At position D, is the speed of the planet increasing or
decreasing? Explain.
12. Provide a concise statement that describes the
relationship that exists between a planet’s orbital speed
and the planet’s distance from its companion star.
E
D
F
C
B
G
A
L
H
K
I
J
Lingering questions
• Kepler’s laws are not so “clean”
• Need to explain
– Why orbits of planets are elliptical
– Why distance from Sun is related to orbital period
– Why planet velocity changes during orbit
• Also want a recipe that gives good predictions of
when eclipses will occur, where the planets will
be in the future.
Lingering questions
• Kepler’s laws are not so “clean”
• Need to explain
–
–
–
–
Why orbits of planets are elliptical
Why distance from Sun is related to orbital period
Why planet velocity changes during orbit
Why people on the south pole don’t fall into space …
• Also want a recipe that gives good predictions of
when eclipses will occur, where the planets will
be in the future.
Isaac Newton
Isaac developed
three principles,
called the laws of
motion, that apply
to the motions of
objects on Earth
as well as in
space
Newt’s “Principles” (Laws of Motion)
1. The law of inertia: a body remains at rest,
or moves in a straight line at a constant
speed, unless acted upon by a net outside
force
2. F = m x a: the force on an object is directly
proportional to its mass and acceleration,
provided the mass does not change
3. The principle of action and reaction:
whenever one body exerts a force on a
second body, the second body exerts an
equal and opposite force on the first body
Group Question
• An object at rest tends to stay at rest.
An object in motion tends to stay in
motion.
–What is wrong with this statement?
–Why don’t we observe “objects in
motion tending to stay in motion”
more often?
Newton’s Law of Universal
Gravitation
A number
(T.B.D.)
 m1m2 
Force  G 2 
 r 
r
Mass m1
Mass m2
• Mass and Weight are not the
same
–Mass refers to how much stuff is
in an object (atoms, molecules,
etc).
–Weight refers to how much that
stuff will push down on a scale.
This depends on what planet
you are on.
Newton’s Law of Universal
Gravitation
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.
How to get Weight = mass x gravity
 m1m2 
Force  G 2 
 r 
Mass of Earth
 Gm1 
Force  m2  2   m2 g
 r 
2
Radius of Earth
m/s
g  9.8
What about Bob Beamon?
• The law of universal gravitation
accounts for planets not falling
into the Sun nor the Moon
crashing into the Earth
v
v
m2
m2
 m2v
Force  
 r
2



(You will need
to take my
word on this
equation)
 m2v
Force  
 r
v
m1
m2
2


v
Now suppose Earth
m2
provides “pull” instead
of string and arm
 m1m2 
Force  G 2 
 r 
 m1m2 
Force  G 2 
 r 
(Force that can be provided)
 m2v

r

 m2v
Force  
 r
(Force needed to keep it in orbit)

 m1m2 
  G 2 
r



Gm1
2
v 
r
2
2



Is this right?
• G = 6.7 x 10-11 N.m2/kg2
• m1 = 2 x 1030 kg
• Mars
– Orbital velocity = 24 km/s
– Distance from Sun = 228 x 109 km
• Earth
– Orbital velocity = 30 km/s
– Distance from Sun = 150 x 109 km
Gm1
v 
r
2
Compare
• Kepler’s 3rd law relates orbital speed and
radius
• Newton’s law of gravitation was used to
derive a relationship between orbital
speed and radius
• Both will give the same answer. Which is
“better”?
To get something in orbit, you need a
special horizontal velocity
• The law of universal
gravitation accounts for
planets not falling into the
Sun nor the Moon crashing
into the Earth
• Paths A, B, and C do not
have enough horizontal
velocity to escape Earth’s
surface whereas Paths D,
E, and F do.
• Path E is where the
horizontal velocity is
exactly what is needed so
its orbit matches the
circular curve of the Earth
Question
• How far would you have to go from Earth
to be completely beyond the pull of
gravity?
• Suppose the Earth was 2x its current
radius (with the same mass). How would
your mass change? How would your
weight change?
• 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? Explain your reasoning.
• Consider the following debate between two students about
their answer to the previous question.
– 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.
– Do you agree or disagree with either or both students? Explain.
• How would the strength of the force between the Moon and
Earth change if the mass of the Moon were somehow made
two times greater than its actual mass?
Earth
Mars
• In the picture, a spaceprobe traveling from Earth to Mars is
shown at the halfway point between the two (not to scale).
• On the diagram, clearly label the location where the
spaceprobe would be when the gravitational force by Earth
on the spacecraft is strongest. Explain.
• On the diagram, clearly label the location where the
spaceprobe would be when the gravitational force by Mars
on the spacecraft is strongest. Explain your reasoning.
• Where would the spaceprobe experience the strongest net
(or total) gravitational force exerted on it by Earth and Mars?
Explain your reasoning.
• When the spacecraft is at the halfway point, how does the
strength and direction of the gravitational force on the
spaceprobe by Earth compare with the strength and
direction of the gravitational force on the spaceprobe by
Mars. Explain your reasoning.
Earth
Mars
• If the spaceprobe had lost all ability to control its motion and was
sitting at rest at the midpoint between Earth and Mars, would the
spacecraft stay at the midpoint or would it start to move.
– If you think it stays at the midpoint, explain why it would not
move.
– If you think it would move, then (a) Describe the direction it
would move; (b) describe if it would speed up or slow down; (c)
describe how the net (or total) force on the spaceprobe would
change during this motion; and (d) identify when/where the
spaceprobe would experience the greatest acceleration.
• Imagine that you need to completely stop the motion of the
spaceprobe and have it remain at rest while you perform a shutdown
and restart procedure. You have decided that the best place to carry
out this procedure would be at the postion where the net (or total)
gravitational force on the spaceprobe by Mars and Earth would be
zero. On the diagram, label the location where you would perform
this procedure. (Make your best guess; there is no need to perform
any calculations here.) Explain the reasoning behind your choice.
• Your weight on Earth is simply the gravitational force that Earth
exerts on you. Would your weight be more, less, or the same on the
Mars. Explain your reasoning.