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Lecture 6

ASTR 111 – Section 002

Ptolmey Copernicus Brahe

Kepler Galileo (Galilei) Newton

Outline

• Exam Results• Finish Chapter 4

– Kepler’s Laws– Newton’s Laws

Exam Results - Average = 89%

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Percent error in student’s guess of their score out of 100 – Average = -0.88.

Error (out of 100)

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 ellipses2. 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 AbbreviationCircle with radius 1. x from -1.0 to 1.0 in steps of 0.1. Compute y using

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2

2

2

=+ry

rx

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Equation for a circle

12

2

2

2

=+ry

rx

Equation for an ellipse

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2

2

2

=+by

ax

Kepler’s First Law

= a

b

Planets orbit the Sun in an ellipseKepler’s Second Law

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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?

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• Suppose that you are looking down on a solar system with one planet orbiting a star. You take a picture every 10 days.

1. Does this planet obey Kepler’s laws? How do you know?

2. How would the speed of this planet change? How would you measure the change in speed?

#1#2

#3

#4

#5

#6#7

#8

#9

#10

#11

#12

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!

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I

A

B

C

D

E

F

GH

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’ssecond 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.

I

A

B

CDE

F

G

H

J

K

L

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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 NewtonIsaac developed three principles, called the laws of motion, that apply to the motions of objects on Earth as well as in space

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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

⎟⎠⎞

⎜⎝⎛= 2

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rmmGForce

Mass m1 Mass m2

A number (T.B.D.)

r

• 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.

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Newton’s Law of Universal Gravitation

⎟⎠⎞

⎜⎝⎛= 2

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rmmGForce

Mass m1

Mass m2

r

A spring

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

⎟⎠⎞

⎜⎝⎛= 2

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rmmGForce

gmr

GmmForce 221

2 =⎟⎠⎞

⎜⎝⎛=

8.9=gRadius of Earth

Mass of Earth

m/s2

What about Bob Beamon?

• The law of universal gravitation accounts for planets not falling into the Sun nor the Moon crashing into the Earth

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⎟⎟⎠

⎞⎜⎜⎝

⎛=

rvmForce

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v v

m2m2

(You will need to take my word on this equation)

v v

m2m2

Now suppose Earth provides “pull” insteadof string and arm

m1

⎟⎠⎞

⎜⎝⎛= 2

21

rmmGForce

⎟⎟⎠

⎞⎜⎜⎝

⎛=

rvmForce

22

⎟⎠⎞

⎜⎝⎛= 2

21

rmmGForce ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

rvmForce

22

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛2

212

2

rmmG

rvm

rGmv 12 =

(Force needed to keep it in orbit)(Force that can be provided)

rGm12 =v

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

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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?

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• 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.

MarsEarth

MarsEarth

• 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.

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