Kepler and Newton, Summaries of Law

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Lecture 6

Kepler and Newton

September 24, 2018

Kepler’s Three Laws

• One: Planetary orbits are ellipses
• Two: Law of equal areas
• Three: P 2

= a

3

• P = sidereal period in years
• a = semi-major axis in AU
• The closer a planet is to the Sun, the less time it takes to go around the Sun.

Isaac Newton (1642-1727)

• Laws of Motion
• Law of Gravity

Definitions

• Velocity = speed and direction
• Acceleration = a change in velocity
  • change in speed and/or direction
• Mass = measure of how much matter is

present

• Weight = the force of gravity on an object

Newton’s Second Law

• The acceleration of an object is proportional

to the force acting on it.

• The larger the force, the faster the velocity increases or decreases or the direction changes.
• Need more force to accelerate massive objects
• Objects traveling in a circle accelerate because direction, not speed, changes. m F a F ma = =^ F^ = Force

m = mass

a = acceleration

• Whenever a body

exerts a force on

another body, the

second body exerts

and equal and

opposite force on the

first body.

Newton’s Third Law “Action-Reaction Principle”

Newton’s Law of Gravitation

• There is an attractive force between all objects

with mass.

1 2 2 m m Force G r =

G = 6.67× 10

− N∙m 2 /kg 2 m 1 and m 2 = masses in kg r = Distance between two objects in meters

• If mass increases, force increases
• If distance increases, force decreases

• Force decreases

rapidly with distance

• Interactive applet 2 1 r Force  Gravitation

How much larger or smaller is F

compared to F

? A. F new is 2 times greater than F old

B. F

new is ½ the force F old

C. F

new is 4 times greater than F old

D. F

new is ¼ the force F old

r r F 1 F 2 m m^2 m m

old (^2) new 2 2 old new old 2 2 2 2 Gmm F r G m m (^) Gmm F F r r F F = = = = =

How much larger or smaller is F

new

compared to F

old

?

A. F

new is 2 times greater than F old

B. F

new is ½ the force F old

C. F

new is 4 times greater than F old

D. F

new is ¼ the force F old

r 2 r F old F new m m m m

Orbits of Bodies

• Balance between force of gravity and inertia

of planet

Acceleration and force Velocity

Newton’s General form of

Kepler’s Third Law

• Useful for determining masses, works

anywhere in the universe!

• When M 2

is small, eq. becomes M P

2

= a

3

Sun 2 3 1 2 in solar masses (1.0 = mass of Sun) in years in astronomical units (AU) M M P a

M + M P = a

A. 2.5 solar masses B. 6.3 solar masses C. 10 solar masses D. 31 solar masses ( ) ( ) ( ) ( ) 2 3 1 2 2 3 2 3 3 (^2 ) 0 5 AU 2 y 125 31 4 M M P a M P a a M P M

• =
• = = = = =

A small planet orbits a star. The semimajor axis of

its orbit is 5 AU and its orbital period is 2 years.

What is the mass of the star?

Two identical planets each have a mass of

0.0010 M

Sun

and orbit each other with a

semimajor axis of 0.0040 AU. What is the

period of their orbit?

A. 0.000011 y

B. 0.0057 y

C. 0.008 y

D. 177 y