How we know what we know

An introduction into orbital mechanics

Matt Hamill

History• Isaac Newton (1643 –

1727)• Did he discover

gravity?– NO!

• What did he do?– He used physics to

connect the force that causes an apple to fall to the force that causes the moon to orbit the Earth

Newton’s Thought Experiment

• If you launch a cannonball it will follow a curved path due to gravitational influences

• If the cannonball is launched at a high enough velocity it will “fall” around the Earth.

Newton’s Hypothesis

• Every particle in the Universe attracts every other particle with a force (gravity) that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

221

d

mmFG

The legendary apple.

• Newton knew a falling apple accelerated toward the Earth around 32 ft/s2 or 10 m/s2

The mysterious Moon

• Newton also knew – the Moon orbited the

Earth in approximately 28 days

– The moon’s distance from the Earth was about 60 times the Earth’s radius

In the 17th Century how can you prove Newton’s hypothesis?

• Newton used his inverse-square prediction to reason that the Moon’s acceleration toward the Earth (centripetal acceleration) should be proportional to:

2

1

MM ra

mrM81084.3

Inverse square continued…

• Using the same logic, the acceleration of the apple toward the Earth should be proportional to: 2

1

ERg

mRE61037.6

Newton’s hypothesis predicts

• Newton predicted the ratio of the Moon’s acceleration (aM) to the apple’s acceleration (g) would be

• Therefore the centripetal acceleration of the Moon should be around

4

2

8

62

2

2

1075.21084.3

1037.6

)1(

)1(

M

E

E

MM

r

R

R

r

g

a

2324 /1070.2)/80.9)(1075.2( smsmaM

What is the Moon’s actual centripetal acceleration?

• We know centripetal acceleration can be calculated with the following formula

• If we assume the orbit of the Moon is circular it travels a distance that is equal to the circumference of a circle.

r

vac

2

rC 2

What is the Moon’s actual centripetal acceleration?

• The Moon completes its orbital period in a time interval T = 27.32 days or 2.36 x 106 s.

26

82

2

222

)1036.2(

)1084.3(44)2(

s

m

T

r

r

Tr

r

va M

M

MM

23 /1072.2 smaM

Predicted vs. Actual

• Predicted centripetal acceleration of the Moon

• Actual centripetal acceleration of the Moon

• Less than 1% difference

23 /1070.2 smaM

23 /1072.2 smaM

Newton’s Law of Universal Gravitation

221

r

mGmFG

• G—gravitational constantG = 6.67 x 10-11 N·m2/kg2

• m1, m2—mass

• r—distance from their centers of mass

Three ways to calculate gravitational force

• Equation1:

• Equation 2:

• Equation 3

mgF

2

r

vmF

221

r

mGmF

Resources

• Serway, Raymond, and John Jewett. Physics for Scientists and Engineers. 6th ed.. USA: Brooks/Cole, 2004. Print.

• http://en.wikipedia.org/wiki/File:GodfreyKneller-IsaacNewton-1689.jpg

• http://scienceiq.com/Images/FactsImages/apple_falling.gif

• http://www.chemheritage.org/women_chemistry/univ/images/clark_moon.jpg

• http://www.astronautix.com/lvs/newannon.htm