phys 20 lessons unit 5: circular motion gravitation lesson 5: gravitation
TRANSCRIPT
Reading Segment #1:
Early TheoriesKepler’s Laws
To prepare for this section, please read:
Unit 5: pp. 16-17
Introduction to Gravitation
Throughout history, people have wondered about how
and why the stars and planets move.
Early Greeks (6th / 7th century B.C.)
They believed that the stars revolved around the Earth
in perfect circles.
This is called the geocentric view of the heavens.
"geo" = Earth
"centric" = centre So, Earth-centred
Copernicus (1543)
Copernicus suggested that the planets revolve about the
sun in perfect circles.
This is called the heliocentric ("sun-centred") view
of the heavens.
This was a key advance for modern understanding.
D1. Kepler's Laws
In the 16th century, Tyco Brahe carefully observed
planetary motion for 20 years.
He then gave all of this data to his student, Johann Kepler
In 1609, Kepler found patterns in the data.
He summarized these patterns into 3 laws.
Law #1: Elliptical Orbits
- the planets revolve about the sun in ellipses
- the sun is one of the two foci of the ellipse
i.e.
F1 F2
Animation
Kepler's 2nd Law:
http://www.walter-fendt.de/ph11e/keplerlaw2.htm
Law #3: Kepler's Equation
The ratio T2 / R3 is constant for all planets revolving about
the sun (or about any central mass).
i.e. k = T12 = T2
2
R13 R2
3
where
R is the average radius of the oribit (in m)
T is the orbital period, or time for 1 lap (in s)
k is Kepler's constant
Ex. 1 An asteroid has a period of 8.1 107 seconds.
What is its mean (average) radius around the sun?
Note: Kepler's constant for objects around the sun
is 2.985 10-19 s2 / m3
Inverse Square Law (Newton, 1687)
Kepler could not explain why the planets moved this way,
but Isaac Newton could.
When an apple fell on Newton's head due to Earth's gravity,
he wondered:
Could the Moon be held in orbit due to Earth's gravity
as well?
Is it not the same kind of force (Fg) ?
Consider the Moon going in uniform circular motion
around the Earth: Fg
Moon (m)
ac
Earth
The force of gravity is the centripetal force,
holding the Moon in orbit.
Combining Fg = m 4 2 r and T2 = k R3
T2
we find
Fg = m 42 R
k R3
or
Fg = m 42
k R2
where R is the distance from the centre of the planet
Newton's Conclusion: Based on Fg = m 42
k R2
Fg has an inverse square relationship with R.
i.e. Fg 1
R2
Newton was able to use this formula to accurately predict
the centripetal acceleration of the Moon (ac = 2.7 10-3 m/s2)
Ex. 2 An object experiences a Fg of 18 N at a distance D
from the centre of the Earth. What would this Fg be
if the distance was 6D ?