Kepler’s Laws
Kepler’s Laws
Kepler’s 3 laws:
1) The planets move
about the sun in
elliptical paths,
with the sun at
one focus of the
ellipse (slight
exaggeration):
http://www.surendranath.org/Applets/Dynamics/Kepler/Kepler3Applet.html
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
Specific Outcome:
i. I can explain, qualitatively, how Kepler’s laws were used in the
development of Newton’s law of universal gravitation.
Kepler’s Laws
Kepler and Newton’s Law of Gravitation
Kepler’s Laws
Kepler’s Laws
Kepler’s 3 laws:
2) A straight line
joining the
sun and a
given planet
sweeps out
equal areas in
equal times:
http://www.surendranath.org/Applets/Dynamics/Kepler/Kepler1Applet.html
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
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Kepler’s Laws
Kepler’s 3 laws:
3) The square of the period of revolution of
a planet about the sun is directly
proportional to the cube of its mean
distance from the sun:
T2 ∝ r3
With the addition of a proportionality
constant (K), we get an equation: T2 = Kr3
Kepler’s Laws
Kepler’s constant applies to any planetary
orbit, so two planets may be compared:
K=
Ta 2
ra3
Ta =
rb3
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
Ta, Tb = orbital periods (s)
ra, rb = mean orbital radii (m)
ex. Mars has an orbital radius of 2.28 x 1011 m. How
long is a Martian year (in days)?
T2 = Kr3
T=
Kr3 = (2.975 x 10-19)(2.28 x 1011 m)3
= 5.938…x 107 s
(1.176 x 108 m)3
= 359 days
K = Kepler’s proportionality
constant (2.975 x 10-19 s2/m3)
Kepler’s Laws
(5.513 x 109 m)3(1.12 days)2
=
rb3
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
Kepler’s Laws
ra3Tb2
where:
a = 1st planet
b = 2nd planet
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
ex. Proteus, one of Neptune’s moons, has an orbital
radius of 1.176 x 108 m and an orbital period of 1.12
Earth days. Nereid, another of Neptune’s moons,
has an orbital radius of 5.513 x 109 m. What is
Nereid’s orbital period?
=
Tb2
T=
5.938…x 107 s
= 687 d
(60 s/min)(60 min/h)(24 h/d)
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
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Kepler and Newton’s Law of Gravitation
Kepler’s laws helped to inspire Newton’s law
of gravitation:
Gm1m2
Fg =
r2
The ellipses of Kepler’s 1st law imply that
planets accelerate towards the Sun,
suggesting to Newton that a force acts on the
planets
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
Kepler and Newton’s Law of Gravitation
From the equal areas of Kepler’s 2nd law,
Newton could show that force is directed
toward the Sun from the planets
Kepler’s 1st law may be used to show that the
magnitude of force must be inversely
proportional to the square of distance (1/r2)
Kepler’s 3rd law implied that force is directly
proportional to the product of masses (m1m2)
Dulku – Physics 20 – Unit 3 (Circular Motion, Work and Energy) – Topic E
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