Newton’s 2
nd
Law
Uniform Circular Motion
planetary motion due to gravity requires understanding of circular motion (though most orbits are
elliptical)
r
A
B
v1
v2
Figure 1
Figure 2
figure 1 shows the circular motion of an object with radius and speed constant
velocity is perpendicular to radius (
tangential
)
v
1
and v
2
are instantaneous velocities at A and B, respectively
figure 2 shows
Δ
v = v
2
– v
1
if A and B are very close together, then arc AB
≈
chord AB
thus, because of similar triangles
in figures 1 & 2…
r
v
t
v
a
t
v
v
v
AB
chord
v
v
2
r
r
=
∆
∆
=
∆
≈
∆
≈
∆
this describes centripetal (or “centerseeking”) acceleration (a
c
)
r
v
a
c
2
=
since
T
r
v
p
2
=
in a circle (where T is the period of revolution), substituting…
2
2
4
T
r
a
c
p
=
force that causes circular motion is called
centripetal force (F
c
)
2
2
4
2
T
r
m
r
mv
ma
F
F
c
net
p
=
=
=
=
V1
V2
Δ
V
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View Full DocumentKepler’s Laws of Planetary Motion
Tycho Brahe (15461601) carefully recorded, tracked & predicted movements of planets and comets
he believed Earth as center of planetary orbits
his student, Johannes Kepler (15711630), believed Sun as center
also believed planetary motion could be studied using geometry & math
Tycho Brahe
Johannes Kepler
Kepler’s 1
st
Law of Planetary Motion
The paths of the planets are ellipses
with the centre of the Sun at one
focus.
Kepler’s 2
nd
Law of Planetary Motion
An imaginary line from the Sun to a
planet sweeps out equal areas in
equal time intervals.
Thus planets
move fastest when closest to the
Sun, slowest when farthest away.
Kepler’s 3
rd
Law of Planetary Motion
The ratio of the squares of the
periods of any two planets revolving
around the Sun is equal to the ratio
of the cubes of their average
distances from the Sun.
Thus if Ta
and Tb are their periods and ra and
rb are their average distances,
then:
3
2
=
b
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 Fall '10
 Jay
 Mass, General Relativity, Gravitational constant

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