Centripetal Force
m
Fc =
Fc
r
v,T or ω
mv2
r
2r
4π
m 2
T
mω2r
Speed
Period
angular speed (radians/s)
These are all equivalent
Period of satellite
in near-earth orbit
g = 9.8 m/s2
T=?
g
R
R = 40 000 km/2π
Period of satellite
in near-earth orbit
g = 9.8 m/s2
R = 4 x 107 m/2π
4π2R
g=
T2
T=
√
4π2R
g
=
√
4π2(4 x 107 m/2π)
9.8 m/s2
= √2.58 x 107 s2 = 5064 s = 84.4 min
Acceleration of moon
r = 60 Re = 60 x 4 x 107/2π m
T = 27.3 days
= 2.3 x 106s
4π2r
a=
T2
a = 2.7 x 10–3 m/s2
Compare
R
a
2
R
1 x Re
9.8 m/s2
1 x Re2
9.8 m/s2 x Re2
3600 x Re2
9.7 m/s2 x Re2
60 x Re 2.7x10–3 m/s2
2
Ra
!
Planets
“Wanderers”
Retograde motion
Closest approach
Tycho Brahe
Johannes Kepler
1546-1601
1571-1630
Brahe’s Observatory in
Ven
Tycho’s Geocentric Model
Tycho’s data disagreed with
models of the planets moving in
circular motion by 8/60 °.
Kepler: “On this 8 minutes I
will build a theory of the
universe!”
Kepler’s Laws
• Planets orbit around sun in ellipses. Nearly circular.
• The area swept out by a planet’s radius arm in a given
time is always the same. It goes faster nearer the sun.
•
3
2
R /T is the same for all the planets
What Kepler’s 3rd Law Means
2R
4π
Fc = m 2
T
3
2
R /T = K
2
2
R/T = K/R
m
Fc
R
2K
4π
Fc = m 2
R
2
The force of gravity scales as 1/R
M
What Kepler’s 3rd Law Means
2K
4π
Fc = m 2
R
Fc = m GM
R2
mM
Fc = G R2
this is a constant— it depends on M :
4π2K = GM
G is the universal gravitational constant
m
Fc
R
This is Newton’s universal gravitational law
M