1

Gravity Notes for PHYS 1121-1131. Joe Wolfe, UNSW

Gravity: where does it fit in?

Gravity Weak Strong
Electric Colour
[general nuclear nuclear
force* force
relativity] force force
intermediate
gravitons photons vector bosons pions gluons

electro-weak

Grand Unified Theories

Some tries
for classical
gravity

Theories Of Everything
* Electromagnetism "unified" by Maxwell, and also by Einstein: Magnetism
can be considered as the relativistic correction to electric interactions which
applies when charges move.

• Only gravity and electric force have macroscopic ("infinite")

range.
m graviton? = mphoton = 0
• Gravity weakest, but dominates on large scales because it is
always attractive

Greeks to Galileo:
i) things fall to the ground ('natural' places)
ii) planets etc move (variety of reasons)
but no connection (in fact, natural vs supernatural)
Newton's calculation: acceln of moon

= rmωm2
2
= (3.8 108 m) (27.3 242π 3600)
= 2.7 10-3 m.s-2
acceln of "apple" = 9.8 ms-2

aapple rm 385000 km
amoon = 3600; Re = 6370km = 60;
 rm  2
 R  = 3600
 e
Newton's brilliant idea: What if the apple and the moon accelerate according to the same law? →
What if every body in the universe attracts every other, inverse square law?
 2

Newton's law of gravity:

m1m2 Negative sign means

F = −G _ // − r_
F
r2
Why is it inverse square? Wait for Gauss' law in electricity.

_ 12 = − F
F _ 21 Newton 3

Newton already knew Kepler's empirical law:

For planets, r3 ∝ Τ2 orbit radius and period
1
Now if F ∝ ac ∝ 2
r
then constant = acr = rω2r2 = r3ω2
2

Planet r from sun T ω rω2 r3ω2

million km Ms rad.s-1 ms-2
Mercury 58 7.62 8.25 10-7 3.95 10-5 1.31 1020 m3s-2
Venus 108 19.4 3.23 10-7 1.13 10-5 1.32 1020 m3s-2
Earth 150 31.6 1.99 10-7 5.94 10-6 1.33 1020 m3s-2
etc
 3
How big is G? Cavendish's experiment (1798)

m1m2
F = −G
r2
From deflection and spring constant, calculate F, know m1 and m2, ∴ can
calculate G. G = 6.67 10-11 Nm2kg-2 ( or m3kg-1s-2
m.Me
Now also weight of m: |W| = mg ≅ G
Re2
∴ Cavendish first calculated mass of the earth:
gr 2 9.8 m.s-2 x (6.37 106 m)2
Me = e = = 6.0 1024 kg
G 6.67 10-11 Nm2kg-2
see http://www.physicscentral.com/action/action-01-5-print.html
and http://physics.usask.ca/~kolb/p404/cavendish/

Some numbers
What is force between two oil tankers at 100 m?
m m
F = − G 12 2
r
What happens when more there are ≥ 3 bodies?
Superposition principle.
_ all objects together = Σ F
F _ individual
_1 = Σ F
force on m 1
or F _ 1i due to masses m i
i
continuous F
_1 = ∫ dF
_
body body
 4
Shell theorem
A uniform shell of mass M causes the same gravitational
force on a body outside is as does a point mass M located at the
centre of the shell, and zero force on a body inside it.

r
dθ Fg
m
F=0 R

M dm GMm
Fg =
R2

Example. If ρearth were uniform (it isn't), how long would it take
for a mass to fall through a hole through the earth to the other
side?

Mr = ρ.43 πr3

mρ.43 πr3
∴ Fr = − G
r2
F = − Kr where K = Gmρ.34 π

√ m
K
∴ motion is SHM with ω = Simple Harmonic Motion: discussed later

2π 2π 2π
Τ = ω = = = .... = 84 minutes
 π
√ Gρ.43 
√ GM/R3

∴ falls through (one half cycle) in 42 minutes (actually faster for real density profile)
 5
Gravity near Earth's surface
Mm
W = |F g | = G 2
Re
Mm
W = mgo = G e2
r
go is acceln in an inertial (non-rotating) frame
M
go = G 2e
r
Usually, r ≅ Re, but
Me  R 2
go = G = gs  R +e h
(Re + h) 2  e 
 1 2 where g s is
= gs  1 + h/R 
 e g o at surface
Other complications:
i) Earth is not uniform (especially the crust) useful for prospecting
ii) Earth is not spherical
iii) Earth rotates (see Foucault pendulum)

(Weight) = − (the force exerted by scales)

At poles, _ − N
F _ = 0
At latitude θ, _F − N
_ = ma _
where a = rω2 = (Re cos θ)ω2
= .... = 0.03 ms-2 at equator
= 0 at poles
N
_ _ − ma_
F
We define − _ g = =
m m
So g_ is greatest at the poles, least at the equator, and does not
(quite) point towards centre.

horizontal _| g_
Earth is flattened at poles
 6
Puzzle: How far from the earth is the point at which the
gravitational attractions towards the earth and that towards the sun
are equal and opposite? Compare with distance earth-moon
(380,000 km)

|Fe|= |Fs|
GMem GMsm
=
d2 (re - d)2
Me(re − d)2 = Msd2
M
re2 − 2red + d2 = M s d2
e
Ms 
 M − 1 d2 + 2red − re2 = 0
 e 
d = .... = ?

Gravitational field. A field is ratio of force on a particle to some

property of the particle. For gravity, (gravitational) mass is the
property:
F
_ grav
m = g_ = g_(r_) is a vector field
_Felec
cf electric field
q =E _ (r_) later in syllabus
Gravitational potential energy. Revision:
Potential energy
For a conservative force _F (i.e. one where work done against it, W =
_)) we can define potential energy U by ∆U = Wagainst. i.e.
W(r
f
∆U = − ∫ F_ . dr
__
i
_ ≅ constant
near Earth's surface, _Fg = mg
f
= − ∫ (-mgk
_) . (dx_i + dyj_ + dzk
_)
i
f
= mg _k . k
_ ∫ dz = mg (zf - zi)
i
choose reference at zi = 0, so U = mgz

Gravitational potential energy of m and M.

M m
r

Fg ds
f f
f Mm 1 1
∆Ug ≡ − ∫ F ds = ∫ Fgdr
_ g . __ = ∫ G 2 dr = − GMm[r − r ]
i i i r f i
GMm
Convention: take ri = ∞ as reference: U(r) = − r
U = work to move one mass from ∞ to r in the field of the other. Always negative.
Usually one mass >> other, we talk of U of one in the field of the other, but it is U of both.
 7
Escape "velocity".
Escape "velocity" is minimum speed ve required to escape, i.e.
to get to a large distance (r → ∞).

v
M
r
R m
Projectile in space: no non-conservative forces so
conservation of mechancial energy
K i + U i = Kf + U f
1
mv 2 − GMm = 0 + 0
2 e R

√R
2GM
vesc =

For Earth: vesc =

√
2 6.67 10-11 m3kg-1s-2 5.98 1024 kg

= 11.2 km.s-1 = 40,000 k.p.h.

6.37 106 m

Put launch sites near equator: veq = Reωe = 0.47 km.s-1

Question In Jules Verne's "From the Earth to the Moon", the

heros' spaceship is fired from a cannon*. If the barrel were 100 m
long, what would be the average acceleration in the barrel?
vf2 − vi2 = 2as
v 2-0 (1.12 104 ms-2)2
a = e2s =
2 x 100 m
= 630,000 ms-2 = 64,000 g

* why? If you burn all the fuel on the ground, you don't have to
accelerate and to lift it. Much more efficient.

Planetary motion
"The music of the spheres" - Plato
Leucippus & Democritus C5 B.C.
heliocentric universe
Hipparchus (C2 BC) & Ptolemy (C2 AD) geocentric universe
Tycho Brahe (1546-1601) - very many, very careful, naked eye
observations.
Johannes Kepler joined him. He fitted the data to these empirical
laws:
Kepler's laws:
1 All planets move in elliptical orbits, with the sun at one
focus.
Except for Pluto and Oort cloud objects, these ellipses are ≅ circles.
M sun >> mplanet, so sun is ≅ c.m.
2 A line joining the planet to the sun sweeps out equal areas in
equal time.
Slow at apogee (distant), fast at perigee (close)
3 The square of the period ∝ the cube of the semi-major axis
Slow for distant, fast for close
 8
Newton's explanations:
Law of areas:

Area = 21 r.rδθ
1
i.e. for same δt, 2 r2δθ = constant
Conservation of angular momentum L
_ . Sun at c.m.
momentum
∴ _ | = |r_ x _p| = |r_ x mv_|
|L = p. see later
= mrvtangential
δθ
= mr.rω = mr2
δt
m
= δt r2δθ = constant.
_ ⇒ Kepler 2.
Conservation of L
Law of periods: (we consider only circular orbits)
Kepler 3: T2 ∝ r3
Newton 2 → F = ma = m rω2
Mm  2π 2
G 2 = mr  T 
r  
 4π2 
T2 =  GM r3 → Kepler 3
(works for elipses with semi-major axis a instead of r)
Newton 2 &
Newton's gravity ⇒ Kepler 3
Newton 2 &
also ⇒ Kepler 1
Newton's gravity

Example What is the period of the smallest earth orbit? (r ≅ Re)

What is period of the moon? (rmoon = 3.82 108 m)

 √
√
 4π2  3 4 π2
T1 =  r = (6.37 10 6 ) 3 s
 GM 6.67 10 -11 5.98 10 2 4
= 84 min
Kepler 3: T2 ∝ r3
T2  r2 3/2  3.82 108 3/2
=   =   = 464
T1  r1  6.37 106
T2 = 464 T1 = 27.2 days

For other planets: most have moons, so the mass of the planet
can be calculated from
 4π2 
T2 =  GM r3
 9
Orbits and energy
No non-conservative forces do work, so mechanical energy is
constant:
E = K + U
GMm
= 1 mv2 −
2 r
Let's remove v. Consider circular orbit:
v2 F GMm
r = ac = m = r2m

∴ 1
mv2 = 1 GMm
2 2 r
E = K + U
GMm GMm
= 1 − r
2 r
GMm
= − 2r

i.e. E = 21 U, or K = − 12 U, or K = − E.
Small r ⇒ U very negative, K large. (inner planets fast, outer slow)

Example A spacecraft in orbit fires rockets while pointing

forward. Is its new orbit faster or slower?

ds ∴ Work done on craft

_ // __
F
W = ∫F _ . __
ds > 0.
GMm
∴ E = − 2r increases, i.e. it becomes less negative. (R is
larger). K = - E, ∴ K smaller, so it travels more slowly.
called "Speeding down"
Quantitatively:
Ki = − Ei Kf = − Ef = − (Ei + ∆E)
Kf = Ki − ∆E
1
mvf2 = 21 mvf2 − W
2
Looks odd, but need lots of work to get to a high, slow orbit.
 10
Manœuvring in orbit.

To catch up, vessel 1 fires engines backwards, and loses energy.

It thus falls to a lower orbit where it travels faster, until it catches
up. It then fires its engines forwards in order to slow down (it
climbs back to the original, slower orbit).

Example: In what orbit does a satellite remain above the same

point on the equator?
Called the Clarke Geosynchronous Orbit
i) Period of orbit = period of earth's rotation
ii) Must be circular so that ω constant
T = 23.9 hours
 4π2 
T2 =  GM r3

√

3
GMT2
r = = .....
4π2
= 42,000 km popular orbit!