This is a condensed version of gravitation for my students.

1.
Kepler’s 3rd
Law (T2
∝ r3
)
Gravitational force provides centripetal force
mrω
2
=
GMm
r
2
⇔
2π
T
⎛
⎝⎜
⎞
⎠⎟
2
=
GM
r
3
⇔ T
2
=
4π
2
GM
r
3
Law of Gravitation
The force of attraction
between any two point
masses is directly
proportional to the product of
their masses and inversely
proportional to the square of
their distance apart.
Fgrav and field strength g
Gravitation field strength g
Gravitational field strength at
a point is defined as the
gravitational force per unit
mass acting on a small test
mass placed at that point.
g =
F
m
GRAVITATIONAL FORCE AND FIELD STRENGTH
Gravitational field near surface of planet
• Earth’s curvature negligible at this scale
• Approx. uniform g-field
o field lines near the surface ⊥ to surface
o equal separation lines ⇒ constant g
Questions you should know how to solve
1. Use law of gravitation and field strength formula for up to a few bodies (include
the use of vector addition).
2. Determine point with zero field strength between 2 masses.
3. Compare field strengths given ratio of radius/density/mass.
4. Circular motion with gravitational force provides the centripetal force.
5. Use Kepler’s 3
rd
law to compare periods of orbits.
6. Recognise that the balance reading is dependent on the position of
measurement due to earth’s self-rotation.
Balance readings at different positions
by considering earth’s self rotation
! Satellites
• Use circular motion concepts (see LHS)
• Geostationary – T=24h, above equator at
fixed distance from earth, moves in same
direction as earth’s rotation

2.
GRAVITATIONAL POTENTIAL
AND POTENTIAL ENERGY
M
(fixed)
m

r

FG
∞

Fext
final
position
initial
position
Gravitational potential energy
…is the work done by an external
agent in bringing the mass from
infinity to its present location (without
any change in kinetic energy).
For 2 point masses:
Ugrav
= −
GMm
r
unit: J( )
Expression is negative:
1. Grav. PE is zero at infinity.
2. External force is opposite to the
displacement of the mass m.
3. W.D by external agent is -ve.
Gravitational potential
Gravitational potential at a point is
defined as the work done in bringing
unit mass from infinity to the point.
φ = Ugrav
/ m
Note: the mass m refers to the mass
that is being moved
For a point mass
φ = −
GM
r
unit: J kg−1
( )
Solving problems using Ugrav and Φ
(Step 1) Δφ between two points
Δφ = φfinal
− φinitial
= φB
− φA
(Step 2) change in gravitational
potential energy of mass m
ΔUgrav
= mΔφ
Caution! Negative sign of φ must be substituted
Caution! When calculating the change in potential
energy of the mass, the positive or negative sign of the
potential difference must be substituted
ΔUgrav Interpretations
Positive • Potential energy of mass increases
o External work done/energy transfer
without change in KE, or,
o Loss in KE
Negative • Potential energy of mass decreases
o Energy transfer out without change in
KE, or,
o Gain in KE
Field strength and potential
Gravitational field strength at a point
is numerically equal to the potential
gradient at the point.
g = −
dφ
dr
or graphically, it is the gradient
Δφ
Δr
⎛
⎝⎜
⎞
⎠⎟
Note: -ve indicates g is in direction of
decreasing φ
!Questions you should know how to solve
1. Compute potential (or GPE) at a point (or mass) due to a few bodies (using scalar addition).
2. Compute change in potential (or GPE) between final and initial position (of a mass).
3. Obtain the field strength at a point from the graph of potential with position for simple situations.
4. Understand total energy ET = EK + Ugrav and its implication and for object/satellite in orbit ( EK
=
GMm
2r
U
grav
= −
GMm
2r
ET
= −
GMm
2r
)
5. Use conservation of energy (ET,initial = ET, final or ΔUgrav = ΔEK) for different situations
a) motion between two points b) escape velociy