Gravitational Fields

1. Planetary Motion &
Gravitational Fields

2. From
• Close to earth the acceleration due to gravity
changes slowly.
• As the distance from earth increases, the
change in the acceleration due to gravity
increases.
Variation of g
2
r
GM
g 
www.physicsclassroom.com/Class/energy/U5L1b.cfm

3. Variation of g
On a line joining the centers of two
point masses
• If m1 > m2 then
Note: when g = 0
the gravitational
fields cancel each
other out

4. Escape Velocity
• The velocity needed to overcome
(escape) the gravitational pull of a
planet.
• As the body is moving away from the
planet, it is losing kinetic energy and
gaining potential energy.
• To completely escape from the
gravitational attraction of the planet, the
body must be given enough kinetic
energy to take it to a position where its
potential energy is zero.

5. Escape Velocity
• The potential energy possessed by a body
of mass m, in a gravitational field is given
by
• If the field is due to a planet of mass M and
radius R, then the escape velocity can be
calculated as follows:
• So,
• as g = GM/R²
GPE = Φm
ΔKE = ΔGPE
E
esc
r
GMm
mv 
2
2
1
R
GM
vesc
2

gR
vesc 2


6. Satellites: Orbits & Energy
using Newton’s second law (F=ma)
where v2/r is the
centripetal acceleration.
Solving for v2
So,
Total Mechanical Energy of a satellite is:
PE = -
GMm
r
r
v
m
r
GMm
F
2
2


r
GMm
mv
KE
2
2
2
1

 KE = -
PE
2
ME = KE+PE =
GMm
2r
-
GMm
r r
GMm
ME
2


r
GM
v 
2

7. Satellites: Orbits & Energy
This is for a circular orbit.
• For a satellite in an elliptical orbit with a
semi-major axis a
where a was substituted for r
a
GMm
E
2


ME = -KE

8. Energy Graphs
• As shown the energies for a satellite are,
• Graphing these,
F = -
GMm
r r
GMm
KE
2

r
GMm
ME
2


http://www.opencourse.info/astronomy/introduction/06.motion_gravity_laws/energy_elliptical.gif

10. Energy Wells
• If we rotate the potential energy
graph, we can generate a 3D picture
known as an energy well.
http://www.opencourse.info/astronomy/introduction/06.motion_gravity_laws/

11. Energy Changes in a Gravitational
Field
• A mass placed in a gravitational field
experiences a force.
– If no other force acts, the total energy will
remain constant but energy might be
converted from g.p.e. to kinetic energy.
• If the mass of the planet is M and the
radius of the orbit of the satellite is r, then
it can easily be shown that the speed of
the satellite, v, is given by
– if r decreases, v must increase.
Physics for the IB Diploma 5th Edition (Tsokos) 2008

12. Energy Changes in a Gravitational
Field
• If the satellite’s mass is m, then the kinetic
energy, K, of the satellite is
• the potential energy of the satellite, U, is
These equations show:
– that if r decreases, K increases but U decreases
(becomes a bigger negative number)
– the decrease in U is greater than the increase in K.
• Therefore, to fall from one orbit to a lower orbit,
the total energy must decrease.
– work must be done to decrease the energy of the
satellite if it is to fall to a lower orbit.








r
GM
m
U







r
GM
m
K
2
1

13. Energy Changes in a Gravitational
Field
• The work done, W, is equal to the change in the
total energy of the satellite, W = ΔK + ΔP.
• This work results in a conversion of energy
from gravitational potential energy to
internal energy of the satellite (it makes it
hot!).
• Air resistance reduces the speed of the satellite
along its orbit. This allows the satellite to fall
towards the planet. As it falls, it gains speed.
• So, if a viscous drag (air resistance) acts on a
satellite, it will
– decrease the radius of the orbit
– increase the speed of the satellite in it’s new orbit.

14. Energy Changes in a Gravitational
Field
• In principle, the satellite could settle in a
lower, faster orbit.
• In practice it will usually be falling to a
region where the drag is greater. It will
therefore continue to move towards the
planet in a spiral path.

15. Physics for the IB Diploma 5th Edition (Tsokos) 2008
Energy Changes in a
Gravitational Field
• Looking at this with a fbd of the satellite
spiraling downward due to air resistance.
– Note that the velocity is no longer perpendicular to
the weight.
– Due to this there is a component of gravity which is
now acting in the direction of the velocity providing
a net force which is accelerating the satellite.
Physics for the IB Diploma 5th Edition (Tsokos) 2008

16. Weightlessness
Astronauts on the orbiting space station are
weightless because...
a. there is no gravity in space and they do
not weigh anything.
b. space is a vacuum and there is no gravity
in a vacuum.
c. space is a vacuum and there is no air
resistance in a vacuum.
d. the astronauts are far from Earth's surface
at a location where gravitation has a
minimal affect.
e. None of the above
Source: http://www.physicsclassroom.com/Class/circles/u6l4d.cfm

17. Weightlessness
Astronauts on the orbiting space station are weightless
because...
The astronaut and the space station are both free-falling
together - Apparent Weightlessness
• The gravitational force on the astronaut provided the
needed centripetal acceleration for the astronaut to
stay in orbit.
• The space station remains in orbit because of the
gravitational force on it.
• However, since there is no contact force between the
satellite and the astronaut here is an apparent
weightlessness
Source: Kirk, Tim, Physics for the IB Diploma, Oxford University Press 2007
Source: Kirk, Tim, Physics for the IB Diploma, Oxford University Press 2007
Weightlessness

18. Weightlessness
Astronauts on the orbiting space station are
weightless because...with calculations
• Summing the forces on the astronaut
• For circular motion
• So, or
• Recall for a satellite,
• Substituting,
• With no reaction force (normal) the astronaut
feels weightless
 

 net
N
g F
F
F
F
r
v
m
Fnet
2

(Will learn in the next unit)
r
v
m
F
r
Mm
G
F N
2
2



 








 2
2
2
v
r
M
G
r
m
r
v
m
r
Mm
G
N
r
GM
v 
2
0









r
M
G
r
M
G
r
m
N
Source: Physics for the IB Diploma 5th Edition (Tsokos) 2008