Transcript Rotational Motion and Equilibrium

-Energy Considerations in
Satellite and Planetary Motion
-Escape Velocity
-Black Holes
AP Physics C
Mrs. Coyle
Tangential Velocity of an Orbiting
Satellite
GmM mv

2
r
r
GM
v
r
2
Satellites
• At a certain r the speeds of the
satellites are the same.
• Geosynchronous: satellite that have
the same period as earth.
Escape Velocity
• The minimum speed
required to launch an
object from the earth’s
surface in order for it to
escape the earth’s pull.
To find the escape velocity of an
object use conservation of energy
Energy at Earth's surface= Energy at Infinity
Ei  E f
For a Earth-Satellite System
• Total energy E = K
+ U
1 2
Mm
E  mv  G
2
r
• Note: in a bound system, E < 0
Escape Velocity
Ei  E f
1
GMm 1
GMm
2
2
mvi 
 mv f 
2
ri
2
rf
1
GMm
2
mvi 
0
2
ri
vesc
2GM

R
Escape Velocity
• For any planet:
vesc
2GM

R
Note:
–According to Newton’s Law of Universal
Gravitation the gravitational field even
at infinity is does not equal to zero but
approaches zero.
–Some planets have atmospheres and
others do not because their escape
velocities vary and some gas molecules
have high enough speeds to escape.
Two Particle
Bound System
Ei  E f
1
GMm 1
GMm
2
2
mvi 
 mv f 
2
ri
2
rf
Energy in a Circular Orbit
1 2
Mm
E  mv  G
2
r
GM
Tangential v 
r
GMm
E
2r
Note: Energy in a Circular Orbit
• K>0 and is equal to half the absolute
value of the potential energy.
• |E| = binding energy of the system.
• The total mechanical energy is
negative.
Energy in an Elliptical Orbit
• r= 2a=the
semimajor axis
GMm
E
2a
• The total mechanical energy,E is negative.
• E is constant if the system is isolated.
Example #63
a) Determine the amount of work (in Joules)
that must be done on a 100kg payload to
elevate it to a height of 1,000km above the
earth’s surface.
b) Determine the additional work required to
put the payload into circular orbit at this
elevation
(The radius of the earth is 6.37x106 m,
G=6.67x10-11 Nm2 / kg2)
Ans: a)8.50x108 J, b) 2.71x109 J
Note: For a Two Particle Bound
System
• Both the total energy
• and
• the total angular momentum are
constant.
Compare the Kinetic Energy and
Angular Momentum of a Satellite
at orbit 1 and 2
1
Earth
2
How does the speed of a satellite at
position 2 compare to the speed at
position 1. The distance r2 =2r1.
(Hint: Use conservation of angular
momentum)
2
Earth
1
Black Holes
• A black hole is the remains of a star that
has collapsed under its own gravitational
force
• The escape speed for a black hole is very
large due to the concentration of a large
mass into a sphere of very small radius
– If the escape speed exceeds the speed of
light, radiation cannot escape and it appears
black
Black Holes
• The radius at which the
escape speed equals the
speed of light, c, is called
the Schwarzschild
radius, RS
• An imaginary surface of
a sphere with this radius
is called the event
horizon.
• If an object is not closer
than the Rs , it can still
escape the black hole.
Accretion Disks
Material from a nearby
star (in a binary
system) can be pulled
into the black hole and
forms an accretion disk
around the black hole.
Black Hole Video Clip
http://www.youtube.com/watch?v=hoLvOv
GW3Tk&feature=player_embedded#!