-Energy Considerations in Satellite and Planetary Motion

-Escape Velocity-Black Holes

AP Physics CMrs. Coyle

Tangential Velocity of an Orbiting Tangential Velocity of an Orbiting SatelliteSatellite

2

2

GmM mv

r r

GMv

r

SatellitesSatellites

• At a certain r the speeds of the satellites are the same.

• Geosynchronous: satellite that have the same period as earth.

Escape VelocityEscape 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

i fE E

For a Earth-Satellite System For a Earth-Satellite System

• Total energy E = K + U

• Note: in a bound system, E < 0

21

2

MmE mv G

r

Escape VelocityEscape Velocity

2 2

2

1 1

2 2

10

2

i f

i fi f

ii

E E

GMm GMmmv mv

r r

GMmmv

r

esc

2GMv

R

Escape VelocityEscape Velocity

• For any planet:

esc

2GMv

R

Note: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 Two Particle Bound SystemBound System

2 21 1

2 2

i f

i fi f

E E

GMm GMmmv mv

r r

Energy in a Circular Orbit Energy in a Circular Orbit

21

2

MmE mv G

r

2

GMmE

r

Tangential GM

vr

Note: 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 OrbitEnergy in an Elliptical Orbit• r= 2a=the

semimajor axis

2

GMmE

a

• The total mechanical energy,E is negative.

• E is constant if the system is isolated.

Example #63a) 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

21

Earth

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)

Earth 12

Black HolesBlack 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 HolesBlack 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 DisksAccretion 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=hoLvOvGW3Tk&feature=player_embedded#!