Motion, Forces and Energy Gravitation Part 2

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Transcript Motion, Forces and Energy Gravitation Part 2

Motion, Forces and Energy
Gravitation Part 2
Initially we assumed that the gravitational force on a body
is constant. But now we know that the acceleration due to
gravity is a function of its height above the Earth’s surface:
g = g(h)
We can find a new expression for the gravitational potential
energy by calculating the work done by gravity when the object’s
position changes from r1 to r2:
GM E m
Fg  
r2
r2
Wgrav  GM E m 
r1
Wgrav 
Usually, we choose r1 to be equal to infinity,
so that generally we can write:
1
dr
2
r
GM E m GM E m

r2
r1
GM E m
U 
r
Total Potential Energy of a System of Particles
m2
 m1m2 m2 m3 m1m3 

U total  G 


r23
r13 
 r12
r23
r12
m3
m1
r13
This Utotal represents the work needed to
Separate all three particles by infinity.
Energy Considerations
v
Etotal  KE  U
1
GMm
mv 2 
2
r
GMm mv 2
But

(centripetal )
2
r
r
1
GMm
So
mv 2 
2
2r
GMm GMm
GMm
Etotal 


2r
r
2r
Etotal 
m
M
r
In an isolated system, Etotal is constant so:
1
GMm
1
GMm
2
2
mvi 
 mv f 
2
ri
2
rf
Escape Speed
1
GM E m
GM E m
2
mvi 

2
RE
rmax
Vf=0
KE + U at launch point
 1
1
vi  2GM E 

 RE rmax
2
Vi
Mass, m
rmax
U at distance rmax
from surface (KE=0)



If we know vi, then we can calculate height h reached
by the object. If rmax tends to infinity, then:
2GM E
vi 
RE
2
Escape speed , Vesc 
2GM E
RE
If vi>Vesc, then the object escapes the Earth’s gravitational
field and carries excess kinetic energy.
Escape Speed and Thermal Physics
Escape speed , Vesc
2GM E

RE
Note that the mass of the “escaping” object does not
figure in the equation above for escape velocity.
Number of
Molecules, N
The escape velocity for Earth is 11.2 km/s.
Vesc
N2
Heavy gases are more abundant than
lighter ones on planets and why nearly
all the hydrogen and helium has escaped
from the Earth’s surface.
H2
Speed (m/s)
Escape speeds for the planets and the sun.
Body
Mercury
Venus
Earth
Moon
Mars
Jupiter
Saturn
Uranus
Neptune
Pluto
Sun
Escape Speed
(km/s)
4.3
10.3
11.2
2.3
5.0
60
36
22
24
1.1
618