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LAW OF UNIVERSAL GRAVITATION
221
r
mGmFG
FG gravitational force (in two directions)
G universal gravitation constant 6.67x10-11 Nm2kg-2
r distance between the objectsm1 mass of the larger object
21
221
2
r
Gmg
r
mGmgm
near the earth’s surface . . .
both of these equations could be applied to the surface of any planet
Planet X has a radius that is 3.5 times the radius of the earth and a mass that is 2.0 times the earth’s. Compare the acceleration due to gravity at the surface of each planet.
21
r
Gmg
ex
e
ex
e
ex
x
xx
gg
r
Gmg
r
mg
r
Gmg
)5.3
2(
)5.3
2(
)5.3(
2
2
22
2
2
163.0e
x
g
g
What happens to the gravitational attraction between two particles if one mass is doubled, the other tripled and the distance between them cut in half?
221
2
21
1
2)
2(
32
rmGm
rmmG
F
F 24
1
2 F
F
read p. 139-142
p. 141 1-6 extra p. 143 8-13
p. 144 1-6
SATELLITES
A satellite is an object or a body that revolves around another object, which is usually larger in mass.Planets, moons, space shuttles, space stations, comets, and “satellites” are satellites.Satellites remain in a constant orbit because they are acted upon by a centripetal force and display centripetal acceleration.
r
Gmv
r
Gmv
r
vm
r
mGm
FF cG
1
12
22
221
remember m1 is the larger mass and the central object
3221
12
12
4
2
rTGm
r
Gm
T
r
r
Gmv
What is the period of rotation of the moon about the earth?
3221 4 rTGm
1
324
Gm
rT
kgkg
Nm
mT
242
211
382
1098.51067.6
)1084.3(4
sT 610367.2
dT 40.27
read 145-146
p. 151 1, 3-6
extra p. 147 2-4, 6
p. 160 14-20
GRAVITATIONAL FIELDS
A force field exists in the space surrounding an object in which a force is exerted on objects (e.g. gravitational, electric, magnetic). The strength of gravitational force fields is deter-mined by the Law of Universal Gravitation. If two or more gravitational fields are acting on an object then the net field is the sum of all the individual fields.
read 274-275 p.276 2-6 p.277 1-8
KEPLER’S LAWS
In 1543 Copernicus proposes the heliocentric model of the solar system in which planets revolve around the sun in circular orbits. Slight irregularities show up over long periods of study.
Tycho Brahe takes painstaking observations for 20 years with large precision instruments but dies (1600) before he can analyze them properly.
A young mathematician continues Brahe’s work.
From his analysis the kinematics of the planets is fully understood.
Kepler’s First Law of Planetary Motion
Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse.
Kepler’s Second Law of Planetary Motion
The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time.
Planets move faster when they are closer to the Sun (centripetal force is stronger).
equal areasequal times
orbits are elliptical but are not very elongated
Kepler’s Third Law of Planetary Motion
The cube of the average radius of a planet is directly proportional to the square of the period of the planet’s orbit.
We have already proved this a few slides back. Recall
r
vm
r
mGm 22
221
rTr
m
r
mGm2
2
221
)2
(
21
2
3
4Gm
T
r
For our solar system m1 is the mass of the sun.
constant
Mars’ average distance from the sun is 2.28 x1011 m while its period of rotation is 5.94 x 107 s. What is Jupiter’s average distance from the sun if its period of rotation is 3.75 x 108 s ?
2
3
2
3
m
m
J
J
T
r
T
r
this equation holds for objects orbiting the same mass
mrJ111079.7
read 278-283
p. 283 10-12 p. 284 4-7, 9
GRAVITATIONAL POTENTIAL ENERGY, AGAIN
Recall the Law of Universal Gravitation
221
r
mGmFG
for constant masses, a graph of force vs. radius would be . . .
The graph above is a F vs. d graph which means the shaded area is the work required to move an object from r1 to r2.
The shaded area is not easy to calculate but can be done with a geometric mean. In this case the work done by the lifter is equal to Ep.
Another method involves calculus and integration over a range from r1 to r2.
2
21
1
21
1221
21
1222
212
1
21
1221
)(
)(
)(
r
mGm
r
mGmE
rrrr
mGmE
rrr
mGm
r
mGmE
rrFFW
p
p
p
geometric mean of force
r
mGmE
r
mGm
r
mGmE
p
p
21
1
21
2
21
Know these two equations, you are not required to know the previous development.Which preceding equation can be simplified to mgh, the potential energy change near the earth’s surface?
Potential energy is a negative function!
It increases until it is zero.
PE stops here because the objects come into contact and cannot get closer.
Recall
)(22
22
12
21
22
2
22
1
21
mghmghmvmv
or
mghmv
mghmv
)(22
22
1
21
2
2121
22
2
2122
1
2121
r
mGm
r
mGmmvmv
or
r
mGmmv
r
mGmmv
so . . .
read p. 285-287 p. 287 1-5
Escape from a Gravitational Field
To escape a gravitational field an object must have at least a total mechanical energy of zero!!
PKM EEE for escape
0ME
Escape energy - the minimum EK needed to pro-ject a mass (m2) from the surface of another mass (m1) to escape the gravitational force of m1
Escape speed - the minimum speed needed to project a mass (m2) from the surface of another mass (m1) to escape the gravitational force of m1
Binding energy - the additional EK needed by a mass (m2) to escape the gravitational force of m1 (similar to escape energy but applies to objects that possess Ek i.e. satellites).
To calculate the escape energy or the escape speed of a mass (m2):
0 PK EE
0 BPK EEE
To calculate the binding energy of a mass (m2):
binding energy
Calculate the escape velocity of any object on the Earth’s surface.
0 PK EE
e
e
r
Gmv
r
mGmvm
12
212
2
2
02
m
kgkgNmv
6
242211
1038.6
)1098.5)(1067.6(2
s
mv 11182
The escape velocity is the same for all objects on the Earth’s surface while the escape energy is different for different massed object.
What is Ek and EM of an orbiting body (satellite)?
22
21
22
221
vmr
mGmr
vm
r
mGm
FF cG
this is always true of satellites
2
22
2221
22
21
pK
EE
vm
r
mGm
vmr
mGm
2
2
pM
pp
M
PKM
EE
EE
E
EEE
for an orbiting satellite !!
Note that the total energy is negative since the satellite is “bound” to the central body.
read p.288-293 p. 293 6-11#12 is interesting!
extra p. 294 1-8 p. 300 1-17 25,26 look fun
a) What is the speed of Earth in orbit about the Sun?b) What is the total energy of Earth?c) What is the binding energy of Earth?d) If Earth was launched from the surface of the Sun to its present orbit then what velocity must it be launched with (Ignore the radius of Earth.)?e) If Earth came to rest and fell to the Sun then what velocity would it have when it hit the Sun (Ignore radius of Earth.)?
me= 5.98x1024 kg
ms= 1.99x1030 kg
re= 1.49x1011 m (of orbit)
rs= 6.96x108 m (of the body)
G= 6.67x10-11 Nm2kg-2
sT
T
rv
710156.3
2
s
mv 410966.2
a)
or
22
21 vmr
mGm
s
mv 410985.2
b)
e
esM
pM
r
mGmE
EE
2
2
JEM3310664.2
c) The binding energy is 2.664 x 1033 J
1
21
2
2121
2
21
1
2121
2
2122
1
2121
22
22
22
r
mGm
r
mGmmv
r
mGm
r
mGmmv
r
mGmmv
r
mGmmv
d)
s
mv
Jmv
51
3621
10169.6
10138.12
s
mv
Jmv
mv
r
mGm
r
mGm
r
mGmmv
r
mGmmv
52
3622
22
2
21
1
21
2
2122
1
2121
10285.6
10181.12
2
22
e)