Download Lecture16_Gravity

With the rockets
we described last
time, we are no
longer earthbound!
We know, if set anywhere
above the earth at rest, this
space capsule would
simply fall.
If propelled in earth’s direction, it
would just build speed and crash.
If propelled directly
opposite the earth, it
might escape
(depending on its
speed),
or it may decelerate,
slowing to a stop,
and falling to earth
anyway.
If given an initial velocity
exactly perpendicular to
the direction of the earth,
it might just orbit!
What conditions
must be met
to orbit
(and not fall)?
To be steered from the straight-line path
that inertia would automatically carry it,
an object needs a continuously applied FORCE
exactly perpendicular to its motion.
Centripetal force:
2
F
F=
And of course, there is
a continuously applied
force acting on this
space capsule!
The gravitational force we’ve been worrying
about bringing it in for a fatal landing!
Fgravity
v
To stay aloft
what v does it need?
How fast would an object have to move horizontally
to orbit just above the earth’s atmosphere?
mv
F
W = mg
R
2
v  gR
2
v  gR
v  (9.8m / sec )(6.4  10 m)
2
= 7,920 m/sec
6
17,716 mph!!
That’s fast enough to complete an
orbit of 2R = 2(6.41024 m) in
d  vt
 t  d /v
t  2 (6.4  10 m) /(7920m / s)
6
= 5077 sec
= 84.6 minutes
Some Planetary Data
RADIUS
OF ORBIT
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
PERIOD OF
REVOLUTION
5.79  1010 meters 7.60 106 seconds
1.08  1011 meters 1.94 107 seconds
1.49  1011
3.16 107
2.28  1011
5.94 107
7.78 
1011
3.74
108
1.43  1012
9.30 108
2.87  1012
2.66 109
4.50  1012
5.20 109
about double
~3
Some Planetary Data
RADIUS
OF ORBIT
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
PERIOD OF
REVOLUTION
5.79  1010 meters 7.60 106 seconds
1.08  1011 meters 1.94 107 seconds
1.49  1011
3.16 107
2.28  1011
5.94 107
7.78  1011
3.74 108
1.43  1012
9.30 108
2.87  1012
2.66 109
4.50  1012
5.20 109
about triple
~6
Some Planetary Data
RADIUS
OF ORBIT
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
PERIOD OF
REVOLUTION
5.79  1010 meters 7.60 106 seconds
1.08  1011 meters 1.94 107 seconds
1.49  1011
3.16 107
2.28  1011
5.94 107
7.78  1011
3.74 108
1.43  1012
9.30 108
2.87  1012
2.66 109
4.50  1012
5.20 109
about ten times
~30
Some Planetary Data
RADIUS
OF ORBIT
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
PERIOD OF
REVOLUTION
5.79  1010 meters 7.60 106 seconds
1.08  1011 meters 1.94 107 seconds
1.49  1011
3.16 107
2.28  1011
5.94 107
7.78  1011
3.74 108
1.43  1012
9.30 108
2.87  1012
2.66 109
4.50 
1012
about 30 times
5.20
109
164
Johannes Kepler (1571-1630)
The square of the periods of all the planets
are proportional to
the cube of their distance from the sun!
2
T 
3
R
2
(0.241)
0.0580
3 =
0.0580
(0.387)
PERIOD T
(YEARS)

0.241

Mercury

Venus
0.615
Earth
1.000
Mars
1.881
Jupiter 11.862
Saturn 29.458
DISTANCE
Ravg(AU)
T2/Ravg3
0.387
0.723
1.000
1.523
5.200
9.540
1.000
1.000
1.000
1.000
1.000
1.000
Distances measured in AUs (Astronomical Unit)
simply use the earth’s orbital radius
as the standard unit of measure.
The Galilean Moons of Jupiter
T2/R3
Io
421,700 km
1.77 days
4.1710-17
Europa
671,034 km
3.55 days
4.1710-17
Ganymede 1,070,412 km
7.15 days
4.1710-17
Callisto 1,882,709 km 16.69 days
4.1710-17
Isaac Newton (1642 – 1727)
Anything traveling in a circle
must be experiencing a continuous where for an
centripetal acceleration:
orbit of period T
2?R
v
T
2
v
a
R
4 R
a
2
T
2
Isaac Newton (1642 – 1727)
4 R
For any circular orbit: a 
23
kTR
2
3
For planets: T  k R
2
1
a 2
R
If the moon were in orbit
twice as far from the earth,
its acceleration toward the
earth would be:
A. 4 what is is now.
B. twice what it is now.
C. the same as it is now.
D. half of what it is now.
E. 1/4th what it is now.
F. 1/8th what it is now.
A satellite orbiting the
earth at half the moon’s
distance has an acceleration
toward the earth:
A. 4 that of the moon.
B. twice that of the moon.
C. the same as the moon’s.
D. half that of the moon.
E. 1/4th that of the moon.
F. 1/8th that of the moon.
Jupiter is ~5 times further from the sun than earth.
Jupiter’s acceleration toward the sun is about
A. 5 that of earth’s.
B. the same as earth’s.
C. 1/5th that of earth’s.
D. 1/10th that of earth’s.
E. 1/25th that of earth’s.
Apples fall
toward the earth
at 9.8 m/sec2.
Something pulls the moon
(60 further away at
6428.8257 kilometers)
into its orbit of 27.321 days.
That requires a centripetal force
accelerating it at
2
4 R
4 2 (3.84399  108 m)

a
2
( 27.321days )(86,400sec / day)
T
= 0.002723 m/sec2
9.8 m/sec2
602
= 0.002723
The mass of the earth is 80 times
greater than the mass of the moon.
The earth pulls gravitationally on the moon
_______ the moon pulls on the earth.
A. just as hard as
B. twice as hard as
C. 80 times harder than
D. 160 times harder than
E. (80)2=6400 times harder than
F
F
The earth is approximately equal to
80 moon-sized chunks of mass.
Each of these moon-sized pieces pulls
on the moon (about equally)
and the moon pulls on each of these
moon-sized chunks…just as hard!
The earth pulls on the moon with
a total force of 80F.
The moon pulls on the earth with
a total force of 80F.
This suggests the force of gravity is also
directly proportional to the masses involved:
Fgrav 
m1m2
Fgrav  G
R
2
m1m2
R
2
G is a universal constant measured to be
6.67  10-11 N·m2/kg2
0.000 000 000 066 7 N·m2/kg2
Henry Cavendish (1731 – 1810)
How irresistible is the gravitational
force of attraction between a pair of us
when 1 meter (center-to-center) apart?
(80 kg) (70 kg)
kg2
= G 5600 2
G (1 meter)2
m
= 0. 000 000 32 N
Fgrav
R
m
F
m
F
R
Two objects of mass, m, separated by a
center-to-center distance R are mutually
attracted to one another by a force F.
How strong is the attractive force between
the other pairs of objects shown?
A. ¼ F
B. ½ F
C. F
D. 2F
m
m
R/2
2R
m
2m
R
R
E. 4F
F. other
m
m
2m
2m