With the rockets we described last time, we are no longer earthbound!

We know, if set anywhere above the earth at rest, thisspace capsule would simply fall.

If propelled in earth’s direction, itwould just build speed and crash.

If propelled directlyopposite the earth, itmight escape (depending on its speed),

or it may decelerate,slowing to a stop,and falling to earthanyway.

If given an initial velocityexactly perpendicular tothe direction of the earth,

it might just orbit!

What conditions must be met

to orbit (and not fall)?

F = 2

Centripetal force:

F

an object needs a continuously applied FORCE exactly perpendicular to its motion.

To be steered from the straight-line paththat inertia would automatically carry it,

And of course, there isa continuously appliedforce acting on thisspace capsule!

The gravitational force we’ve been worrying about bringing it in for a fatal landing!

To stay aloft what v does it need?

v

Fgravity

How fast would an object have to move horizontallyto orbit just above the earth’s atmosphere?

R

mvF

2

=W = mg

gRv =2

gRv =

)104.6)(/8.9( 62 msecmv ×=

= 7,920 m/sec

That’s fast enough to complete anorbit of 2R = 2(6.41024 m) in

vdt vtd /=⇒=)/7920/()104.6(2 6 smmt ×=

= 5077 sec

= 84.6 minutes

17,716 mph!!

Some Planetary Data

RADIUSOF ORBIT

PERIOD OFREVOLUTION

MercuryVenusEarthMarsJupiterSaturnUranusNeptune

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

RADIUSOF ORBIT

PERIOD OFREVOLUTION

MercuryVenusEarthMarsJupiterSaturnUranusNeptune

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

RADIUSOF ORBIT

PERIOD OFREVOLUTION

MercuryVenusEarthMarsJupiterSaturnUranusNeptune

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

RADIUSOF ORBIT

PERIOD OFREVOLUTION

MercuryVenusEarthMarsJupiterSaturnUranusNeptune

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 30 times

164

T2 R3

The square of the periods of all the planets are proportional to

Johannes Kepler (1571-1630)

the cube of their distance from the sun!

Mercury 0.241 0.387Venus 0.615 0.723Earth 1.000 1.000Mars 1.881 1.523Jupiter 11.862 5.200Saturn 29.458 9.540

PERIOD T(YEARS)

DISTANCE Ravg(AU) T2/Ravg

3

1.0001.0001.0001.0001.0001.000

Distances measured in AUs (Astronomical Unit) simply use the earth’s orbital radius

as the standard unit of measure.

1.000

1.000

€

(0.241)2

(0.387)3

€

0.0580

€

0.0580=

The Galilean Moons of Jupiter

Io

Europa

Ganymede

Callisto

421,700 km

671,034 km

1,070,412 km

1,882,709 km

1.77 days

3.55 days

7.15 days

16.69 days

4.17 10-17

T2/R3

4.17 10-17

4.17 10-17

4.17 10-17

R

va

2

=T

Rv

2=

Isaac Newton (1642 – 1727)

Anything traveling in a circle must be experiencing a continuous

centripetal acceleration:where for an

orbit of period T

?

2

24

T

Ra

=

Isaac Newton (1642 – 1727)

2

24

T

Ra

=For any circular orbit:

For planets: 32 R k T = R k 3

2

1

Ra ∝

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 at6428.8257 kilometers)

into its orbit of 27.321 days.

That requires a centripetal force accelerating it at

2

24

T

Ra

=)/400,86)(321.27(

)1084399.3(4 82

daysecdays

m×=

= 0.002723 m/sec2

9.8 m/sec2

602= 0.002723

The mass of the earth is 80 timesgreater than the mass of the moon.

A. just as hard asB. twice as hard asC. 80 times harder thanD. 160 times harder thanE. (80)2=6400 times harder than

The earth pulls gravitationally on the moon_______ the moon pulls on the earth.

The earth is approximately equal to80 moon-sized chunks of mass.

Each of these moon-sized pieces pullson the moon (about equally)

and the moon pulls on each of thesemoon-sized chunks…just as hard!

F F

The earth pulls on the moon with a total force of 80F.

The moon pulls on the earth with a total force of 80F.

This suggests the force of gravity is also directly proportional to the masses involved:

221

R

mmF

grav∝

221

R

mmGF

grav=

G is a universal constant measured to be6.67 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?

G(80 kg) (70 kg)

(1 meter)2 = G 5600kg2

m2

= 0. 000 000 32 N

Fgrav

R

2R

R

Rm mF F

R

R/2

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 C. F E. 4FB. ½ F D. 2F F. other

m m

m m

m 2m

2m 2m