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)?

F = 2 Centripetal force: F an object needs a continuously applied FORCE exactly perpendicular to its motion. To be steered from the straight-line path that inertia would automatically carry it,

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! To stay aloft what v does it need? v F gravity

How fast would an object have to move horizontally to orbit just above the earth’s atmosphere? W = mg

= 7,920 m/sec That’s fast enough to complete an orbit of 2  R = 2  (6.4  m) in = 5077 sec = 84.6 minutes 17,716 mph!!

Some Planetary Data RADIUS OF ORBIT PERIOD OF REVOLUTION Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 5.79  meters 7.60  10 6 seconds 1.08  meters 1.94  10 7 seconds 1.49             10 9 about double ~3 

Some Planetary Data RADIUS OF ORBIT PERIOD OF REVOLUTION Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 5.79  meters 7.60  10 6 seconds 1.08  meters 1.94  10 7 seconds 1.49             10 9 about triple ~6 

Some Planetary Data RADIUS OF ORBIT PERIOD OF REVOLUTION Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 5.79  meters 7.60  10 6 seconds 1.08  meters 1.94  10 7 seconds 1.49             10 9 about ten times ~30 

Some Planetary Data RADIUS OF ORBIT PERIOD OF REVOLUTION Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 5.79  meters 7.60  10 6 seconds 1.08  meters 1.94  10 7 seconds 1.49             10 9 about 30 times 164 

T 2  R 3 The square of the periods of all the planets are proportional to Johannes Kepler ( ) the cube of their distance from the sun!

Mercury Venus Earth Mars Jupiter Saturn PERIOD T (YEARS) DISTANCE R avg (AU) T 2 /R avg Distances measured in AU s (Astronomical Unit) simply use the earth’s orbital radius as the standard unit of measure =

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 days 4.17  T 2 /R 

Isaac Newton (1642 – 1727) Anything traveling in a circle must be experiencing a continuous centripetal acceleration: where for an orbit of period T ?

Isaac Newton (1642 – 1727) For any circular orbit: For planets:

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. 1/4 th what it is now. F. 1/8 th 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. 1/4 th that of the moon. F. 1/8 th 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. 1/5 th that of earth’s. D. 1/10 th that of earth’s. E. 1/25 th that of earth’s.

Apples fall toward the earth at 9.8 m/sec 2. Something pulls the moon (60  further away at kilometers) into its orbit of days. That requires a centripetal force accelerating it at = m/sec m/sec =

The mass of the earth is 80 times greater than the mass of the moon. 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 The earth pulls gravitationally on the moon _______ the moon pulls on the earth.

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! F F 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: G is a universal constant measured to be 6.67  N·m 2 /kg N·m 2 /kg 2

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 5600 kg 2 m 2 = N F grav R

2R R R mm FF 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 FC. F E. 4F B. ½ FD. 2F F. other mm mm m2m2m 2m2m2m2m