Gravitation Part II

One of the very first telescopic observations ever was Galileo’s discovery of moons orbiting Jupiter. Here two moons are visible, along with the shadow of another on the surface of the planet.

Orbits

When in a circular orbit, an object is continually falling ( under the influence of the earth’s gravity).

However, it is continuing to move tangent to the earth, so it continues in a circular path at a constant speed.

Orbit Speed

Notice how the projectile must have enough speed so that it can continually fall around the earth.

When just enough speed is reached, 8 km/s, a circular orbit will result.

The first man made object to accomplish this task of orbiting the earth was Sputnik, launched on October 4th 1957. Sputnik II even carried a dog as a passenger!

The audio is a transmission from

Sputnik

Gravity Changes Direction

Notice that gravity does not pull the satellite forward or backward. Gravity simply acts as the centripetal force to keep it going in a circular orbit.

Tangential velocity

Fc gravity

Gravity Equals Centripetal Force

Since the centripetal force is provided by gravity, we can equate the two forces:

cG FF

2

2 r

mv

r

mMG

2vr

MG

r

GMv

Notice the mass of the satellite cancels

out!

The speed of an circular orbiting satellite depends only on the radius, gravitational constant and mass of the earth!

Correct Distance Value

The radius used in the previous equation is measured from the center of the orbit ( center of the earth). Shell theorem!

Don’t just plug in the distance above the surface of the earth!

Importance of Mass

This means that any mass satellite will have the same orbital speed for any particular radius.

A giant orbiting satellite will have the same speed as a tiny satellite in the same orbit.

However, it can be much more difficult to get that large satellite into orbit in the first place…

Energy Costs

It turns out that it takes 62,000,000 J of energy to put 1kg outside of the Earth’s orbit. (62 MJ)

This is a large amount of energy, which is why it is so costly and difficult to put people and objects into space!

Elliptical OrbitsIf an object is fired faster than 8 km/s, then it will follow an elliptical orbit.

< 8km/s

8 km/s> 8 km/s

foci

In an elliptical orbit, the sum of the distances from the foci is constant. As the foci get closer together, the orbit becomes more circular, and less elliptical.

Energy is still conserved at any point in the orbit, KE+PE=constant

Apogee = smallest velocity

Perigee = largest velocity

Escape SpeedIf an object travels fast enough, it may have sufficient kinetic energy to overcome the gravitational potential energy of its location.

For this situation, the speeding object would not fall back to the surface of the planet. Instead, it would escape the surface and even escape orbiting the planet!

Escape Speed Derivation

The escape speed of any body can be found if the gravitation potential energy balances out the kinetic energy. This total would equal zero.

0PEKE grav

We have formulas for both of these quantities.

0d

mGm

2

mv 212

Canceling out the mass, and rearranging a bit, we can get a formula for the velocity needed to escape the gravity of a body.

Escape Speed Result

0d

Gm

2

v 12

d

Gm

2

v 12

d

2Gmv 12

d

2Gmv 1

Notice the mass of the satellite cancels out! The

remaining mass is the planet being escaped.

Solve for v

This is the speed needed to completely escape any

orbit of a body.

Weight and Weightlessness

You can feel weightless even though gravity is acting on you.

Astronauts in free fall are still being pulled around the earth by gravity.

Gravity Effects

When astronauts live long time periods in space, this impacts their bodies. Bones may weaken, muscles may lose mass, etc...

In the future, humans may design space ships that create “artificial” gravity through circular motion...

Rotating Space Habitats

We can’t create a gravitational force, but we can use centripetal force to act like gravity.

If a round space ship is large enough, and spins at the correct rate, the centripetal force would simulate gravity.

Kepler’s 1st Law:

Each planet moves in an elliptical orbit with the sun at one focus of the ellipse.

foci

Kepler’s 2nd Law:The line from the sun to any planet sweeps out equal areas of space in equal time intervals. This is a restatement of conservation of angular momentum.

Area covered in 1 month of time.

An equal area covered in a different month of time.

Kepler’s 3rd Law:

The squares of the times of revolutions (periods) of the planets are proportional to the cubes of their average distances from the sun.

32 r~T

32

2 rGM

4πT

The second format gives the correct units and scaling.

Questions?

Your physics assignment is:Page 353+ P # 23,43,44,52