Weight and Gravitational Acceleration:

The gravitational pull that an object experiences is proportional to the mass of the planet divided by its distance from the planet squared. If the object is sitting on the planet's surface, then the distance is the planet's radius. We call this pull the gravitational acceleration, g, such that:

The weight of an object is its mass times the gravitational acceleration. Note that g can vary from planet to planet by the planets mass and radius.

Kinetic and Potential Energy:

Huygens (1650's) was the first to develop the terminology, stating that:

Instead, it was defined that energy is a measure of the ability of a physical system to perform work (i.e. to change the system) and is measured in ergs.

The law of the conservation of energy means that energy can neither be created or destroyed, only transformed from one form to another.

Escape Velocity:

A perfect circular orbit occurs if the centrifugal force exactly balances the gravitational force. At the Earth's surface, this value is 7.86 km/s.

A circular orbit for object A occurs when its burnout velocity is vc. When the velocity is v=1.44vc, then a parabolic orbit is achieved and we say the object has reached escape velocity. For a greater velocity, the orbit is hyperbolic.

When the velocity between 1.44vc and vc, then the orbit is an ellipse with perigee at E2. For less than vc, the elliptical orbit has an apogee at E1.

Differential Gravitational Forces:

Most applications of Newton's law of gravity assume the objects are point sources. If a body is extended, then differential gravitational forces are important. The most common application of this effect is the twice daily tides on the Earth's oceans. Tides are due to the gravitational effects of the Moon, where all points on the Earth's surface feel a gravitational pull towards the center of the Moon.

There is also a gravitational force from the Earth, and an acceleration associated with that force. If we subtract the acceleration from the Moon and the Earth's, the resulting vectors push in from 90 degree points and pull out towards and away from the Moon.

Precession:

If the Earth were a solid, then its shape would be spherical. However, the interior of the Earth is only semi-solid, so its rotation causes it to take on the shape of an oblate spheroid, large at the equator than the poles. This means that the differential tidal forces will be stronger on the equator sections than the poles. Since the Moon's orbit is inclined with respect to the Earth's equator, the results is a tidal torque on the Earth.

Gravitation pull of the Sun and Moon causes a "wobble" in the Earth's axis with a period of 25,000 years (like pushing a gyroscope). This wobble is called precession and has the result of changing the point in the sky where the celestial poles are located and, therefore, changes the "pole star".

This motion was first recorded by Hipparchus in 100 B.C. who noticed differences between ancient Babylonian observations and his own. When the Babylonians were the world power in 2000 B.C., the vernal equinox was in the constellation Aries and the star Thuban (in Draco) was the closest bright star to the north celestial pole. When the Pyramids were built, the north star was Vega.