Short Summary

Gravitational Fields

Gravitational Field Strength

• Gravitational field strength (g) is the force per unit mass on a small test mass in the field.
• Defined mathematically as: $g = \frac{F}{m}$
• Unit of strength: N kg

Types of Gravitational Fields

• Radial Field: Field lines directed towards a central mass; strength decreases with distance.
• Uniform Field: Same gravitational field strength throughout; lines are parallel.

Free Fall and Weight

• Weight (W) of an object is the gravitational force acting on it: $W = mg$, where $g$ is strength.
• Free fall acceleration is equal to $g$.

Gravitational Potential Energy

• Gravitational potential (V) is work done per unit mass to move an object from infinity to a point in a field: $V = \frac{W}{m}$.
• Change in gravitational potential energy $\Delta E<em>{gpe} = m(V</em>2 - V_1)$.

Escape Velocity

• Minimum velocity needed to escape a gravitational field is derived as: $v_{esc} = \sqrt{\frac{2GM}{R}}$.

Newton's Law of Gravitation

• States that every two masses exert an attractive force proportional to their masses and inversely proportional to the square of the distance between them: $F = \frac{G m<em>1 m</em>2}{r^2}$.
• Universal gravitational constant ($G = 6.67 \times 10^{-11} \text{ Nm}^2 \text{kg}^{-2}$).

Variation of Gravitational Field

• Gravitational field strength decreases with distance from the center of a spherical mass according to $g = \frac{GM}{r^2}$.

Equipotential Surfaces

• Surfaces of constant potential where no work is done to move along them.
• Equipotential lines indicate areas of equal gravitational potential.

Satellite Motion

• Satellites in stable orbits require a balance between gravitational force and centripetal acceleration.
• Geostationary satellites maintain a fixed position above the equator with a period of 24 hours. Radius of orbit $r = 4.2 \times 10^7 ext{ m}$.

Energy of an Orbiting Satellite

• Kinetic energy ($E<em>k = \frac{1}{2}mv^2$) and potential energy ($E</em>p = - \frac{GMm}{r}$) combine for total energy in orbit: $E = E<em>k + E</em>p = - \frac{GMm}{2r}$.