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=Fmg = \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=mgW = mg, where gg is strength.
  • Free fall acceleration is equal to gg.

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=WmV = \frac{W}{m}.
  • Change in gravitational potential energy ΔE<em>gpe=m(V</em>2−V1)\Delta E<em>{gpe} = m(V</em>2 - V_1).

Escape Velocity

  • Minimum velocity needed to escape a gravitational field is derived as: vesc=2GMRv_{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=Gm<em>1m</em>2r2F = \frac{G m<em>1 m</em>2}{r^2}.
  • Universal gravitational constant (G=6.67×10−11 Nm2kg−2G = 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=GMr2g = \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=4.2×107extmr = 4.2 \times 10^7 ext{ m}.

Energy of an Orbiting Satellite

  • Kinetic energy (E<em>k=12mv2E<em>k = \frac{1}{2}mv^2) and potential energy (E</em>p=−GMmrE</em>p = - \frac{GMm}{r}) combine for total energy in orbit: E=E<em>k+E</em>p=−GMm2rE = E<em>k + E</em>p = - \frac{GMm}{2r}.