7. Gravitational Field Concise Notes

Gravitational Field Newton’s Law of Gravitation – mutual force of attraction, FG, between two point masses is proportional to the product of their masses ( m1 & m2) and inversely proportional to the square of their separation, r. Gravitational Potential Energy – work done, U, by an external agent on a mass in moving it from infinity to that point. 12 G 2 Gm mF r 1 2 Gmg r 12Gm mU r 1Gm r  Note: There is NO mass at P. Gravitational Field Strength, g – at a point in a gravitational field is defined as the gravitational force per unit mass acting on a small mass placed at that point. Gravitational Potential, ϕ – at a point in a gravitational field is defined as the work done per unit mass by an external agent in bringing a point mass from infinity to that point. Field Lines  Tangent of the lines points in the direction of the force on a test mass  Point in the direction of decreasing gravitational potentials  Closer lines indicate larger g  No two lines intersect one another  Perpendicular to Equipotential Lines Equipotential Lines  All points along a line has the same gravitational potential  No work done when moving a mass along a equipotential line  Perpendicular to Field Lines Escape Speed: (Motion of an object that eventually escapes the gravitational pull – contrast with Orbital motion) The minimum speed for object at the surface of a planet to reach infinity. By conservation of energy, total energy = 0, 21G M mTE mv 02r    2GMv r r  m1 m2 FG FG r  m1 E Point P Earth v? ∞ Ek = 0 Ep = 0 G dUF dr G 2 Fg m dg dr  r gdr    r GUF d r    FG = m2g  U m  U = mφ

Satellites & Planets in Orbit – Circular Motion (Motion of an object ‘bound’ by gravity) Equation Examples Gravitational Force provides for centripetal force on Satellites, GCFF Satellite / Moon orbiting Earth 2 2 GMm mv rr  or 2 2 GMm mrr  Binary Stars 2 2 Gmm mv rr '  or 2 2 Gmm mr 'r  Period / Frequency of Satellite Use: 2 2 2 GMm 2 mr mrrT    2 23 4Tr GM   Kepler’s Third Law, 23Tr  Energies of Satellite  KE  GPE  TE Use: 2 2 GMm mv rr  2 k 1G M mmv E22 r  p GMmE r kp GMm GMm GMmTE E E 2r r 2r       Geostationary Satellite – in a geostationary orbit in which the orbiting object remains stationary relative to an observer on the Earth. Conditions for Geostationary Orbit:  Centre  of  the  Earth  coincides  with  the  centre  of  orbit  of  the  object/satellite,  and,  axis  of  rotation of the Earth coincides with the axis of orbit of the object/satellite. [OR The orbit of  the object/satellite lies in the equatorial plane.]   The object/satellite orbits from West to East.   The period of one orbit of the object/satellite is 24 hours.  M r m r  m m r' E r Ek TE Ep