Title: InkPhysics / Gravitational fields
On every note of this deck, unless a note says otherwise:
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Front: EQUATION — NOT ON THE DATA SHEET
Kepler's third law, from Newton:
T^2 = ?
Back: T^2 = 4π^2GM r^3

Front: CHECK YOURSELF
Show that the geostationary orbit has a radius of about 4.2 × 10^7 m. (M = 5.97 × 10^24 kg; take the period as 24 hours.)
Show a hint
Kepler's derived law, rearranged for r cubed.
Back: T = 24 × 3600 = 8.64 × 10^4 s. Rearranging the derived law: r^3 = GMT^2/4π^2.
r^3 = (6.67 × 10^−11 × 5.97 × 10^24 × (8.64 × 10^4)^2) / 4π^2 = 7.5 × 10^22 m^3.
r = 4.2 × 10^7 m from the Earth's centre, about 36,000 km above the surface, in the equatorial plane. One radius, shared by every geostationary satellite ever flown.

Front: CHECK YOURSELF
The ISS orbits about 400 km above the Earth's surface. Taking M = 5.97 × 10^24 kg and R_E = 6.37 × 10^6 m, calculate g at the station, and explain why astronauts float despite your answer.
Show a hint
Build r from the centre first, then ask what free fall feels like from inside.
Back: r = 6.37 × 10^6 + 4.0 × 10^5 = 6.77 × 10^6 m.
g = GM/r^2 = (6.67 × 10^−11 × 5.97 × 10^24) / (6.77 × 10^6)^2 = 8.7 N kg^−1, about 89% of the surface value.
Astronauts float because station and crew are both in free fall, accelerating identically under that g while perpetually missing the ground. Gravity up there is nearly full strength; support forces are what vanished.

Front: EQUATION — NOT ON THE DATA SHEET
Escape velocity:
v_esc = ?
Back: v_esc = √2GMr

Front: CHECK YOURSELF
State two similarities and one difference between gravitational and electrostatic forces, and give one situation where each field would be drawn as uniform.
Show a hint
Think of the shared mathematics first, then who is allowed to repel.
Back: Similarities: both obey an inverse-square force law, and both are described with the same apparatus of field lines, potentials and equipotential surfaces.
Difference: masses always attract; charges may attract or repel.
Uniform cases: the gravitational field near the Earth's surface over small distances, and the electric field between parallel charged plates.

Front: DEFINITION
Define: Escape velocity.
Back: The minimum launch speed at which an object leaves a body's gravitational field without further propulsion.

Front: DEFINITION
Define: Force field.
Back: A region in which a body experiences a non-contact force.

Front: DEFINITION
Define: Gravitational potential.
Back: The work done per unit mass to move a small test mass from infinity to that point; zero at infinity and negative everywhere else.

Front: EQUATION — NOT ON THE DATA SHEET
Orbit condition: gravity as the centripetal force:
GMmr^2 = ?
Back: GMmr^2 = mv^2r

Front: DEFINITION
Define: Gravitational field strength.
Back: The force per unit mass on a small test mass placed in the field: g = F/m, in N kg^−1.

Front: DEFINITION
Define: Geostationary orbit.
Back: An equatorial orbit with a 24-hour period in the direction of the Earth's spin, so the satellite stays above one point on the surface.

Front: CHECK YOURSELF
At the Earth's surface V = −6.26 × 10^7 J kg^−1, and at a satellite's orbit V = −0.94 × 10^7 J kg^−1. Find the energy needed to lift a 1200 kg satellite between the two, ignoring its kinetic energy.
Show a hint
The energy needed is m times the potential difference.
Back: ΔV = (−0.94 × 10^7) − (−6.26 × 10^7) = +5.32 × 10^7 J kg^−1: the climb is out of the well, so ΔV is positive.
ΔW = mΔV = 1200 × 5.32 × 10^7 = 6.4 × 10^10 J.
Sixty-four gigajoules just to be there, before a single joule of orbital kinetic energy: launch costs are mostly the well's depth.