Gravitational Fields

ADD GRAV FIELDS IS1 NOTES

Examples of natural satellites:

Moon, Earth, Sun

How do natural satellites stay in orbit?

• As the satellite is launched into space, it will constantly be attracted to the Earth

• If the Earth and satellite fall at the same rate, the satellite will orbit

Equation for orbital speed:

v = sqrt(GM/r)

Low Earth orbit satellites

• Satellites with a low distance from the Earth

• r is low, implying that orbital speed is high

Uses of low Earth orbit satellites:

Observation and remote sensing

Geostationary satellites

• Far away from the Earth

• Appear to be stationary on Earth (in one fixed position)

• Must be above the equator, rotate in the same direction and have a time period of 24 hours

Uses for geostationary satellites:

Communications, weather, television

Gravitational Potential

The work done per unit mass to move an object from infinity to that point

Gravitational Potential equation

V_g = -Gm/r as r → ∞, Vg → 0

Gravitational Potential energy

The work done to move a mass from infinity to a point in a gravitational field

Uniform grav. field vs radial grav. field

A uniform grav. field is an approximation for where field lines are parallel, but radial grav. field lines point towards a central mass

Change in Gravitational Potential energy equation

• ΔE = mΔV_g

• In radial fields, E = -GMm/r

The area of a force-radius graph

The change in G.P.e

Escape velocity

The minimum speed an object needs to escape the gravitational field of a body

Escape velocity equation and derivation

• E_k = E_p

• ½mv² = GMm/r

• v = sqrt(2GM/r), where r is the distance from the surface of the body

Kepler’s Third Law derivation

• Assume a circular planetary orbit

• Centripetal force to keep the planet in orbit is provided by the Gravitational Force between the Sun and Planet

• F = mv²/r = GMm/r²

• r³=(GM/4π²)T²

• Therefore, r³ ∝ T²

K3’s Law for linear velocity:

• v² = GM/r

• v = 2πr/T

How does gravitational potential vary between the Earth and Moon?

Change in gravitational potential formula + derrivation

• Distance from r₁ to r₂

• V_g1 = -GM/r₁

• V_g2 = -GM/r₂

• ΔV_g = V_g2 - V_g1

• = -GM/r1 + GM/r2

Total energy equation

Total energy = E_k + E_p

A satellite with mass 200kg has a velocity of 5.6 × 10³ ms^-1 and has a circular orbit of 1.28 × 10^7m around the Earth. Given the mass of the Earth is 6 × 10^24 kg, what is the potential energy of the satelite?

• E_k = 1/2mv² = 3.14 × 10^9J

• E_p = -GMm/r = -6.25 × 10^9J

• T = E_k + E_p = -3.11 × 10^9J

Escape velocity is independent of:

Escape velocity is independent of mass