PHYS2250 Chap.10 Gravitation and Orbital Motion
Gravitation and Orbital Motion
Combine central force with an energy reference
Newtonian gravitation is an inverse-square central force with negative potential when zero is chosen at infinity. Spherical symmetry supplies shell results, while circular orbit speed follows from radial force balance.
Energy separates bound orbits from escape and explains why a higher circular orbit has greater total energy but lower speed.
Reasoning route and exam method
Newtonian gravitation gives an attractive force GMm/r² along the line joining two masses. Gravitational field is force per unit test mass, and potential is potential energy per unit mass; this relationship belongs in the diagnostic checklist for the chapter.
Outside a spherical mass distribution, spherical symmetry makes the field behave as though all mass were concentrated at the centre; this is a useful checkpoint before any numerical substitution.
Inside a uniform shell the field is zero, a result that requires vector cancellation and cannot be inferred by inserting a smaller radius into the exterior formula; the same result should survive the relevant limiting-case test.
For a circular orbit, gravity supplies the radial resultant, so GMm/r² = mv²/r and v = √(GM/r); that distinction determines which quantities may be combined.
The total energy is negative, equal to −GMm/(2r), because kinetic energy is half the magnitude of potential energy; this observation fixes the physical interpretation of the algebra. Escape speed sets total energy to zero at infinity and is √(2GM/r), larger than circular speed by √2 at the same radius; the statement supplies a direct check on the final result.
A satellite moved to a higher circular orbit has greater total energy but lower speed; this condition should remain visible beside the governing equation.
A satellite orbits a body with GM = 3.99 × 10¹⁴ m³ s⁻² at radius 7.00 × 10⁶ m. The circular speed is √(GM/r) = √(3.99 × 10¹⁴ / 7.00 × 10⁶) = 7.55 × 10³ m s⁻¹.
The period is circumference divided by speed: T = 2πr/v = 2π(7.00 × 10⁶)/(7.55 × 10³) = 5.83 × 10³ s, or 97.1 minutes. Radius is measured from the centre, not from the surface.
BOUND OR UNBOUND Energy signs. Choose U = 0 at infinity.
Then U = −GMm/r, bound circular orbits have negative total energy, and zero total energy is the escape threshold.
Gravity points along the instantaneous radius while orbital velocity is tangent in a circular orbit. The inward force changes direction continuously without changing speed, so it does no work on that circular path.
A higher circular orbit is slower and less tightly bound.
Do not confuse greater total energy with greater kinetic energy.
What this chapter covers
- 01
inverse-square gravity
- 02
field and potential
- 03
spherical shell results
- 04
circular orbits
- 05
escape energy
Worked example · free
Calculate a circular-orbit period
- 3Balance gravity with radial dynamics.
- 2Calculate circular speed.
- 2Use circumference to obtain period.
Key terms
- Gravitational Field
- Force per unit test mass at a position.
- Escape Speed
- The minimum launch speed giving zero total energy at infinity.
- Bound Orbit
- Motion with negative total gravitational energy that cannot reach infinite separation without added energy.
Gravitation and Orbital Motion FAQ
Why is gravitational potential negative?
Zero is chosen at infinite separation. Bringing masses together lowers potential energy below that reference because gravity is attractive and positive work is released during inward motion.
Is a higher circular orbit faster?
No. Circular speed decreases with the inverse square root of radius even though total energy becomes less negative, so energy must be supplied to reach the higher orbit.
What is the field inside a uniform shell?
It is zero everywhere inside by vector cancellation, while the potential is constant rather than necessarily zero because potential values accumulate without a local gradient.
Exam move
Derive circular speed, total energy and escape speed from force and energy statements. Compare their radius dependence instead of memorising isolated formulas. Draw a table of how field, potential, circular speed, period and total orbital energy scale with radius. Use it to predict the direction of every change before inserting numbers.
Re-derive the shell result conceptually from symmetry and keep field distinct from potential inside the shell. For a transfer between circular orbits, explain why the higher orbit is slower even though energy must be supplied. Calculate one two-body period using separation and total mass, then test the small-satellite limit.
End with a reference check: radius is from the attracting centre and potential zero is at infinity, so both geometry and sign convention are explicit. Compare circular and escape conditions at one radius on an energy diagram, making their square-root factor and different total-energy levels visually traceable.
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