Radially Varying Density and Enclosed Mass
For spherical symmetry M(r)=4 pi integral_0^r rho(s)s^2 ds and g(r)=GM(r)/r^2; a circular test orbit then satisfies v^2=GM(r)/r and T=2 pi r/v.
Why this shows up in the exam
Nonuniform planets · Galaxy rotation laws · Locating maximum interior field
Learn the idea
For spherical density rho(r), integrate enclosed mass first; only that mass sets the local field. Different density profiles create different gravity and orbital-speed curves, so the familiar point-mass powers need not apply inside or through extended systems.
🧠 Memory hook: Density to enclosed mass, then field, then orbit.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- M(r) = 4 pi integral₀^r rho(s) s² ds — mass enclosed within radius r
- g(r) = G M(r)/r² — field magnitude from enclosed mass
- v² = G M(r)/r — circular speed in a spherical extended system
How to approach it
- 1Integrate rho to obtain M(r)
- 2Build g(r) and differentiate if an extremum is asked
- 3Use circular balance only after the field is known
Common slip-ups that cost marks
- •Using total mass at an interior point
- •Applying uniform-density linear gravity to every rho(r)
- •Forgetting exterior shells cancel only with spherical symmetry
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A satellite moves in a circular orbit where GM = 4 x 10^14 m^3/s^2 and orbital radius is 2 x 10^6 m. Find its orbital speed.
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