Gravity Inside a Uniform Planet
At centre distance r inside a uniform sphere, M(r)=M r^3/R^3 and g(r)=GM r/R^3; at depth d below the surface this becomes g_d=g0(1-d/R).
Why this shows up in the exam
Weights in mines · Gravity through a planet · Height-depth equality
Learn the idea
Inside a uniform sphere only enclosed mass contributes, making gravity proportional to centre distance. As a body descends through a uniform planet, the outer shells cancel and the shrinking enclosed sphere weakens the pull linearly.
🧠 Memory hook: Inside uniform Earth, gravity tracks distance from the centre.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- M(r) = M r³/R³ — enclosed mass for uniform density
- g(r) = g0 r/R — interior gravity versus centre distance
- g_d = g0(1-d/R) — gravity at depth d below the surface
How to approach it
- 1Convert depth to r=R-d
- 2Use enclosed mass or the linear result
- 3Check zero gravity at the centre
Common slip-ups that cost marks
- •Using inverse square inside
- •Treating depth d as centre distance
- •Applying the linear law to nonuniform density
🌟 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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