Cavities by Negative-Mass Superposition
Because Newtonian field and potential are linear in source mass, a removed region may be represented by the same region carrying density -rho and superposed with the original complete distribution.
Why this shows up in the exam
Off-centre spherical cavities · Perforated uniform spheres · Far-field effects of removed mass
Learn the idea
A cavity equals the original full body plus an overlapping body of negative density. Instead of integrating an awkward leftover shape, calculate the easy complete body and subtract the gravitational effect of the removed piece.
🧠 Memory hook: Removed matter is a negative copy, not missing algebra.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- g_left = g_full - g_removed — field of a body after material is removed
- V_left = V_full - V_removed — potential of the remaining body
How to approach it
- 1Draw both component bodies and centres
- 2Compute full and removed contributions at the same point
- 3Subtract vectorially and check symmetry
Common slip-ups that cost marks
- •Subtracting magnitudes instead of vectors
- •Forgetting the removed mass scales with volume
- •Using the wrong centre for the negative sphere
🌟 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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