Gravity of Continuous Mass Distributions
For a continuous density, dm is written from the appropriate linear, surface, or volume density and d g = -G dm r_hat/r^2 is integrated over the complete source with geometric limits.
Why this shows up in the exam
Fields of rods and arcs · Potential of discs and rings · Nonuniform density integrations
Learn the idea
Replace a distributed source by small mass elements and integrate their vector contributions. A rod, arc, ring, disc, or nonuniform body is a collection of tiny masses whose pulls must be resolved before integration.
🧠 Memory hook: Choose dm, resolve, then integrate.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- d g = -G dm r_hat/r² — field contribution of one mass element
- dm = lambda dl = sigma dA = rho dV — mass element for one-, two-, or three-dimensional sources
How to approach it
- 1Choose coordinates adapted to symmetry
- 2Write dm and the required component
- 3Integrate and test dimensions or limiting cases
Common slip-ups that cost marks
- •Integrating magnitudes before resolving
- •Using incorrect geometric limits
- •Treating a finite source as a point mass too early
🌟 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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