Oscillation Through a Uniform Earth
For uniform density and radius R, the interior field is g(r) = g_surface r/R toward the centre; its component along a straight frictionless tunnel is linear in tunnel displacement.
Why this shows up in the exam
Diameter-tunnel oscillation · Off-centre chord tunnels · Comparing gravitational SHM with a spring
Learn the idea
Inside a uniform spherical Earth, gravity is linear in radius and produces SHM along any straight tunnel. Only the sphere inside the current radius contributes to the net pull, so the force falls steadily to zero at Earth's centre.
🧠 Memory hook: Uniform Earth turns gravity into a giant linear spring.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- g(r) = g_surface r/R — interior gravitational field magnitude for uniform density
- omega = sqrt(g_surface/R) — angular frequency in a straight tunnel
- T = 2 pi sqrt(R/g_surface) — period, independent of tunnel chord for the ideal model
How to approach it
- 1State the uniform-density assumption
- 2Resolve the interior field along the tunnel
- 3Match the coefficient of displacement to omega squared
Common slip-ups that cost marks
- •Using inverse-square gravity inside Earth
- •Changing the period for a different straight chord
- •Applying the result to a nonuniform Earth without qualification
🌟 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 mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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