Mechanics: circular motion, oscillations and gravity
This domain links angular motion, centripetal force, simple harmonic motion and Newtonian gravitation. Questions often move between a conical pendulum, an oscillating mass and a satellite orbit.
Circular motion
Angular speed ω = dθ/dt, linear speed v = rω, and period T = 2π/ω. The centripetal acceleration is v²/r = rω² directed towards the centre, so the resultant inward force is mv²/r. For a conical pendulum, vertical equilibrium gives T cosθ = mg and horizontal circular motion gives T sinθ = mv²/r; dividing gives tanθ = v²/(rg).
Simple harmonic motion
SHM satisfies a = −ω²x, with x measured from the centre. If motion starts at maximum displacement, x = A cosωt and v = ±ω√(A² − x²). The period is T = 2π/ω; for a simple pendulum making small oscillations, T = 2π√(l/g).
Gravity
Newton's law is F = GMm/r². At a planet's surface g = GM/R². For a circular orbit, GMm/r² = mv²/r, so v = √(GM/r) and T = 2π√(r³/GM). Here r is measured from the planet's centre, not its surface.
Worked example
A 0.5 kg mass on a 0.8 m string moves in a horizontal circle with the string at 30° to the vertical. Find its speed. Radius r = 0.8 sin30° = 0.4 m. Using tanθ = v²/(rg): v² = 0.4 × 9.8 × tan30° = 2.263, so v ≈ 1.50 m/s.
Traps
Using the string length as the circle radius; inventing a separate centripetal force instead of identifying the resultant inward force; mixing degrees and radians; using altitude instead of orbital radius; writing a = +ω²x.