Conical Pendulum
T sin θ = mω²r = mv²/r (horizontal, provides Fc)
T cos θ = mg (vertical equilibrium)
tan θ = ω²r/g → r = L sin θ → cos θ = g/(ω²L)
Period T = 2π√(L cos θ / g) = 2π/ω
Banked Curve (No Friction)
N sin θ = mv²/r (horizontal centripetal)
N cos θ = mg (vertical equilibrium)
tan θ = v²/(rg) → v_design = √(rg tan θ)
Banked Curve (With Friction)
v_max = √[rg(tan θ + μ)/(1 - μ tan θ)] (prevent sliding up)
v_min = √[rg(tan θ - μ)/(1 + μ tan θ)] (prevent sliding down)
Vertical Circle
At top: T + mg = mv²/r → T = m(v²/r - g)
At bottom: T - mg = mv²/r → T = m(v²/r + g)
Minimum speed at top: v_min = √(gr) (when T = 0)
Energy conservation: ½mv²_bottom = ½mv²_top + mg(2r)
Key Concepts
- Centripetal force is always directed toward the centre of circular motion
- In a conical pendulum, tension provides both vertical support and horizontal centripetal force
- The "design speed" on a banked curve requires no friction at all
- At the top of a vertical circle, both tension and gravity point downward
- If speed at top < √(gr), the object falls (string goes slack)