Motion in a horizontal circle — A/L Combined Mathematics

What must I remember about Motion in a horizontal circle?

• Horizontal Circular Motion: Particle moves along a circle in a horizontal plane with radius r at constant angular velocity ω and constant speed v = r*ω.
• Equation of Motion along radius: Inward radial resultant force provides centripetal acceleration: ∑ F_radial = m * v² / r = m * r * ω².
• Equation of Motion vertically: Since there is zero vertical acceleration: ∑ F_vertical = 0 (vertical equilibrium).
• Conical Pendulum: Particle of mass m suspended by a light inextensible string of length L rotating in a horizontal circle of radius r = L sin θ (θ is semi-vertical angle with vertical): T cos θ = mg and T sin θ = m*r*ω² => tan θ = r*ω² / g => period T_p = 2π √(h/g) = 2π √[(L cos θ)/g].
• Banked Tracks and Curves: Road banked at angle θ to horizontal allows vehicle to negotiate curve of radius r without relying on friction: N sin θ = m*v² / r and N cos θ = mg => tan θ = v² / (rg).

Test yourself (1 minute)

1. Friction on flat unbanked curved road

   Maximum safe turning speed without skidding on flat curve of radius r: friction force f ≤ μ N = μ mg => m*v²/r ≤ μ mg => v_max = √(μ g r).

2. Banked track with friction

   Maximum speed before slipping up: tan(θ + λ) = v_max² / (rg) where λ is angle of friction (tan λ = μ); minimum speed before slipping down: tan(θ - λ) = v_min² / (rg).

3. Rotor / Wall of Death ride

   Rider of mass m pressed against vertical cylinder wall of radius r spinning at ω: normal reaction N = m*r*ω²; friction balances weight f = mg; no slipping if f ≤ μ N => mg ≤ μ(m*r*ω²) => ω_min = √(g / (μ r)).

Practise Motion in a horizontal circle questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 3

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Next in the syllabus

Where does Motion in a horizontal circle fit in the syllabus?

Year 2 · Term 3