Motion in a vertical circle — A/L Combined Mathematics

Motion of a particle of mass m in a vertical circle of radius a involves non-uniform speed due to gravity. At angular position θ with the downward vertical, resolving along the inward radius gives Newton's Second Law: T - mg cos θ = m v² / a (where T is tension or normal reaction). Applying the Principle of Conservation of Mechanical Energy between the lowest point (speed u) and angle θ gives: 1/2 m u² = 1/2 m v² + mg a(1 - cos θ) ⟹ v² = u² - 2ga(1 - cos θ). For a particle on a string to complete full vertical circles without slackening (T ≥ 0 at top θ = π), the minimum bottom speed must be u ≥ √(5ga); for a particle on a light rigid rod, speed can drop to zero at top, requiring u ≥ √(4ga) = 2√(ga).

Check before you answer (A/L Applied Maths Paper 2)

1. Energy Conservation Equation: Writing 1/2 m v² + mg h = constant correctly between benchmark points.
2. Inward Radial Equation of Motion: Formulating T - mg cos θ = m v² / a without omitting gravitational weight components.
3. Critical Condition Execution: Setting T = 0 at the highest point θ = π to derive the critical threshold speed u = √(5ga), and distinguishing between leaving the circle and oscillating.