Motion in a vertical circle — A/L Combined Mathematics

What must I remember about Motion in a vertical circle?

• Vertical Circular Motion: Non-uniform circular motion in a vertical plane under gravity where speed v and tension/normal reaction T vary continuously with angular position θ.
• Conservation of Mechanical Energy: Total mechanical energy is conserved: (1/2)m u² = (1/2)m v² + m g h, where u is speed at lowest point and h = a(1 - cos θ) is height above lowest point for circle of radius a => v² = u² - 2ga(1 - cos θ).
• Radial Equation of Motion: Tension T at angle θ to downward vertical: T - mg cos θ = m * v² / a => T = mg cos θ + (m/a)[u² - 2ga(1 - cos θ)] = (m/a)[u² - ga(2 - 3 cos θ)].
• Tension at Extreme Positions: Lowest point (θ = 0): T_bottom = mg + m*u² / a (maximum tension); Highest point (θ = π): T_top = (m/a)(u² - 5ga) (minimum tension).
• Condition for complete vertical loops (String): Particle completes a full circle without the string going slack if T_top ≥ 0 at the highest point, requiring speed at lowest point u ≥ √(5ga) and speed at highest point v_top ≥ √(ga).

Test yourself (1 minute)

1. Motion inside a smooth vertical hollow sphere

   Normal reaction R replaces tension T; particle leaves the surface where R = 0 (cos θ = -(u² - 2ga)/(3ga)); if u ≤ √(2ga), particle oscillates in lower hemisphere; if u ≥ √(5ga), completes full loops; if √(2ga) < u < √(5ga), leaves surface in upper hemisphere and moves as a free parabola under gravity.

2. Particle moving on a light rigid rod

   A rigid rod can support thrust (compression) as well as tension; particle completes loop provided speed at top v_top > 0, requiring speed at lowest point u > √(4ga) = 2√(ga).

3. Tension difference between bottom and top

   In any complete vertical circular revolution: T_bottom - T_top = 6mg, completely INDEPENDENT of initial speed u.

Practise Motion in a vertical circle questions today

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

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

Year 2 · Term 3