Combine Energy Conservation with the Condition at the Top
In a vertical circle the speed changes, so energy conservation links two points, and a separate condition at the top decides whether the motion continues. The condition is different for a string and for a rigid rod.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Minimum speeds
For a string, the tension at the top can be zero at minimum, so m v_top²/L = m g gives v_top ≥ sqrt(g L), and energy conservation then gives v_bottom ≥ sqrt(5 g L). For a rigid rod or a tube, the support can push as well as pull, so the speed at the top can be zero and (1/2) m v² = m g (2L) gives v_bottom ≥ sqrt(4 g L).
A block released from height h on a track with a loop of radius r completes the loop if h ≥ 5r/2.
Changes in velocity and acceleration
A stone whirled in a vertical circle starting at the bottom with speed u has speed sqrt(u² - 2 g ℓ) when it reaches the level of the centre, directed vertically. The change in velocity between the bottom and that point has magnitude sqrt(u² + (u² - 2 g ℓ)) = sqrt(2(u² - g ℓ)). For a bead released from the horizontal position and sliding down a circular track, v² = 2 g ℓ sin θ at angle θ below the horizontal, so a_c = 2 g sin θ and aₜ = g cos θ, and the total acceleration is g sqrt(1 + 3 sin² θ).
When the string breaks
A body whirled with the minimum speed sqrt(g L) at the top that has its string cut flies off horizontally from a height 2L.
OlympIQ exam lens
- ✓String: v_top ≥ sqrt(g L) and v_bottom ≥ sqrt(5 g L)
- ✓Rod or tube: v_top can be zero, so v_bottom ≥ sqrt(4 g L)
- ✓To rise to the suspension level: v_bottom = sqrt(2 g L)
- ✓Track loop: release height h ≥ 5r/2
- ✓Total acceleration on a track = sqrt(a_c² + aₜ²); aₜ = g cos θ, a_c = v²/r
Do not use the string condition for a rod. A string goes slack when the tension would be negative, so it needs sqrt(g L) at the top; a rigid rod can hold the mass with zero speed at the top, so it needs only sqrt(4 g L) at the bottom.
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