Completing the circle and leaving a surface

Conditions for complete circles

• String, or inside a smooth surface: the tension (or reaction) must stay at least zero at the top. T ≥ 0 at the top needs v² ≥ ag there, so with speed u at the bottom, u² - 4ag ≥ ag, that is u² ≥ 5ag.
• Light rod, or bead on a smooth circular wire: the rod can push, so it only needs to reach the top with positive speed: u² > 4ag.

When a string goes slack

Find v² at angle θ by energy, then T from the radial equation, and set T = 0 to find where it goes slack. After that the particle is a projectile until the string is taut again.

Leaving the outside of a sphere

A particle sliding on the outside of a smooth sphere of radius a, at angle θ from the upward vertical: mg cos θ - R = mv²/a. It leaves the surface when R = 0. Starting from rest at the top, v² = 2ga(1 - cos θ), so cos θ = 2/3. With speed u at the top, cos θ = (u² + 2ag)/(3ag). Pearson's June 2025 report says candidates used R = 0 to reach the given result, but the projectile part that followed was often poorly presented and error-prone; draw a simple diagram and use exact trig values.

Worked example

A particle is slightly displaced from rest at the top of a smooth fixed sphere of radius a. Find the angle with the upward vertical at which it leaves the sphere.

1. Energy: v² = 2ga(1 - cos θ).
2. Radially: mg cos θ - R = mv²/a.
3. R = 0: g cos θ = 2g(1 - cos θ), so cos θ = 2/3.

Answer: θ = 48.2°

Check yourself

1. A particle on a string of length 0.5 m moves in a vertical circle. Find the least speed at the lowest point for it to complete the circle, to 3 significant figures.
   Show answer

   4.95 m/s. u² = 5ag = 24.5, u = 4.95 m/s.

2. A particle on a light rod of length 0.5 m, free to rotate in a vertical plane about one end, is given a speed u at the lowest point. Find the value of u above which it completes the circle, to 3 significant figures.
   Show answer

   4.43 m/s. It must reach the top with positive speed: u² > 4ag = 19.6, u > 4.43 m/s.

3. In the worked example, find the angle at which the particle leaves the sphere, to 1 decimal place.
   Show answer

   48.2 °. cos θ = 2/3, θ = 48.2°.

4. A particle slides from rest at the top of a smooth sphere of radius 2 m. Find its speed when it leaves the surface, to 3 significant figures.
   Show answer

   3.61 m/s. v² = 2ga(1 - 2/3) = 2ga/3 = 13.07, v = 3.61 m/s.

5. A particle is projected horizontally from the top of a smooth sphere of radius a with speed u, where u² = ga/2. Find the angle with the upward vertical at which it leaves the surface, to 1 decimal place.
   Show answer

   33.6 °. cos θ = (u² + 2ag)/(3ag) = 2.5/3, θ = 33.6°.

6. A particle on a string of length 1 m is given a speed of 6 m/s at the lowest point of a vertical circle. Find the height above the lowest point at which the string becomes slack, to 3 significant figures.
   Show answer

   1.56 m. At angle φ from the lowest point: v² = 36 - 2g(1 - cos φ), and T = mv²/1 + mg cos φ. T = 0 gives cos φ = (2g - 36)/(3g) = -0.558. Height = 1 - cos φ = 1.56 m.

7. A bead threaded on a smooth vertical circular wire of radius a is projected from the lowest point with speed u. What condition makes it go all the way round?
   • u² ≥ 5ag
   • u² > 2ag
   • u² > 4ag
   • u² ≥ ag
   Show answer

   u² > 4ag. The wire can push in either direction, so the bead only has to reach the top with some speed.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M3
• Pearson Edexcel IAL Mechanics M3 (WME03) question paper, June 2025: front-cover instruction on g and accuracy
• Pearson Edexcel IAL Mechanics M3 (WME03) examiners' report, June 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.