Applied Maths: Motion in a Circle (Strand 7) — Higher Level Notes

1. Constant speed is not constant velocity

A body going round a circle at a steady speed is accelerating, because velocity is a vector and its direction is changing every instant. That acceleration points towards the centre.

Its magnitude has constant size but a continuously changing direction, so the acceleration is not itself constant.

2. The core formulas

v = ωr  |  a = ω²r = v²/r  |  F = mω²r = mv²/r

Since v = ωr, the expressions ωv, v²/r and ω²r are three ways of writing the same acceleration. If revolutions per minute are given, convert with ω = 2πn/60 first.

3. Horizontal circles

Two equations, always:

• Vertically, nothing accelerates: the upward forces balance the weight.
• Horizontally, the resultant is the centripetal force: F = mv²/r.

That pair solves the conical pendulum, the car on a bend and the banked-track question alike.

Banking is worth understanding rather than memorising. Tilting the road tilts the normal reaction, so part of it now points towards the centre of the bend and friction no longer has to do all the work. It does not increase friction, and it does not reduce the centripetal force required — that is fixed by the speed and the radius. At the design speed, no sideways friction acts at all and the horizontal component of R supplies the whole centripetal force. Note that R is then larger than the weight, not equal to it.

4. Vertical circles

Here the speed varies, because the body changes height and gravity does work on it. The tension never does work — it is always perpendicular to the motion. So combine conservation of energy with the circular-motion equation.

• At the lowest point the tension is greatest: it must both supply the centripetal force and support the weight, T = mg + mv²/r.
• At the highest point, if the speed is exactly √(gr), the weight alone provides the whole centripetal force and the tension is zero — the string is on the point of going slack.

For a string, completing a full circle needs v² ≥ 5gr at the bottom. For a light rigid rod the condition is easier, because a rod can push as well as pull: the bead need only just reach the top.

5. Hooke's law

T = λx / l

Tension is proportional to the extension beyond the natural length — not to the total stretched length. At natural length the tension is zero.

And an elastic string is not a spring. Compress a spring and it pushes back; shorten a string below its natural length and it simply goes slack, exerting nothing at all until it is stretched again.