Why can energy conservation be used in a vertical circle?

Tension or a smooth normal reaction is perpendicular to the motion and does no work, so only gravity does work.

Condition to complete a circle on a string?

T\ge0 at the top, so u^2\ge5gr.

Condition to complete a circle for a bead on a wire?

v\ge0 at the top, so u^2\ge4gr.

When does the particle oscillate rather than leave the circle?

When u^2\le2gr, so it never rises above the level of O.

What defines the point where a string goes slack?

T=0 (the speed is still positive).

Vertical circular motionEdexcel A-Level Further Maths: Flashcards

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Radial and tangential acceleration components?

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Radial and tangential acceleration components?: Radial $\frac{v^2}{r}$ towards the centre; tangential $\frac{dv}{dt}$ along the path.
Why can energy conservation be used in a vertical circle?: Tension or a smooth normal reaction is perpendicular to the motion and does no work, so only gravity does work.
Speed at height $h$ above the lowest point, starting at speed $u$ ?: $v^2=u^2-2gh$
Radial equation at the lowest point?: $T-mg=\frac{mv^2}{r}$
Radial equation at the highest point?: $T+mg=\frac{mv^2}{r}$
Radial equation at angle $\theta$ from the downward vertical?: $T-mg\cos\theta=\frac{mv^2}{r}$
Condition to complete a circle on a string?: $T\ge0$ at the top, so $u^2\ge5gr$ .
Condition to complete a circle for a bead on a wire?: $v\ge0$ at the top, so $u^2\ge4gr$ .
When does the particle oscillate rather than leave the circle?: When $u^2\le2gr$ , so it never rises above the level of $O$ .
What defines the point where a string goes slack?: $T=0$ (the speed is still positive).
What defines the point where a particle leaves a smooth surface?: $R=0$ .
Radial equation with $T=0$ at angle $\alpha$ above horizontal?: $mg\sin\alpha=\frac{mv^2}{r}$
Where is the tension in a vertical circle greatest?: At the lowest point.

Exam questions on Vertical circular motion

A particle $P$ of mass 0.5 kg is attached to one end of a light inextensible string of length 2 m. The other end of the string is fixed at a point $O$ . $P$ moves in a vertical circle with centre $O$ , with the string taut. At the lowest point $A$ of the circle, $P$ has speed 12 m s⁻¹. Take $g=9.8$ m s⁻².
Find the tension in the string when $P$ is at $A$ .2 marks
A smooth circular wire of radius 0.9 m is fixed in a vertical plane with centre $O$ . A small bead $B$ of mass 0.2 kg is threaded on the wire and is projected from the lowest point $A$ of the wire with speed $u$ m s⁻¹. Take $g=9.8$ m s⁻².
Given that $u=7$ , find the magnitude of the force exerted by the wire on $B$ when $OB$ is horizontal.2 marks
A particle $P$ of mass 0.4 kg is attached to one end of a light inextensible string of length 0.6 m. The other end of the string is fixed at a point $O$ . $P$ is at rest at the point $A$ vertically below $O$ and is projected horizontally with speed 4 m s⁻¹. $P$ then moves along a circular path until the string becomes slack. Take $g=9.8$ m s⁻².
Find the tension in the string when $OP$ is horizontal.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).