Link between v, r and \omega?

v=r\omega (with \omega in radians per second).

Period of uniform circular motion?

T=\frac{2\pi}{\omega}=\frac{2\pi r}{v}

Direction of the acceleration in uniform circular motion?

Towards the centre of the circle (radially inwards).

Resultant force needed for circular motion?

mr\omega^2=\frac{mv^2}{r} directed towards the centre.

Conical pendulum: vertical and horizontal equations?

T\cos\theta=mg and T\sin\theta=mr\omega^2 with r=l\sin\theta.

Convert 60 rev min⁻¹ to rad s⁻¹.

60\times\frac{2\pi}{60}=2\pi rad s⁻¹

Horizontal circular motionEdexcel A-Level Further Maths: Flashcards

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Define angular speed.

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Define angular speed.: The rate of change of angle, $\omega=\frac{d\theta}{dt}$ , in rad s⁻¹.
Link between $v$ , $r$ and $\omega$ ?: $v=r\omega$ (with $\omega$ in radians per second).
Period of uniform circular motion?: $T=\frac{2\pi}{\omega}=\frac{2\pi r}{v}$
Direction of the acceleration in uniform circular motion?: Towards the centre of the circle (radially inwards).
Two forms of the magnitude of radial acceleration?: $r\omega^2$ and $\frac{v^2}{r}$
Resultant force needed for circular motion?: $mr\omega^2=\frac{mv^2}{r}$ directed towards the centre.
Conical pendulum: vertical and horizontal equations?: $T\cos\theta=mg$ and $T\sin\theta=mr\omega^2$ with $r=l\sin\theta$ .
Conical pendulum: $\omega^2$ in terms of $l$ and $\theta$ ?: $\omega^2=\frac{g}{l\cos\theta}$
Hooke's law for an elastic string?: $T=\frac{\lambda x}{l}$
Radius of a horizontal circle with an elastic string of natural length $l$ and extension $x$ ?: $r=l+x$
Smooth banked track: condition on speed?: $\tan\theta=\frac{v^2}{rg}$
Flat bend: condition for no slipping?: $\mu\ge\frac{v^2}{rg}$
Convert 60 rev min⁻¹ to rad s⁻¹.: $60\times\frac{2\pi}{60}=2\pi$ rad s⁻¹

Exam questions on Horizontal circular motion

A particle of mass 0.4 kg moves in a horizontal circle of radius 0.5 m with constant angular speed 6 rad s⁻¹.
Find the magnitude and direction of the resultant force on the particle.2 marks
A cyclist and bicycle, of total mass 90 kg, travel at constant speed round a circular track of radius 40 m. The track is banked at $20^\circ$ to the horizontal and the cyclist experiences no sideways frictional force. The cyclist and bicycle are modelled as a particle and $g=9.8$ m s⁻².
Find the normal reaction between the track and the bicycle.2 marks
A particle $P$ of mass 0.5 kg is attached to one end of a light inextensible string of length 1.2 m. The other end of the string is attached to a fixed point $O$ . $P$ moves in a horizontal circle with constant angular speed, with the string taut and making a constant angle of $60^\circ$ with the downward vertical through $O$ . Take $g=9.8$ m s⁻².
Find the tension in the string and the radius of the circle.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).