Resultant force on a particle in horizontal circular motion?

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

What can you say about vertical forces in horizontal circular motion?

They balance, because there is no vertical acceleration.

Conical pendulum: two equations?

T\cos\theta=mg and T\sin\theta=mr\omega^2

Conical pendulum: formula for \omega^2?

\omega^2=\frac{g}{L\cos\theta}, independent of mass

Direction of friction on a bank when v is above the design speed?

Down the slope, since the vehicle tends to slide up.

Condition for a coin not to slip on a turntable?

mr\omega^2\le\mu mg, so \omega^2\le\frac{\mu g}{r}

Horizontal circular motionEdexcel International A Level Further Maths: Flashcards

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Acceleration of a particle in uniform circular motion?

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Acceleration of a particle in uniform circular motion?: $\frac{v^2}{r}=r\omega^2$ towards the centre
Relationship between $v$ , $r$ and $\omega$ ?: $v=r\omega$
Period of circular motion in terms of $\omega$ ?: $\frac{2\pi}{\omega}$
Resultant force on a particle in horizontal circular motion?: $\frac{mv^2}{r}=mr\omega^2$ towards the centre
What can you say about vertical forces in horizontal circular motion?: They balance, because there is no vertical acceleration.
Conical pendulum: two equations?: $T\cos\theta=mg$ and $T\sin\theta=mr\omega^2$
Radius of a conical pendulum circle, string length $L$ ?: $r=L\sin\theta$
Conical pendulum: formula for $\omega^2$ ?: $\omega^2=\frac{g}{L\cos\theta}$ , independent of mass
Hooke's law?: $T=\frac{\lambda x}{l}$
Elastic string on a smooth table: radius of the circle?: $r=l+x$ , the stretched length
Design speed on a smooth bank?: $v^2=rg\tan\alpha$
Direction of friction on a bank when $v$ is above the design speed?: Down the slope, since the vehicle tends to slide up.
Condition for a coin not to slip on a turntable?: $mr\omega^2\le\mu mg$ , so $\omega^2\le\frac{\mu g}{r}$

Exam questions on Horizontal circular motion

A particle 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 to a point $O$ on a smooth horizontal table. The particle moves on the table in a circle with centre $O$ , at a constant speed of $4$ m s $^{-1}$ .
The speed of the particle is increased until the tension in the string is three times its original value. Find the new speed.2 marks
A particle of mass $0.4$ kg is attached to one end of a light inextensible string of length $2.45$ m. The other end of the string is fixed. The particle moves in a horizontal circle with constant angular speed, with the string inclined at an angle $\theta$ to the vertical, where $\cos\theta=0.8$ . Take $g=9.8$ m s $^{-2}$ .
Find the angular speed of the particle.2 marks
A car of mass $800$ kg is modelled as a particle moving in a horizontal circle of radius $60$ m on a road banked at an angle $\alpha$ to the horizontal, where $\tan\alpha=\frac14$ . Take $g=9.8$ m s $^{-2}$ .
Find the speed at which the car can travel round the bend with no tendency to slip sideways.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).