Direction of the acceleration in uniform circular motion?

Towards the centre of the circle

Why does a particle moving at constant speed in a circle accelerate?

Its velocity changes direction.

Direction of the velocity in circular motion?

Along the tangent, perpendicular to the radius

Particles on one rotating string: what is common?

The angular speed \omega (speeds differ).

Is there a centrifugal force on the particle?

No. Only real forces act; their resultant points to the centre.

Angular speed and radial accelerationEdexcel International A Level Further Maths: Flashcards

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Definition of angular speed?

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Definition of angular speed?: $\omega=\frac{d\theta}{dt}$ , the rate of change of angle, in rad s $^{-1}$
Relationship between speed and angular speed?: $v=r\omega$
Period of circular motion?: $T=\frac{2\pi}{\omega}$
Convert $n$ revolutions per minute to rad s $^{-1}$ ?: $\omega=\frac{2\pi n}{60}$
Magnitude of radial acceleration?: $r\omega^2$ or $\frac{v^2}{r}$
Direction of the acceleration in uniform circular motion?: Towards the centre of the circle
Why does a particle moving at constant speed in a circle accelerate?: Its velocity changes direction.
Direction of the velocity in circular motion?: Along the tangent, perpendicular to the radius
Resultant force towards the centre?: $mr\omega^2=\frac{mv^2}{r}$
Tension in a string for a particle on a smooth table?: $T=mr\omega^2=\frac{mv^2}{r}$
Particles on one rotating string: what is common?: The angular speed $\omega$ (speeds differ).
Is there a centrifugal force on the particle?: No. Only real forces act; their resultant points to the centre.

Exam questions on Angular speed and radial acceleration

A particle moves in a horizontal circle of radius $0.8$ m with constant speed, completing $30$ revolutions per minute.
Find the magnitude and direction of the acceleration of the particle.2 marks
A particle $P$ of mass $0.4$ kg is attached to one end of a light inextensible string of length $0.5$ m. The other end of the string is fixed to a point $O$ on a smooth horizontal table, and $P$ moves on the table in a circle with centre $O$ at a constant speed of $3$ m s $^{-1}$ .
Find the angular speed of $P$ and the time taken for $P$ to complete one revolution.2 marks
A stone of mass $0.2$ kg is attached to one end of a light inextensible string of length $0.6$ m. The other end of the string is fixed to a point $O$ on a smooth horizontal table, and the stone moves in a horizontal circle on the table with centre $O$ . The string will break if the tension exceeds $30$ N.
Find the greatest angular speed at which the stone can move without the string breaking.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).