Velocity vector for \mathbf{r}=R\cos\omega t\,\mathbf{i}+R\sin\omega t\,\mathbf{j}?

\mathbf{v}=-R\omega\sin\omega t\,\mathbf{i}+R\omega\cos\omega t\,\mathbf{j}

What is the angle between \mathbf{v} and \mathbf{r}?

90^\circ; show \mathbf{v}\cdot\mathbf{r}=0.

In which direction is the acceleration?

Towards the centre, opposite to \mathbf{r}.

Why is the velocity not constant when the speed is?

Its direction changes continuously.

Position vector for clockwise motion starting at (0,R)?

\mathbf{r}=R\sin\omega t\,\mathbf{i}+R\cos\omega t\,\mathbf{j}

Resultant force on a particle of mass m?

\mathbf{F}=-m\omega^2\mathbf{r}, towards the centre

Vectors in circular motionAQA A-Level Further Maths: Flashcards

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Position vector for anticlockwise circular motion from $(R,0)$?

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Position vector for anticlockwise circular motion from $(R,0)$ ?: $\mathbf{r}=R\cos\omega t\,\mathbf{i}+R\sin\omega t\,\mathbf{j}$
Velocity vector for $\mathbf{r}=R\cos\omega t\,\mathbf{i}+R\sin\omega t\,\mathbf{j}$ ?: $\mathbf{v}=-R\omega\sin\omega t\,\mathbf{i}+R\omega\cos\omega t\,\mathbf{j}$
Acceleration vector in terms of $\mathbf{r}$ ?: $\mathbf{a}=-\omega^2\mathbf{r}$
What is the speed in terms of $R$ and $\omega$ ?: $v=R\omega$
Magnitude of the acceleration in circular motion?: $R\omega^2=\frac{v^2}{R}$
What is the angle between $\mathbf{v}$ and $\mathbf{r}$ ?: $90^\circ$ ; show $\mathbf{v}\cdot\mathbf{r}=0$ .
In which direction is the acceleration?: Towards the centre, opposite to $\mathbf{r}$ .
Why is the velocity not constant when the speed is?: Its direction changes continuously.
Period in terms of $\omega$ ?: $T=\frac{2\pi}{\omega}$
Position vector for clockwise motion starting at $(0,R)$ ?: $\mathbf{r}=R\sin\omega t\,\mathbf{i}+R\cos\omega t\,\mathbf{j}$
Resultant force on a particle of mass $m$ ?: $\mathbf{F}=-m\omega^2\mathbf{r}$ , towards the centre
$\mathbf{r}=2\cos3t\,\mathbf{i}+2\sin3t\,\mathbf{j}$ : speed?: $6$ m s⁻¹

Exam questions on Vectors in circular motion

A particle $P$ moves anticlockwise on a circle of radius $2$ m with centre at the origin $O$ . At time $t$ seconds its position vector, in metres, is $\mathbf{r}=2\cos3t\,\mathbf{i}+2\sin3t\,\mathbf{j}$ .
Find the acceleration of $P$ , as a vector, when $t=\frac{\pi}{6}$ .2 marks
A particle $Q$ moves on a circle of radius $4$ m with centre at the origin $O$ . At time $t$ seconds its position vector, in metres, is $\mathbf{r}=4\sin2t\,\mathbf{i}+4\cos2t\,\mathbf{j}$ .
Show that the speed of $Q$ is constant and state its value.2 marks
A particle $P$ moves anticlockwise with constant speed $4$ m s⁻¹ on a circle of radius $6$ m with centre at the origin $O$ . At time $t=0$ it is at the point $(6,0)$ . Position vectors are in metres relative to $O$ and $t$ is in seconds.
Find the angular speed of $P$ and write down its position vector at time $t$ .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).