Vectors in circular motionAQA A-Level Further Maths: Subtopic test

1. 1
   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 $\mathbf{r}=2\cos3t\,\mathbf{i}+2\sin3t\,\mathbf{j}$.

   (a)

   Find the speed of $P$.

   [1 mark]

   • A$2$ m s⁻¹
   • B$6$ m s⁻¹
   • C$18$ m s⁻¹
   • D$3$ m s⁻¹

   (b)

   Find the magnitude of the acceleration of $P$.

   [1 mark]

   • A$6$ m s⁻²
   • B$9$ m s⁻²
   • C$12$ m s⁻²
   • D$18$ m s⁻²

   (c)

   Find the acceleration of $P$, as a vector, when $t=\frac{\pi}{6}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   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 $\mathbf{r}=4\sin2t\,\mathbf{i}+4\cos2t\,\mathbf{j}$.

   (a)

   Find the velocity of $Q$ when $t=0$.

   [1 mark]

   • A$-8\mathbf{i}$ m s⁻¹
   • B$8\mathbf{j}$ m s⁻¹
   • C$8\mathbf{i}$ m s⁻¹
   • D$4\mathbf{i}$ m s⁻¹

   (b)

   Find the acceleration of $Q$ when $t=0$.

   [1 mark]

   • A$-16\mathbf{j}$ m s⁻²
   • B$16\mathbf{j}$ m s⁻²
   • C$-8\mathbf{j}$ m s⁻²
   • D$-4\mathbf{j}$ m s⁻²

   (c)

   Show that the speed of $Q$ is constant and state its value.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   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.

   (a)

   Find the angular speed of $P$ and write down its position vector at time $t$.

   [3 marks]

   (b)

   The particle has mass $1.5$ kg. Find, as a vector, the resultant force on $P$ when $t=\frac{3\pi}{4}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves anticlockwise with constant angular speed $\omega$ rad s⁻¹ on a circle of radius $R$ m with centre at the origin $O$. At time $t=0$ it is at the point $(R,0)$, so at time $t$ seconds its position vector is $\mathbf{r}=R\cos\omega t\,\mathbf{i}+R\sin\omega t\,\mathbf{j}$.

   (a)

   (i) Find $\mathbf{v}$ and $\mathbf{a}$ as vectors in terms of $R$, $\omega$ and $t$.
   (ii) Show that $\mathbf{v}$ is perpendicular to $\mathbf{r}$ and that $\mathbf{a}=-\omega^2\mathbf{r}$.

   [6 marks]

   (b)

   At a certain instant the speed of the particle is $12$ m s⁻¹ and the magnitude of its acceleration is $18$ m s⁻². Find $\omega$ and $R$, and find the first time $t$ at which the particle is at the point $(0,-R)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions