Circular motion Problem set A

A bead is on a spinning wheel. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} −12\sin 4t \\ 12\cos 4t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) m. Find its position vector at time \(t\).

A dot moves round a dial. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 6\cos 3t \\ −6\sin 3t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 0 \\ 2 \end{pmatrix}\) m. Find its position vector at time \(t\).

A satellite model and a toy car are tracked. Each position vector gives the position of an object at time \(t\) seconds, in metres. Show that each object moves in a circle about the origin, and state the radius.

1. (a) \(\mathbf{r} = \begin{pmatrix} 5\cos 2t \\ 5\sin 2t \end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix} 1.5\sin 7t \\ 1.5\cos 7t \end{pmatrix}\)

Each position vector gives the position of an object at time \(t\) seconds, in metres. Show that each object moves in a circle, and state the centre and the radius.

1. (a) \(\mathbf{r} = \begin{pmatrix} 8 + 1.5\cos 4t \\ −2 + 1.5\sin 4t \end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix} −3 + 2\sin 3t \\ 5 + 2\cos 3t \end{pmatrix}\)

Each position vector gives the position of an object at time \(t\) seconds, in metres, about the origin. State whether the object goes round the circle clockwise or anticlockwise, and find the time it takes to go round once.

1. (a) \(\mathbf{r} = \begin{pmatrix} 2\cos 3t \\ 2\sin 3t \end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix} 5\cos \dfrac{t}{2} \\ −5\sin \dfrac{t}{2} \end{pmatrix}\)
3. (c) \(\mathbf{r} = \begin{pmatrix} 1.5\sin 4t \\ 1.5\cos 4t \end{pmatrix}\)

A small light is fixed to the edge of a turntable. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} −10\sin 5t \\ 10\cos 5t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\) m, relative to the centre of the wheel.

1. (a) Find the position vector of the light at time \(t\).
2. (b) Find its distance from the centre of the wheel, and describe its path.
3. (c) Find the first time at which it is back at its starting position.

A seat on a fairground ride moves in a vertical plane. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 15\cos 3t \\ −15\sin 3t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\) m, relative to the centre of the ride.

1. (a) Find the position vector of the seat at time \(t\).
2. (b) Show that it moves in a circle about the centre and state the radius.
3. (c) Find its position when \(t = 1\). Give the coordinates to 3 significant figures.

The diagram shows an observation wheel of radius 15 m. Its centre is 17 m above level ground. Take the origin O on the ground directly below the centre, with \(\mathbf{i}\) horizontal and \(\mathbf{j}\) vertically up. A seat starts at the right-hand point of the wheel when \(t = 0\), and the wheel turns anticlockwise once every 30 seconds.

1. (a) Write down the position vector of the seat at time \(t\) seconds.
2. (b) Find its height above the ground when \(t = 5\).
3. (c) Find the speed of the seat.

A sensor is fixed to a slowly turning dish. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} −3\sin \dfrac{t}{2} \\ 3\cos \dfrac{t}{2} \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 6 \\ 0 \end{pmatrix}\) m.

1. (a) Find its position vector at time \(t\).
2. (b) Show that it moves in a circle about the origin and state the radius.
3. (c) Show that its speed is constant and state it.