Circular motion Problem set B

A marker is on a rotating disc. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} −14\sin 7t \\ 14\cos 7t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\) m. Find its position vector at time \(t\).

A point moves round a clock face. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 20\cos 4t \\ −20\sin 4t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 0 \\ 5 \end{pmatrix}\) m. Find its position vector at time \(t\).

A drone and a model train 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) \(\mathbf{r} = \begin{pmatrix} 9\cos 3t \\ 9\sin 3t \end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix} 2.5\sin 4t \\ 2.5\cos 4t \end{pmatrix}\)

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) \(\mathbf{r} = \begin{pmatrix} −5 + 2.4\cos 3t \\ 6 + 2.4\sin 3t \end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix} 7 + 4\sin 2t \\ −1 + 4\cos 2t \end{pmatrix}\)

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) \(\mathbf{r} = \begin{pmatrix} 3\cos 2t \\ 3\sin 2t \end{pmatrix}\)
2. (b) \(\mathbf{r} = \begin{pmatrix} 4\cos \dfrac{t}{3} \\ −4\sin \dfrac{t}{3} \end{pmatrix}\)
3. (c) \(\mathbf{r} = \begin{pmatrix} 0.8\sin 5t \\ 0.8\cos 5t \end{pmatrix}\)

A reflector is fixed to the rim of a bicycle wheel. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} −18\sin 6t \\ 18\cos 6t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) m, relative to the centre of the wheel.

1. (a) Find the position vector of the reflector 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 cabin on a giant wheel moves in a vertical plane. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} 24\cos 4t \\ −24\sin 4t \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 0 \\ 6 \end{pmatrix}\) m, relative to the centre of the ride.

1. (a) Find the position vector of the cabin 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 a water wheel of radius 2.4 m. Its centre is 3.1 m above the bed of a stream. Take the origin O on the bed directly below the centre, with \(\mathbf{i}\) horizontal and \(\mathbf{j}\) vertically up. A paddle starts at the right-hand point of the wheel when \(t = 0\), and the wheel turns clockwise once every 8 seconds.

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

A pointer on a large, slow-turning dial. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix} −2\sin \dfrac{t}{4} \\ 2\cos \dfrac{t}{4} \end{pmatrix}\) m s⁻¹. When \(t = 0\) its position vector is \(\begin{pmatrix} 8 \\ 0 \end{pmatrix}\) 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.