Circular motion with changing speed Problem set A

A stone is whirled on a string. Its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}4.5\cos(0.15t^2)\\4.5\sin(0.15t^2)\end{pmatrix}\) m. Show that the object moves in a circle with its centre at the origin and state the radius.

A ball is spun on a string. Its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}0.8\cos(0.5t^2)\\0.8\sin(0.5t^2)\end{pmatrix}\) m. Find an expression for its velocity.

A car on a ride moves in a circle. Its position vector is \(\mathbf{r} = \begin{pmatrix}6\cos(0.04t^2)\\6\sin(0.04t^2)\end{pmatrix}\) m and its velocity is \(\mathbf{v} = \begin{pmatrix}-0.48t\sin(0.04t^2)\\0.48t\cos(0.04t^2)\end{pmatrix}\) m s⁻¹. Show that the velocity is always perpendicular to the position vector.

A mass on a cord moves in a circle. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}-0.44t\sin(0.2t^2)\\0.44t\cos(0.2t^2)\end{pmatrix}\) m s⁻¹. Find an expression for its acceleration.

A hammer is swung on a chain. Its acceleration at time \(t\) seconds is \(\mathbf{a} = \begin{pmatrix}-0.6\sin(0.2t^2) − 0.24t^2\cos(0.2t^2)\\0.6\cos(0.2t^2) − 0.24t^2\sin(0.2t^2)\end{pmatrix}\) m s⁻². Use your calculator to find the value of \(t\) for which the magnitude of the acceleration is 7 m s⁻².

A fairground cup spins. It turns through an angle of \(0.3t^2\) radians in \(t\) seconds. How many complete revolutions does it make in the first 15 seconds?

A conker is attached to the end of a cord and spun horizontally. Its position relative to a point \(O\) at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}2.4\cos(0.05t^2)\\2.4\sin(0.05t^2)\end{pmatrix}\), where all displacements are in metres and \(t \geq 0\).

1. (a) Show that the object moves in a circle with its centre at \(O\) and state the radius.
2. (b) Find an expression for the velocity at time \(t\).
3. (c) Hence show that the velocity is always perpendicular to the position vector.
4. (d) Find an expression for the acceleration at time \(t\).
5. (e) The cord breaks when the magnitude of the acceleration exceeds 10 m s⁻². Find the value of \(t\) at the instant the cord breaks.
6. (f) How many complete revolutions has the object made from \(t = 0\) to the instant the cord breaks?

A model plane is flown clockwise on a control line. Its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}1.6\sin(0.3t^2)\\1.6\cos(0.3t^2)\end{pmatrix}\) m.

1. (a) Show that the object moves in a circle about the origin and state the radius.
2. (b) Find the velocity vector.
3. (c) Show that the velocity is perpendicular to the position vector.
4. (d) Find the time at which the speed is 4.8 m s⁻¹.