Circular motion with changing speed Problem set B

A bucket is swung on a rope. Its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}0.7\cos(0.4t^2)\\0.7\sin(0.4t^2)\end{pmatrix}\) m. Show that the object moves in a circle with its centre at the origin and state the radius.

A toy aeroplane is flown on a wire. Its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}1.4\cos(0.25t^2)\\1.4\sin(0.25t^2)\end{pmatrix}\) m. Find an expression for its velocity.

A satellite dish pointer moves in a circle. Its position vector is \(\mathbf{r} = \begin{pmatrix}2.5\cos(0.6t^2)\\2.5\sin(0.6t^2)\end{pmatrix}\) m and its velocity is \(\mathbf{v} = \begin{pmatrix}-3t\sin(0.6t^2)\\3t\cos(0.6t^2)\end{pmatrix}\) m s⁻¹. Show that the velocity is always perpendicular to the position vector.

A weight on a spring arm moves in a circle. Its velocity at time \(t\) seconds is \(\mathbf{v} = \begin{pmatrix}-0.18t\sin(0.1t^2)\\0.18t\cos(0.1t^2)\end{pmatrix}\) m s⁻¹. Find an expression for its acceleration.

A tether ball moves around its pole. Its acceleration at time \(t\) seconds is \(\mathbf{a} = \begin{pmatrix}-0.4\sin(0.1t^2) − 0.08t^2\cos(0.1t^2)\\0.4\cos(0.1t^2) − 0.08t^2\sin(0.1t^2)\end{pmatrix}\) m s⁻². Use your calculator to find the value of \(t\) for which the magnitude of the acceleration is 5 m s⁻².

A record turntable speeds up. It turns through an angle of \(0.12t^2\) radians in \(t\) seconds. How many complete revolutions does it make in the first 22 seconds?

A ball is attached to the end of a rope and spun horizontally. Its position relative to a point \(O\) at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}0.9\cos(0.2t^2)\\0.9\sin(0.2t^2)\end{pmatrix}\), 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 rope breaks when the magnitude of the acceleration exceeds 12 m s⁻². Find the value of \(t\) at the instant the rope breaks.
6. (f) How many complete revolutions has the object made from \(t = 0\) to the instant the rope breaks?

A weight on a rope is swung clockwise. Its position vector at time \(t\) seconds is \(\mathbf{r} = \begin{pmatrix}2\sin(0.3t^2)\\2\cos(0.3t^2)\end{pmatrix}\) 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 6 m s⁻¹.