Circular motion with changing speed Problem set B Answers

\(|\mathbf{r}| = 0.7\): a circle of radius 0.7 m

\(\mathbf{v} = \begin{pmatrix}-0.7t\sin(0.25t^2)\\0.7t\cos(0.25t^2)\end{pmatrix}\) m s⁻¹

\(\mathbf{v}\cdot\mathbf{r} = 0\), so they are perpendicular

\(\mathbf{a} = \begin{pmatrix}-0.18\sin(0.1t^2) − 0.036t^2\cos(0.1t^2)\\0.18\cos(0.1t^2) − 0.036t^2\sin(0.1t^2)\end{pmatrix}\) m s⁻²

\(t = 7.89\) s

9 complete revolutions

1. (a) \(|\mathbf{r}| = 0.9\), a circle of radius 0.9 m
2. (b) \(\mathbf{v} = \begin{pmatrix}-0.36t\sin(0.2t^2)\\0.36t\cos(0.2t^2)\end{pmatrix}\) m s⁻¹
3. (c) \(\mathbf{v}\cdot\mathbf{r} = 0\)
4. (d) \(\mathbf{a} = \begin{pmatrix}-0.36\sin(0.2t^2) − 0.144t^2\cos(0.2t^2)\\0.36\cos(0.2t^2) − 0.144t^2\sin(0.2t^2)\end{pmatrix}\) m s⁻²
5. (e) \(t = 9.13\) s
6. (f) 2 complete revolutions

1. (a) \(|\mathbf{r}| = 2\), a circle of radius 2 m
2. (b) \(\mathbf{v} = \begin{pmatrix}1.2t\cos(0.3t^2)\\−1.2t\sin(0.3t^2)\end{pmatrix}\) m s⁻¹
3. (c) \(\mathbf{v}\cdot\mathbf{r} = 0\)
4. (d) \(t = 5\) s