Circular motion with changing speed Problem set A Answers

\(|\mathbf{r}| = 4.5\): a circle of radius 4.5 m

\(\mathbf{v} = \begin{pmatrix}-0.8t\sin(0.5t^2)\\0.8t\cos(0.5t^2)\end{pmatrix}\) m s⁻¹

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

\(\mathbf{a} = \begin{pmatrix}-0.44\sin(0.2t^2) − 0.176t^2\cos(0.2t^2)\\0.44\cos(0.2t^2) − 0.176t^2\sin(0.2t^2)\end{pmatrix}\) m s⁻²

\(t = 5.39\) s

10 complete revolutions

1. (a) \(|\mathbf{r}| = 2.4\), a circle of radius 2.4 m
2. (b) \(\mathbf{v} = \begin{pmatrix}-0.24t\sin(0.05t^2)\\0.24t\cos(0.05t^2)\end{pmatrix}\) m s⁻¹
3. (c) \(\mathbf{v}\cdot\mathbf{r} = 0\)
4. (d) \(\mathbf{a} = \begin{pmatrix}-0.24\sin(0.05t^2) − 0.024t^2\cos(0.05t^2)\\0.24\cos(0.05t^2) − 0.024t^2\sin(0.05t^2)\end{pmatrix}\) m s⁻²
5. (e) \(t = 20.4\) s
6. (f) 3 complete revolutions

1. (a) \(|\mathbf{r}| = 1.6\), a circle of radius 1.6 m
2. (b) \(\mathbf{v} = \begin{pmatrix}0.96t\cos(0.3t^2)\\−0.96t\sin(0.3t^2)\end{pmatrix}\) m s⁻¹
3. (c) \(\mathbf{v}\cdot\mathbf{r} = 0\)
4. (d) \(t = 5\) s