Question

Share

Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in Cartesian co-ordinates $\text{A}=\text{A}_\text{x}\hat{\text{i}}+\text{A}_\text{y}\hat{\text{j}}$ where $\hat{\text{i}}$ and $\hat{\text{j}}$ are unit vector along x and y directions, respectively and A_x and A_y are corresponding components of A Fig. Motion can also be studied by expressing vectors in circular polar co-ordinates as $\text{A}=\text{A}_\text{r}\hat{\text{r}}+\text{A}_\theta\hat{\theta}$ where $\hat{\text{r}}=\frac{\text{r}}{\text{r}}=\cos\theta\hat{\text{i}}+\sin\theta\hat{\text{j}}$ and $\hat{\theta}=-\sin\theta\hat{\text{i}}+\cos\theta\hat{\text{j}}$ are unit vectors along direction in which ‘r’ and ‘$\theta$’ are increasing.

1. Express $\hat{\text{i}}$ and $\hat{\text{j}}$ in terms of $\hat{\text{r}}$ and $\hat{\theta}$.
2. Show that both $\hat{\text{r}}$ and $\hat{\theta}$ are unit vectors and are perpendicular to each other.
3. Show that $\frac{\text{d}}{\text{dt}}(\hat{\text{r}})=\omega\hat{\theta}$ where $\omega=\frac{\text{d}\theta}{\text{dt}}$ and $\frac{\text{d}}{\text{dt}}(\hat{\text{r}})=-\omega\hat{\text{r}}$
4. For a particle moving along a spiral given by $\text{r}=\text{a}\theta\hat{\text{r}}$ , where a = 1 (unit), find dimensions of ‘a’.
5. Find velocity and acceleration in polar vector represention for particle moving along spiral described in (d) above.