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Students derive expressions for the velocity and acceleration in polar coordinates
Estimated Time: 30 minutes
Carry out calculations of velocity and acceleration in polar coordinates
The activity begins by asking the students to write on whiteboard what ${\bf{v}} = \frac{d{\bf{r}}}{dt}$ is. Students propose two alternatives, ${d{\bf{r}}\over{d{t}}} = {d{r}\over{d{t}}} {\bf\hat{r}}$ and ${d{\bf{r}}\over{d{t}}} = {d{r}\over{d{t}}} {\bf\hat{r}} + {d{\phi}\over{d{t}}} {\bf\hat{\phi}}$. A discussion ensues about which is correct. The proper formula is derived using $\frac{d{\bf{r}}}{dt} = \frac{d{r}}{dt} {\bf\hat{r}} + r \frac{d{\bf\hat{r}}}{dt}$. This result is further justified by drawing the velocity vector and discussing the fact that ${\bf\hat{r}}$ changes as $\phi$ changes.
Students are then asked to find values for ${d{ {\bf\hat{r}}}\over{d{t}}}$ and ${d{ {\bf\hat{\phi}}}\over{d{t}}}$. Students are given the handout cfvpolarhand.pdf and asked to carry out calculations of $\bf{v}$ and $\bf{a}$ in polar coordinates.
The most straightforward way to find the time dependence of ${\bf\hat{r}}$ and ${\bf\hat{\phi}}$ is to write them in terms of ${\bf\hat{x}}$ and ${\bf\hat{y}}$. Since ${\bf\hat{x}}$ and ${\bf\hat{y}}$ to not vary with position and therefore do not vary with time (as the particle moves around), there time derivatives are zero. All of the time dependence is in the coefficients.
Most students will not realize that they need to recognize that the derivatives of ${\bf\hat{r}}$ and ${\bf\hat{\phi}}$ that they find can be rewritten in terms of ${\bf\hat{r}}$ and ${\bf\hat{\phi}}$. You will need to check all the groups and prompt this, where necessary. Interrupt the activity for a mini-lecture, if necessary.
One common student confusion is to believe that since there is no explicit time dependence in the equation ${\bf{r}} = r {\bf\hat{r}}$ then there is no time dependence and $\frac{d\bf{r}}{dt} =0$. This provides the teacher an opportunity to remind the students that
to save time and writing, the time dependence of a variable is often not stated explicitly and
it is important to think carefully about what variables we think will vary with time and only set their time dependence equal to zero if we have a good reason.
Students often forget that ${\bf\hat{r}}$ and ${\bf\hat{\phi}}$ vary with $\phi$. It is useful to point out that Cartesian coordinates and polar coordinates are different because the unit vectors in Cartesian coordinates have fixed directions but the directions of polar coordinates are not fixed. It may be useful to emphasize at this time that they will encounter unit vectors whose direction is dependent on the position coordinates very often in this and future physics classes.
If you haven't talked about it before, sometimes students will want to pattern match from $\vec{r}=x\hat{x}+y\hat{y}+z\hat{z}$ to $\vec{r}=r\hat{r}+\phi\hat{\phi}+\theta\hat{\theta}$. Some of them may realize that this is the wrong dimensions and add in $r$'s, so it is important to have them take a step back and think about how to describe a position in spherical coordinates.
Depending on the background of your students, this can be a very time consuming activity. Sometimes it is best to start the activity in class so that you can help students get started, but then to have them finish the calculations at home.
After students have completed the calculations, it is important to quickly review the answers and the procedure for calculating them.
$$ {\bf{v}} = {\bf{\dot{r}}} = \dot{r} {\bf\hat r} + r \dot \phi {\bf\hat \phi}$$ $$ {\bf{a}} = {\bf{\dot{v}}} = {\bf{\ddot{r}}} = \left(\ddot{r}- r \dot{\phi}^2\right) {\bf\hat r} + \left( r \ddot{\phi} + 2 \dot{r}\dot{\phi}\right) {\bf\hat \phi}$$
Notes on these derivations can be found in the Central Forces course notes Section 8
This activity leads smoothly into a mini-lecture defining the kinetic energy and angular momentum in polar coordinates.