How know which direction a particle is moving on a polar curve

Question

I have being doing problems from the released AP BC Calcululs Free-Response questions, and I have come to realize that I don't have a very good idea of explain or a deep understanding of how to tell if a particle is moving to the orgion (pole?).

Example 2005 AP BC Free-Response Question #2.

The curve above is drawn in the $xy$-plane and is described by the equation in polar coodinates $r =\theta +\sin\left(2\theta\right)$ for $0\leq \theta \leq \pi$, where $r$ is measured in meters and $\theta$ is measured in radians. The derivative of $r$ with respect to $\theta$ is given by $\frac{\mathrm{d}r}{\mathrm{d}\theta} = 1 + 2\cos\left(2\theta\right)$.

(c) for $\frac{\pi}{3} < \theta < \frac{2\pi}{3}$, $\frac{\mathrm{d}r}{\mathrm{d}\theta}$ is negative. What does this fact say about $r$? What does this fact say about the curve?

I have learned from doing problems and reading the answers that if the derivative is negative its getting closer to the origin. However, I have no clear understanding why?

I don't trust my justification:

Just as with the Cartesian coordinate system, when the 1st derivative is negative the particle is decreasing, moving downward, thus in the polar coordinate system that is moving to the origin.

Answer 1

First, let's visualize the curve traced out in the $x$-$y$ plane by $r=\theta+\sin(2\theta)$, as $\theta$ varies over $[0,\pi]$:

Now, for the question posed in (c), let's focus on the part of the curve traced out for ${\pi\over 3}<\theta<{2\pi\over 3}$:

The dashed line is showing the length of $r$ as $\theta$ varies. In particular, note that for these $\theta$ values, $r$ is decreasing as a function of $\theta$.

I drew these to help you "see" the geometry in this polar form, but we could also accomplish this much more simply by just plotting $r$ as a function of $\theta$ for ${\pi\over 3}<\theta<{2\pi\over 3}$:

$r$ is certainly looks like it decreases with $\theta$, and a little calculus verifies this since ${dr\over d\theta}=1+2\cos(2\theta)<0$ for ${\pi\over 3}<\theta<{2\pi\over 3}$.

Answer 2

The key here is that the particle gets closer to the origin as it moves counterclockwise along the curve. As $\theta$ increases, $r$ decreases. That is the meaning of the derivative. If the particle moved clockwise instead, $r$ would be increasing. Without a specific parameterization, you cannot know which is the true behavior of the particle, but it's safe to say that as the angle increases, the distance decreases. One tends not to worry about whether that behavior is forwards or backwards in time.

Answer 3

There is a subtlety missed here. If the sign of the radius' rate of change is the same as the sign of the radius at a given point, the particle is moving away from the pole. If, however, the sign of the radius' rate of change and the sign of the radius are different, the particle is moving towards the pole.