Central Forces: Spring-2026
HW 04 Practice (SOLUTION): Due W2 D5

1. Polar vs. Spherical Coordinates S1 5535S Show that the plane polar coordinates are equivalent to spherical coordinates if we make the choices:
   1. The direction of \(\theta=0\) in spherical coordinates is the same as the direction of out of the plane in plane polar coordinates.
   2. Given the correspondance above, then if we choose the \(\theta\) of spherical coordinates is to be \(\pi/2\), we restrict to the equatorial plane of spherical coordinates.

   In spherical coordinates, we have \begin{align} x&=r\sin\theta\cos\phi\\ y&=r\sin\theta\sin\phi\\ z&=\cos\theta \end{align} If we look at \(\theta=\frac{\pi}{2}\), then \(\sin\theta=1\) and \(\cos\theta=0\). With these restrictions, we have: \begin{align} x&=r\cos\phi\\ y&=r\sin\phi\\ z&=0 \end{align} These are plane polar coordinates.

2. Central Forces are Conservative S1 5535S

   (Quick) Purpose: Recall the relationship between conservative forces and potentials.

   Show that a central force is always conservative. Find the scalar potential \(U\) corresponding to the central force \(\vec{F}=f(r)\,\hat{r}\) and show that it depends only on the distance from the center of mass, i.e. \(U=U(r)\).