In the story about Kendall’s painting falling during a small earthquake and being put back on the wall, when was the gravitational potential energy between the painting and Earth the least (assume the painting’s mass did not change)?

What the question is really asking

You are comparing the painting’s gravitational potential energy at different moments in the story. Because the painting’s mass stays the same, the only thing that changes its gravitational potential energy is its height above the floor (relative to Earth).

Using the gravitational potential energy idea

Near Earth’s surface, gravitational potential energy can be modeled by $$U = mgh$$ where $m$ is mass, $g$ is gravitational field strength (about $9.8\,\text{m/s}^2$), and $h$ is height.

Since $m$ and $g$ do not change here, $U$ is smallest when $h$ is smallest.

Mapping the story to heights

• Hanging on the wall (before the earthquake): higher $h$, so larger $U$.
• On the floor after falling: lowest $h$, so smallest $U.
• Back on the wall after cleanup: higher $h$ again, so larger $U$.

So the least gravitational potential energy was stored when the painting was on the floor.