8. A marble with radius \( r \) rolls in a L-shaped track. How far is the center of the marble from the corner of the track? \( \sqrt{2} r \)

When the marble rolls in an L-shaped track, the center of the marble maintains a distance of its radius \( r \) from the walls of the track. At the corner, the center of the marble will be positioned diagonally, forming a right triangle with legs equal to its radius. By applying the Pythagorean theorem, we find that the distance from the corner to the center of the marble becomes \( \sqrt{r^2 + r^2} = \sqrt{2} r \). It's a neat little trick that geometry plays! Visualizing the marble's journey might also help! Imagine the marble's center tracing a perfect diagonal line from the corner of the track as it rolls smoothly around. This picture encourages you to think in 3D, appreciating how every curve and angle plays a role in the path, making physics both fun and relatable. Who knew marbles could lead to such geometrical excitement?