Final answer:
The smallest moment of inertia for a stick occurs when it is spun around its center, as this minimizes the distance (r) from the axis of rotation to the mass of the stick, according to the formula I = Mr²/3.
Step-by-step explanation:
The moment of inertia is smallest when the stick is rotated around the point where it is held fixed, as this point acts as the axis of rotation. According to the formula I = Mr²/3, where I is the moment of inertia, M is the mass of the stick, and r is the distance from the axis of rotation to the mass of the stick, the moment of inertia is directly proportional to the square of the distance.
Therefore, holding the stick at its center would result in a shorter distance, r, to all parts of the stick and consequently the smallest moment of inertia.
An example often used to illustrate this concept is a figure skater executing a spin. The conservation of angular momentum dictates that when a figure skater pulls their arms in, they reduce their moment of inertia and thus spin faster. Much like the skater, if the juggler wants to minimize the moment of inertia, they should spin the stick around its center rather than the ends.