Question

Why is the moment of inertia of a long rod spun around an axis through one end perpendicular to its length greater than if spun with a point mass at the center of mass?

a) The center of mass has less mass.
b) Rod shape amplifies moment of inertia.
c) Point mass distributes weight evenly.
d) The long rod has less rotational energy.

Answer 1

The moment of inertia of a long rod spun around one end perpendicular to its length is greater than if a point mass is spun at the center of mass due to the center of mass having less mass and the rod shape amplifying the moment of inertia.

Step-by-step explanation:

Rod spun around one end perpendicular to its length

The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is ML²/3. This moment of inertia is greater than if a point mass M is spun at the location of the center of mass of the rod (at L/2), which would be ML²/4. The greater moment of inertia in the case of the rod is due to two reasons:

The center of mass of the rod contains less mass compared to the entire rod, resulting in a higher moment of inertia when spun about an axis through one end.

The rod shape amplifies the moment of inertia since mass is distributed farther from the axis of rotation when compared to a point mass at the center of mass.

Therefore, both options a) and b) are correct explanations for why the moment of inertia of a long rod spun around one end perpendicular to its length is greater than if spun with a point mass at the center of mass.