Moments of Inertia and Rotational Motion

Definition: Moment of inertia is a measure of an object's resistance to rotational motion about an axis. It quantifies how difficult it is to change the rotational state of an object.
Formula: For a rigid body, the moment of inertia $I$ about an axis is given by:
I = rac{1}{n} imes ext{mass} imes ext{distance}^2, where $n$ depends on the distribution of mass relative to the axis of rotation.

Spherical Shell:
- Moment of Inertia: In general, the moment of inertia for a thin spherical shell of mass $m$ and radius $R$ about an axis through its center is given by:
  I = rac{2}{3} m R^2
Solid Sphere:
- Moment of Inertia: The moment of inertia for a solid sphere about an axis through its center is:
  I = rac{2}{5} m R^2
Thin Disk:
- Moment of Inertia: For a thin disk of mass $m$ and radius $R$ (when the axis of rotation is perpendicular to the plane through its center), it is:
  I = rac{1}{2} m R^2
Hoop:
- Moment of Inertia: For a hoop (thin circular ring) of mass $m$ and radius $R$ about an axis through its center perpendicular to the plane, it is:
  $I = m R^2$

To determine which object is easiest to spin for a given mass and radius, consider the formula for the moment of inertia for each shape:
- Spherical Shell: I = rac{2}{3} m R^2
- Solid Sphere: I = rac{2}{5} m R^2
- Thin Disk: I = rac{1}{2} m R^2
- Hoop: $I = m R^2$
Easiest to Spin: The shape with the lowest moment of inertia is the easiest to rotate. Comparing the values:
- Solid Sphere has the lowest moment of inertia $rac{2}{5} m R^2$ , followed by the spherical shell, thin disk, and finally the hoop which has the highest moment of inertia.

Answer to Question: Given a mass and radius, the solid sphere would be the easiest object to spin up due to its lower moment of inertia compared to the spherical shell, thin disk, and hoop.