Torque and Rotational Dynamics

Three equal masses (m) are located at the vertices of an equilateral triangle with sides of length 'a'. If the axis of rotation passes through one of the vertices and is perpendicular to the plane of the triangle, what is the rotational inertia of the system?

A dumbbell consists of two equal masses, $m$, separated by a distance $2r$ and connected by a massless rod. It rotates about its center. What happens to the rotational inertia if the axis of rotation is shifted a distance $r$ away from the center, parallel to the original axis?

Two point masses, each with mass $m$, are placed at distances $r$ and $2r$ from the axis of rotation. What is the total rotational inertia of this system?

A system consists of two masses: a 1 kg mass at a distance of 0.5 m from the axis of rotation and a 2 kg mass at a distance of 1 m from the axis of rotation. What is the total rotational inertia of the system?

A solid sphere has a rotational inertia $I$ about an axis through its center. If the axis is shifted to a distance $d$ away from the center, what is the new rotational inertia?

A solid sphere and a hollow sphere have the same mass and radius. Which one has a greater rotational inertia about an axis through its center?

A figure skater spins faster when they pull their arms in close to their body. This is because pulling their arms in:

A hoop and a disk have the same mass and radius. Which has a greater rotational inertia about an axis through its center?

Objects of different shapes (solid sphere, solid cylinder, and a rod) have the same mass and radius. If they are all rotating about an axis through their center, which has the smallest rotational inertia?

Which of the following scenarios results in the lowest rotational inertia for an object?