Motion of System of Particles and Rigid Body — Prelims Questions

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined plane of angle $heta$. The acceleration of its center of mass is:

A uniform rod of mass $M$ and length $L$ is pivoted at one end and allowed to swing freely. Its moment of inertia about the pivot is:

An ice skater is spinning with an angular velocity of $omega_1 = 10,\text{rad/s}$ with her arms outstretched, having a moment of inertia $I_1 = 6,\text{kg m}^2$. She then pulls her arms in, reducing her moment of inertia to $I_2 = 2,\text{kg m}^2$. What is her new angular velocity $omega_2$?

A particle of mass $m$ is moving in a circular path of radius $r$ with a constant speed $v$. The magnitude of its angular momentum about the center of the circle is:

A uniform circular disc of mass $M$ and radius $R$ is rotating about an axis passing through its center and perpendicular to its plane with angular velocity $omega$. If another identical disc is gently placed on the first disc, and they rotate together, what is the new angular velocity of the combined system?

Three particles, each of mass $m$, are placed at the vertices of an equilateral triangle of side length $a$. The moment of inertia of this system about an axis passing through one vertex and perpendicular to the plane of the triangle is: