Mechanics Rotational Motion Mock Tests

A uniform rod of mass M and length L is pivoted at one end and released from the horizontal position. The angular velocity when the rod passes through the vertical position is:

A particle of mass m moves in one dimension under the potential V(x) = (1/2)kx^2 + lambda/(2x^2) where k > 0 and lambda > 0. For small oscillations about the equilibrium position x_0, the angular frequency of oscillation is:

A uniform rod of mass M = 2 kg and length L = 1 m is free to rotate about a frictionless pivot at one end. The rod is released from rest in a horizontal position. The speed of the free end when the rod passes through the vertical position is (g = 9.8 m/s²):

A solid sphere of mass m and radius R rolls without slipping down an incline of angle θ. The coefficient of static friction is μ_s. Find the minimum value of μ_s required to prevent slipping.

A uniform disc of mass M and radius R rolls without slipping down an incline of angle θ. At the bottom, it collides elastically with a vertical wall. After rebound, it rolls back up the incline. If the coefficient of restitution is e = 1, the maximum height h reached by the disc is related to the initial height H by:

A uniform disc of mass M and radius R is rotating with angular velocity omega about its central axis. A second identical disc, initially at rest, is gently placed on top of the rotating disc. After friction equalizes their angular velocities, the final kinetic energy is:

A uniform rod of mass M and length L is pivoted at one end and released from rest in a horizontal position. When the rod reaches the vertical position, the speed of the free end is (taking g = 9.8 m/s²):

A uniform disc of mass M and radius R is rotating with angular speed ω₀ about its symmetry axis. A second uniform disc of mass m and radius r (where r < R), initially at rest, is gently placed on top of the first disc such that their axes coincide. Friction between the discs causes them to eventually rotate together with a common angular speed. Find the final angular speed ω_f.