Mechanics Rotational Motion Mock Tests

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 thin circular ring of mass M and radius R has a small mass m attached to its rim. The ring is placed on a horizontal rough surface and given a slight push so that it rolls without slipping. The speed of the combined system when the attached mass m is at the lowest point, starting from rest with m at the highest point, is:

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 length L and mass M is pivoted at one end and released from rest in a horizontal position. The angular acceleration of the rod when it makes an angle theta with the horizontal is:

A thin circular ring of radius R and linear mass density lambda rotates with angular velocity omega about its axis. The tension in the ring is:

A uniform rod of mass M and length 2L lies along the x-axis with its center at the origin. A particle of mass m moving with velocity v in the +y direction strikes the rod at x = L and sticks to it. The angular momentum of the system about the origin just after collision is:

A solid sphere of mass M and radius R is rotating with angular velocity omega about its axis. A second identical sphere rotating with the same angular speed but in the opposite direction about the same axis is gently brought into contact. After they reach a common angular velocity due to friction between their surfaces, the final kinetic energy is what fraction of the initial total kinetic energy?

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: