Mechanics Rotational Motion Mock Tests

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 particle of mass m moves in a central potential V(r) = −k/r + λ/r² where k > 0 and λ > 0. The angular momentum is L. For bounded orbits, the effective potential has a minimum at r₀. If λ = L²/(2m), the orbit becomes unbounded. The critical value of λ_c above which no circular orbit exists is:

A solid disc of mass M and radius R rotates about its central axis with angular velocity omega. A second identical disc, initially at rest, is dropped coaxially onto the first. After friction brings them to a common angular velocity, the fractional loss of 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 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 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 solid cylinder of mass M and radius R is placed on a rough inclined plane at angle θ to the horizontal. A string wrapped around the cylinder is pulled with force F parallel to the incline upward. The minimum coefficient of friction required to prevent slipping is μ_min. If F = Mg sin θ, then μ_min equals:

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.