The uniform rectangular door panel has a mass of and is held in equilibrium  above the horizontal at the position by rod . Determine the required stiffness of the torsional spring at , so that the door’s  angular velocity becomes zero when the door reaches the closed position () once the supporting rod is removed. The spring is undeformed when . Express your answer in N * m / rad.

The space shuttle is located in ''deep space'', where the effects of gravity can be neglected.  It has a mass of , a center of mass at , and a radius of gyration about the axis. It is originally traveling forward at when the pilot turns on the engine at , 

At the instant shown, the bar rotates clockwise at .  The spring attached to its end always remains vertical due to the roller guide at . 

The assembly consists of two 8-lb bars which are pin-connected to the two 10-lb disks. If the bars are released from rest when , determine their angular velocities at the instant . Assume the disks roll without slipping. Express your answer in rad/s.

For safety reasons, the supporting leg of a sign is designed to break away with negligible resistance at when the leg is subjected to the impact of a car. Assuming that the leg is pinned at and approximates a thin rod, determine the impulse the car bumper exerts on it, if after the impact the leg appears to rotate clockwise to a maximum angle of . Express your answer in N*s.

The slender rod is pinned at and is initially at rest.  If a bullet is fired into the rod with a velocity of , as shown in the figure, determine the angular velocity of the rod just after the bullet becomes embedded in it. Express your answer in rad/s.

A dumbbell, consisting of two massive small spheres and a light rigid rod of length L, rests on a smooth horizontal surface. A third identical sphere collides with one of its spheres. The velocity    of the incoming sphere is directed at an angle of to the rod. A head-on perfectly inelastic collision occurs. Find the angular velocity of the “heavier dumbbell” after the impact.

A system of two identical point masses m, attached to a light rod of length l, rotates with angular velocity    about a fixed axis passing through the center of the rod at an angle to it. Find the magnitude of the angular momentum vector of the system at an arbitrary moment in time.

A light rod of length l is suspended from the ceiling at its ends by two vertical strings. Two small masses are attached to the rod at distances l/4 from its ends: m_1 = 7m and m_2 = m (see figure). The right string suddenly breaks. Find the tension in the left string immediately after this.

A dumbbell, consisting of two massive small spheres and a light rigid rod of length L, rests on a smooth horizontal surface. A third identical sphere collides with one of its spheres. The velocity of the incoming sphere is directed at an angle of  to the rod. A head-on perfectly elastic collision occurs. Find the angular velocity of the dumbbell after the impact.