A particle having a mass of 1.5 kg, moves along a three-dimensional path defined by the equations r = 94 + 3t) m, = (t² + 2) rad, and z = (6 - t³) m, where t is in seconds, and the z-axis is vertical. Determine the r, , and z components of force which the path exerts on the particle when t = 2 s.

Each of the three barges has a mass of 30 Mg, whereas the tugboat has a mass of 12 Mg. As the barges are being pulled forward with a constant velocity of 4 m/s, the tugboat must overcome the frictional resistance of the water, which is 2 kN for each barge and 1.5 kN for the tugboat. If the cable between A and B breaks, determine the acceleration of the tugboat.

The spool, which has a weight of 2lb, slides along the smooth horizontal spiral rod, r = (2) ft, where is in radians. If its angular rate of rotation is constant and equals = 4 rad/s, determine the tangential force P needed to cause the motion and the normal force that the spool exerts on the rod at the instant = 90°.

A smooth can C, having a mass of 2 kg, is lifted from a feed at A to a ramp at B by a forked rotating rod. If the rod maintains a constant angular motion of = 0.5 rad/s, determine the force which the rod exerts on the can at the instant = 30°. Neglect the effects of friction in the calculation. The ramp from A to B is circular, having a radius of 700 min.

A truck T has a weight of 8,000 lb and is traveling along a portion of a road defined by the lemniscate r² = 0.2(106) cos 2 , where r is measured in feet and is in radians. If the truck maintains a constant speed of vr = 4 ft/s, determine the magnitude of the resultant frictional force which must be exerted by all the wheels to maintain the motion when = 0.