15.10 : Relative Motion Analysis using Rotating Axes-Problem Solving

Consider a crane whose telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s². Along with the rotation, the boom also extends linearly with a uniform speed of 5 m/s. The extension of the boom is measured at point D, which is measured with respect to the fixed point C on the other end of the boom. For the given instant, the distance between points C and D is 60 meters.

Here, in order to determine the magnitude of velocity and acceleration for point D, relative motion analysis is used. For the given situation, point D is translating and rotating with respect to point C, so the relative motion analysis can be done using a rotating frame of reference. At the considered instant, the linear velocity and linear acceleration of point C are zero. At the same time, the angular velocity and angular acceleration of point D are along the negative z-axis as the boom is rotating in the clockwise direction.

The relative velocity and relative acceleration equations for the rotating frame of reference for point D are used to calculate the magnitudes of linear velocity and linear acceleration of point D.

Substituting the known values, magnitudes of linear velocity and linear acceleration of point D are found to be 5.55 m/s and 1.4 m/s², respectively.