Relative Motion Analysis using Rotating Axes-Problem Solving

A rotating frame of reference is used because point D undergoes both translation and rotation relative to point C. This approach simplifies the analysis by decomposing complex motion into manageable components. The relative motion analysis using rotating axes allows engineers to apply velocity and acceleration equations that account for the boom's simultaneous extension and rotation.

The crane boom rotates with an angular velocity of 0.04 rad/s and an angular acceleration of 0.02 rad/s². These parameters describe the rotational motion of the boom about point C. At the given instant, both angular quantities are directed along the negative z-axis, indicating clockwise rotation when viewed from above.

The boom extends at a constant speed of 5 m/s relative to point C, measured along the boom's length. This linear extension is independent of the boom's rotation and contributes directly to point D's velocity. At the instant analyzed, the distance between points C and D is 60 meters, which is used in calculating the total velocity and acceleration of point D.

Using the relative velocity equation in the rotating frame of reference and substituting the known quantities, the magnitude of velocity of point D is 5.55 m/s. This velocity results from combining the linear extension speed of 5 m/s and the rotational velocity component generated by the boom's angular velocity and the 60-meter distance from point C.

Using the relative acceleration equation in the rotating frame of reference, the magnitude of linear acceleration of point D is 1.4 m/s². This acceleration accounts for the angular acceleration of the boom and the centripetal acceleration generated by the rotation at the 60-meter radius, combined with the constant extension rate.

Point C is the fixed reference point on the crane boom where the rotation occurs. Although point C rotates with the boom, it does not translate linearly; it remains stationary in space. Therefore, its linear velocity and acceleration are zero, making it an ideal reference point for analyzing the relative motion of point D using a rotating frame of reference.

The relative velocity and acceleration equations for a rotating frame are applied by substituting known values: angular velocity (0.04 rad/s), angular acceleration (0.02 rad/s²), extension speed (5 m/s), and radial distance (60 m). These equations decompose point D's motion into components relative to the rotating boom, yielding the total velocity of 5.55 m/s and acceleration of 1.4 m/s².