15.10 : 使用旋转轴对相对运动进行分析-解决问题

The crane's telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s².

Simultaneously the boom extends with a constant speed of 5 m/s, measured relative to point C. At the given instant, the distance between points C and D is 60 meters.

Determine the magnitudes of the velocity and acceleration of point D at the same instant.

Here, point D is translating and rotating with respect to point C, so relative motion analysis can be done using a rotating frame of reference.

At the instant, the linear velocity and the linear acceleration of point C are zero. The angular velocity and angular acceleration of point D are along the negative z-axis.

Writing the relative velocity equation for point D in the rotating frame of reference, and substituting the known quantities, gives the magnitude of the velocity of point D.

Similarly, using the relative acceleration equation in a rotating frame and substituting the known quantities gives the acceleration of point D.