At the instant shown, the shaft and plate rotates with an...

At the instant shown, the shaft and plate rotates with an angular velocity of $\omega=14 \mathrm{rad} / \mathrm{s}$ and angular acceleration of $\alpha=7 \mathrm{rad} / \mathrm{s}^{2} .$ Determine the velocity and acceleration of point $D$ located on the corner of the plate at this instant. Express the result in Cartesian vector form.

Key Concepts

Angular Velocity and Angular Acceleration

Angular velocity is a vector quantity that describes how fast an object rotates and the direction of the rotation axis, often determined using the right-hand rule. Angular acceleration, on the other hand, is the rate at which the angular velocity changes with time. These quantities provide essential information for determining the kinematics of rotating bodies.

Acceleration Components in Rotational Motion

The acceleration of a point in a rotating body has two main components: the tangential acceleration, which arises from the change in the rate of rotation (i.e., angular acceleration), and the centripetal (or radial) acceleration, which is directed towards the center of the circular path due to the change in direction of the velocity vector. Expressing these components in Cartesian form is critical for solving problems that require detailed spatial analysis.

Rigid Body Rotation

Rigid body rotation refers to the motion where all parts of the object move in circles about a fixed axis while maintaining constant distances relative to each other. This principle simplifies the analysis of motion because the rotational effects (such as velocity and acceleration) at any point in the body can be related directly to the rotation about the fixed axis.

Velocity of a Point on a Rotating Body

In rotational dynamics, the velocity of any point in a rigid body is obtained by taking the cross product between the angular velocity vector and the position vector (from the rotation axis to the point). This cross product yields a velocity vector that is tangential to the circle described by the point, encapsulating both the magnitude and the correct direction of the motion.

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