Kinematics of Rigid Bodies

A rigid body is an ideal body that does not deform. Its motion is defined by the translation of its center of mass and rotation about its center of mass.

Types of Rigid Body Motion

Translation: All lines on the body remain parallel to their original positions.
Rotation about a Fixed Axis: All particles move in circular paths about the axis of rotation.
General Plane Motion: A combination of translation and rotation.

Rotation about a Fixed Axis

Angular Variables

Angular Position ( $\theta$ ): Measured in radians.
Angular Velocity ( $\omega$ ): $\omega = \frac{d\theta}{dt}$ (rad/s)
Angular Acceleration ( $\alpha$ ): $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$ (rad/s $^2$ )
Relation: $\alpha \, d\theta = \omega \, d\omega$

Motion of a Point on a Rotating Body

For a point $P$ at distance $r$ from the axis:

Velocity: $v = r \omega$ (tangent to path)
Tangential Acceleration: $a_t = r \alpha$
Normal Acceleration: $a_n = r \omega^2$

General Plane Motion

General plane motion can be analyzed as the sum of a translation and a rotation.

Relative Motion Analysis (Velocity): $\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{B/A}$ $\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}_{B/A}$

Where $\mathbf{v}_{B/A}$ is the velocity of $B$ relative to $A$ due to rotation about $A$ . Its magnitude is $v_{B/A} = r_{B/A} \omega$ .

Instantaneous Center of Rotation (IC): The point about which the body appears to rotate at a given instant. The velocity of the IC is zero. $v = r_{IC} \omega$

Example: Rolling Wheel

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