Rigid bodies

A rigid body is an extended area of material that includes all the points inside it, and which moves so that the distances and angles between all its points remain constant. The location of a rigid body can be described by the position of one point $P$ inside it, together with the rotation angle of the body (one angle in 2D, three angles in 3D).

Neither point masses nor rigid bodies can physically exist, as no body can really be a single point with no extent, and no extended body can be exactly rigid. Despite this, these are very useful models for mechanics and dynamics.

Rotating rigid bodies

All points on a rigid body have the same angular rotation angles, as we can see on the figure below. Because the angular velocity is the derivative of the rotation angles, this means that every point on a rigid body has the same angular velocity \( \vec{\omega} \), and also the same angular acceleration \( \vec{\alpha} \).

In 2D the angle \( \theta \) of a rigid body the angle of rotation from a fixed reference (typically the \( \hat\imath \) direction), measured positive counter-clockwise. The angular velocity is \( \omega = \dot\theta \) and the angular acceleration is \( \alpha = \dot\omega = \ddot\theta \). The vector versions of these are \( \vec\omega = \omega \, \hat{k} \) and \( \vec\alpha = \alpha\,\hat{k} \), where \( \hat{k} \) is the out-of-plane direction.

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