Rigid Bodies and Plane Motion
Rigid bodies are any system of particles in which the distance between individual particles do not change relative to each other. Generally an assumption of a rigid body is sufficient for most solid objects that are not undergoing extreme deformations.
For the purposes of simplicity we will be dealing primarily with plane motion of rigid bodies. Plane motion occurs when all particles in a rigid body move in parallel planes, making motion effectively two-dimensional. Plane motion can be broken into two components: translation and rotation.
- Translation: Translation is a motion in which lines between particles in the body remain parallel to their initial position.
- Rotation: Rotation is a motion in which particles in a rigid body move in a circular path around a fixed axis.
- General Plane Motion: The combination of translation and rotation can be described as the general case of plane motion.
Rotation
Rotation about a fixed axis is most easily described in polar coordinates about the axis of rotation. The angular velocity, \bm{\overrightarrow{\omega}}, is a measure of how fast a particle is orbiting an axis and is measured in \left[\dfrac{rad}{s}\right]. Angular acceleration, \bm{\overrightarrow{\alpha}}, measured in \left[\dfrac{rad}{s^2}\right], is a measure of how a particles angular velocity is changing. The vectors associated with angular velocity and angular acceleration act in the direction perpendicular to the plane of rotation and follow the right-hand rule. Rotation in the counter-clockwise direction is taken as positive, whereas rotation in the clockwise direction is denoted as negative.
The velocity of a particle can be found by knowing the radius from the axis of rotation and the angular velocity of the particle.
\bm{\overrightarrow{v}}=\bm{\overrightarrow{\omega}}\times\bm{\overrightarrow{r}}
The acceleration of a particle can be defined in terms of angular acceleration and angular velocity:
\bm{\overrightarrow{a}}=\bm{\overrightarrow{\alpha}}\times\bm{\overrightarrow{r}}+\bm{\overrightarrow{\omega}}\times(\bm{\overrightarrow{\omega}}\times\bm{\overrightarrow{r}})
\bm{\overrightarrow{a}}_t=(\alpha r)\bm{\hat{e}}_t
\bm{\overrightarrow{a}}_n=(\omega^2r)\bm{\hat{e}}_n
Angle, angular velocity, and angular acceleration are all related to one another in the same way that position, velocity, and acceleration are related.
\displaystyle\bm{\overrightarrow{\theta}}=\int\bm{\overrightarrow{\omega}} dt+\bm{\overrightarrow{\theta}}_0
\displaystyle\bm{\overrightarrow{\omega}}=\dfrac{d\bm{\overrightarrow{\theta}}}{dt}
\displaystyle\bm{\overrightarrow{\omega}}=\int\bm{\overrightarrow{\alpha}} dt+\bm{\overrightarrow{\omega}}_0
\bm{\overrightarrow{a}}=\dfrac{d\bm{\overrightarrow{\omega}}}{dt}
Absolute Motion
Problems in a stationary frame are generally straightforward. We must first determine the geometric relationship between points of interest within the rigid body of question. Once this relationship is established (mathematically with an equation) it is a simple process of applying common sense and the definitions of position, velocity, and acceleration using calculus.
Relative Velocity
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