Plane Motion of a Rigid Body

Plane motion of a rigid body is motion in which all points of the body move in planes parallel to a fixed plane. Its kinematics can be studied through the motion of a plane figure representing a section of the body parallel to the plane of motion.

Kinematic decomposition

At any instant, plane motion of a rigid body can be represented as translation with an arbitrarily selected reference point $A$ plus rotation about that point:

$$\vec r_B=\vec r_A+\vec r_{B/A}.$$

The position of the plane figure is specified by two coordinates of the reference point, such as $x_A(t)$ and $y_A(t)$, together with the orientation angle $\varphi(t)$.

The reference point can be selected arbitrarily. The translational component depends on that choice, but the angular velocity and angular acceleration of the plane figure at an instant do not.

Equations of motion of a plane figure

The three functions $x_A=x_A(t)$, $y_A=y_A(t)$, and $\varphi=\varphi(t)$ completely specify the figure position in the plane. The first two describe translation of the reference point and the third describes change of body orientation.

Translational and rotational components

Decomposing the motion into translation and rotation does not mean the body physically performs them one after another. It is a kinematic representation of one actual motion: the reference point moves while the figure simultaneously changes orientation.

Angular characteristics

Angular velocity is $\omega=\dot\varphi$ and angular acceleration is $\varepsilon=\ddot\varphi$. For plane motion, their vectors are perpendicular to the plane of motion.

Special cases

If $\omega=0$ throughout an interval, the motion reduces to translation. If the reference point is fixed, the motion reduces to rotation about a fixed axis perpendicular to the plane.

Example

A wheel rolling along a straight path undergoes plane motion: its center translates while the wheel simultaneously rotates. The motion of any point on the rim combines these two components.

Common mistakes

• treating the reference point as a physically fixed point;
• assuming $\omega$ depends on the selected reference point;
• describing plane-figure position only by one point’s coordinates and omitting $\varphi$;
• confusing plane motion with pure translation.