2D kinematics of a rigid body

Rigid body

A solid body that doesn’t deform.

Degrees of freedom

In the motion of a rigid body in 2D kinematics, there are $3$ degrees of freedom. $$

• Movement along $x$ direction
• Movement along $y$ direction
• Rotation about $z$ direction

In 3D, there are $6$ degrees of freedom: movement and rotation along each direction. $$

Translation

Movement that changes the position of an object. Translation can be done through a rectilinear or curvilinear path. Axes of the body always stays parallel.

Rotation

Circular movement of an object about a fixed axis that is perpendicular to the plane.

General 2D motion

Mixture of translation and rotation.

$$v_\text{B}= v_\text{A}+v_{\text{B}/\text{A}}= v_\text{A}+\underline{\dot{\theta}}\times r_{\text{B}/\text{A}}$$ $$a_\text{B}= a_\text{A}+a_{\text{B}/\text{A}}= a_\text{A}+\underline{\ddot{\theta}}\times r_{\text{B}/\text{A}}+\underline{\dot{\theta}}\times (\underline{\dot{\theta}}\times r_{\text{B}/\text{A}})$$

Here:

• $\underline{\dot{\theta}}$ - Angular velocity of $\text{B}$ relative to $\text{A}$
• $v_{\text{B}/\text{A}}$ - Velocity of $\text{B}$ relative to $\text{A}$
• $a_{\text{B}/\text{A}}$ - Acceleration of $\text{B}$ relative to $\text{A}$
• $r_{\text{B}/\text{A}}$ - Position vector of $\text{B}$ relative to $\text{A}$. It’s constant.

In general motion, each particle of the body has a different velocity at every instance.

Instantaneous centre of rotation

The point that has $0$ velocity at a particular instant of time. This point might be changing throughout the motion. Denoted by $I$.

It can be imagined that the object is momentarily having a pure rotation about this centre $I$. $$

$I$ can be found by drawing a line perpendicular at the velocity vectors at 2 different points and finding their intersection point. $$

Centrode

The locus of instantaneous centres during the motion.