9.1: Rigid bodies
After having discussed dynamics of point masses, we now turn to the analysis of rigid bodies. In this textbook we focus on the planar kinematics of rigid bodies in the \(x y\)-plane. That means that the point masses in the rigid body all move in the \(x y\)-plane with \(z=0\). However, we note that much of the presented theory is also applicable in 3D and unless explicitly indicated, e.g. with a subscript \({ }_{2 D}\), the equations in this textbook are also valid in \(3 \mathrm{D}\).
\(A\) rigid body \(F\) is a set of point masses (atoms) \(m_{i}\) which have the special property that their relative position vectors are fixed in time, such that the body is undeformable.
Every point mass \(m_{i}\) in the rigid body can be identified by a position vector \(\overrightarrow{\boldsymbol{r}}_{i}\). Since the body is rigid and undeformable, the distance between every two point masses in the rigid body is constant which relates their dynamics by the following relative constraint equation (see Sec. 5.2.4):
\[\left|\overrightarrow{\boldsymbol{r}}_{i / j}\right|=\left|\overrightarrow{\boldsymbol{r}}_{i}-\overrightarrow{\boldsymbol{r}}_{j}\right|=\text { constant } \tag{9.1} \label{9.1}\]
In the next section we discuss how the orientation and position of a rigid body can be specified.