9.3: Velocities in a rigid body
By taking the time derivative of the position vector \(\overrightarrow{\boldsymbol{r}}_{B}\) in Equation 9.4, we can determine the velocity vector of point \(B\) in the rigid body as follows:
\[\begin{align} \overrightarrow{\boldsymbol{v}}_{B, 2 D} & =\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{r}}_{B}=\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{r}}_{A}+\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.6} \label{9.6}\\[4pt] & =\overrightarrow{\boldsymbol{v}}_{A}+\dot{\phi}_{B / A}\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right|\left(-\sin \phi_{B / A} \hat{\boldsymbol{\imath}}+\cos \phi_{B / A} \hat{\boldsymbol{\jmath}}\right) \tag{9.7} \label{9.7}\\[4pt] & =\overrightarrow{\boldsymbol{v}}_{A}+\dot{\phi}_{B / A}\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A} \tag{9.8} \label{9.8}\end{align}\]
In the last step we used Equation 5.71 to replace the terms in brackets by \(\hat{\phi}_{A}\), which represents the unit vector at point \(B\) of a cylindrical coordinate system with origin \(A\), which is why we add the subscript \(A\) to the unit vector. The velocity of point \(B\) in Equation 9.8 can be split up in two parts: a vector \(\overrightarrow{\boldsymbol{v}}_{B, \text { trans }}\) related to translation and a vector \(\overrightarrow{\boldsymbol{v}}_{B, \text { rot }}\) related to rotation:
\[\begin{align} \overrightarrow{\boldsymbol{v}}_{B} & =\overrightarrow{\boldsymbol{v}}_{B, \text { trans }}+\overrightarrow{\boldsymbol{v}}_{B, \text { rot }} \tag{9.9} \label{9.9}\\[4pt] \overrightarrow{\boldsymbol{v}}_{B, \text { trans }} & =\overrightarrow{\boldsymbol{v}}_{A} \tag{9.10} \label{9.10}\\[4pt] \overrightarrow{\boldsymbol{v}}_{B, \text { rot }, 2 D} & =\dot{\phi}_{B / A}\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A} \tag{9.11} \label{9.11}\end{align}\]
In Figs. 9.2, 9.3 and 9.4 we show these three velocity vectors \(\overrightarrow{\boldsymbol{v}}_{B, \text { trans }}, \overrightarrow{\boldsymbol{v}}_{B, \text { rot }}\) and \(\overrightarrow{\boldsymbol{v}}_{B}\). Let us first discuss two special types of rigid body motion: pure translation and pure rotation.
9.3.1 Pure translation
When the angle \(\phi_{B / A}\) is kept constant, like in Figure 9.2, all points in the rigid body move with the same velocity vector:
\[\overrightarrow{\boldsymbol{v}}_{B}=\overrightarrow{\boldsymbol{v}}_{B, \text { trans }}=\overrightarrow{\boldsymbol{v}}_{A} \tag{9.12} \label{9.12}\]
This type of motion is called pure translation. The rotational component of velocity is zero, as follows from Equation 9.11 and \(\dot{\phi}_{B / A}=0\). It is important to note that pure translation can happen along any path curve \(\overrightarrow{\boldsymbol{r}}_{A}(s)\), even a circular path. The word translation thus only means that the shape of the path is identical for all points in the rigid body because \(\phi_{B / A}\) is constant.
9.3.2 Pure rotation
When the position vector of a point in the rigid body is constant in time, and all other points make circular paths around it, e.g. because it rotates around an axle at that position, the motion of the rigid body is a pure rotation. We choose point \(A\) to be the point with constant position vector \(\overrightarrow{\boldsymbol{r}}_{A}\), as shown in Figure 9.3. Then according to Equation 9.11 the velocity vector of point \(B\) is given
by:
\[\overrightarrow{\boldsymbol{v}}_{B}=\overrightarrow{\boldsymbol{v}}_{B, \mathrm{rot}, 2 D}=\dot{\phi}_{B / A}\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A} \tag{9.13} \label{9.13}\]
To facilitate the analysis of rotations and generalise it to \(3 \mathrm{D}\), we now define the angular velocity \(\omega\) and angular velocity vector \(\overrightarrow{\boldsymbol{\omega}}\).
9.3.3 Angular velocity of a rigid body
The kinematics of a rigid body is closely linked to the time derivative \(\dot{\phi}_{B / A}(t)\), which is defined as its angular velocity.
The time derivative \(\dot{\phi}_{B / A}\) is the angular velocity \(\omega\) of the rigid body:
\[\omega \equiv \dot{\phi}_{B / A} \tag{9.14} \label{9.14}\]
The unit of angular velocity is rad/s. Interestingly, the angular velocity \(\omega\) of the rigid body is independent of the choice of the points \(A\) and \(B\) on the rigid body and is therefore a general property of a rigid body as can be shown as follows.
The angular velocity \(\omega\) of a rigid body is independent of the choice of points \(A\) and \(B\) on the rigid body.
Note that the angular velocity \(\omega\) of a rigid body is different from the orbital angular velocity \(\omega_{o}\) of a single point mass around an axis. Orbital angular velocity can depend on the position of the rotation axis or origin (Sec. 5.9.4). To distinguish angular velocity of a rigid body from orbital angular velocity, the angular velocity of a rigid body is therefore sometimes called its spin angular velocity.
9.3.4 Angular velocity vector
It can be seen in Figure 9.3 that for pure rotation all points in the rigid body move in circular paths around point \(A\). The velocity vector \(\overrightarrow{\boldsymbol{v}}_{B}\) and all other points in the rigid body, lie in the plane of those circles. To describe that plane we define the angular velocity vector to be perpendicular to that plane.
The angular velocity vector \(\overrightarrow{\boldsymbol{\omega}}\) of a rigid body is a vector with magnitude \(|\omega|=\left|\dot{\phi}_{B / A}\right|\) and a direction that is perpendicular to the plane in which the rigid body rotates. Its direction can be determined using the right hand rule.
The angular velocity vector (unit rad/s) of a rigid body that rotates in the \(x y\) plane is:
\[\overrightarrow{\boldsymbol{\omega}}_{2 D}=\omega \hat{\boldsymbol{k}}=\dot{\phi}_{B / A} \hat{\boldsymbol{k}} \tag{9.15} \label{9.15}\]
The direction of the vector can be determined using the right-hand rule by curving the fingers of your right-hand around the curved arrow in Figure 9.3, which indicates the direction of rotational motion. Then your thumb points in the \(\hat{\boldsymbol{k}}\) direction, in agreement with Equation 9.15.
9.3.5 Determining velocities with the angular velocity vector
From Figure 9.3 we see that the velocity vector \(\overrightarrow{\boldsymbol{v}}_{B, \text { rot }}\) lies in the plane in which the rigid body moves. It is therefore perpendicular to \(\overrightarrow{\boldsymbol{\omega}}\). It is also perpendicular to the vector \(\overrightarrow{\boldsymbol{r}}_{B / A}\), since this vector is the radius of the circular motion. To obtain a vector that is perpendicular to two other vectors we take the cross product of these vectors:
The rotational velocity of a point \(B\) in a rigid body is given by:
\[\overrightarrow{\boldsymbol{v}}_{B, \text { rot }}=\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.16} \label{9.16}\]
For the 2D case, where \(\overrightarrow{\boldsymbol{\omega}}_{2 D}=\dot{\phi}_{B / A} \hat{\boldsymbol{k}}\) and \(\overrightarrow{\boldsymbol{r}}_{B / A}=\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\rho}}_{A}\), it is straightforward to check the correctness of this equation by comparison with Equation 9.11 and by using that \(\hat{\boldsymbol{k}} \times \hat{\boldsymbol{\rho}}_{A}=\hat{\boldsymbol{\phi}}_{A}\) as follows from the right-hand rule.
9.3.6 General motion
In general, as shown in Figure 9.4, the motion of a point in a rigid body is a sum of translational and rotational motion. By combining Eqs. (9.9) and (9.16) we
obtain the most general equation and important equation for the velocity in a rigid body:
\[\overrightarrow{\boldsymbol{v}}_{B}=\overrightarrow{\boldsymbol{v}}_{A}+\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.17} \label{9.17}\]
We note that this equation is valid in \(3 \mathrm{D}\) and for any choice of the points \(A\) and \(B\) as long as both points move along with the rigid body. However, a smart choice of point \(A\) can simplify the analysis.