9.4: Angular acceleration of a rigid body
After having determined the velocity vector of a point in a rigid body, it is now of interest to also determine the acceleration vector of the points in the rigid body. We take the time derivative of Equation 9.17, and use the product rule on the vector cross product to determine the acceleration vector \(\overrightarrow{\boldsymbol{a}}_{B}\) in a rigid body.
\[\begin{align} \frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{v}}_{B} & =\frac{\mathrm{d}}{\mathrm{d} t} \overrightarrow{\boldsymbol{v}}_{A}+\frac{\mathrm{d} \overrightarrow{\boldsymbol{\omega}}}{\mathrm{d} t} \times \overrightarrow{\boldsymbol{r}}_{B / A}+\overrightarrow{\boldsymbol{\omega}} \times \frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{B / A}}{\mathrm{~d} t} \tag{9.18} \label{9.18}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B} & =\overrightarrow{\boldsymbol{a}}_{A}+\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\boldsymbol{r}}_{B / A}+\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{v}}_{B, \mathrm{rot}} \tag{9.19} \label{9.19}\end{align}\]
In this derivation we defined the angular acceleration vector \(\overrightarrow{\boldsymbol{\alpha}}\) of the rigid body.
The angular acceleration vector \(\overrightarrow{\boldsymbol{\alpha}}\) of a rigid body is the time derivative of its angular velocity vector.
\[\overrightarrow{\boldsymbol{\alpha}} \equiv \frac{\mathrm{d} \overrightarrow{\boldsymbol{\omega}}}{\mathrm{d} t} \tag{9.20} \label{9.20}\]
We note that in planar kinematics this expression can be simplified:
\[\overrightarrow{\boldsymbol{\alpha}}_{2 D}=\dot{\omega} \hat{\boldsymbol{k}}=\alpha \hat{\boldsymbol{k}} \tag{9.21} \label{9.21}\]
The unit of angular acceleration is \(\mathrm{rad} / \mathrm{s}^{2}\). We now substitute Equation 9.16 in Equation 9.19 and obtain the general expression for the acceleration.
The general vector expression for the acceleration vector of a point \(B\) in a rigid body that has translational and rotational acceleration:
\[\overrightarrow{\boldsymbol{a}}_{B}=\overrightarrow{\boldsymbol{a}}_{A}+\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\boldsymbol{r}}_{B / A}+\overrightarrow{\boldsymbol{\omega}} \times\left(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A}\right) \tag{9.22} \label{9.22}\]
This equation shows that the acceleration of a point \(B\) on a rigid body consists of three contributions that are shown in Figure 9.5.
- The translational acceleration \(\overrightarrow{\boldsymbol{a}}_{B, \text { trans }}=\overrightarrow{\boldsymbol{a}}_{A}\) due to the acceleration of point \(A\).
- The angular acceleration \(\overrightarrow{\boldsymbol{a}}_{B, \text { ang }}=\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\boldsymbol{r}}_{B / A}\) due to the angular acceleration vector.
- The centripetal acceleration \(\overrightarrow{\boldsymbol{a}}_{B, \mathrm{cptl}}=\overrightarrow{\boldsymbol{\omega}} \times\left(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A}\right)\), due to the angular velocity vector \(\overrightarrow{\boldsymbol{\omega}}\).
Acceleration in planar kinematics
In planar kinematics we can simplify the expressions for the angular acceleration somewhat. By using that \(\overrightarrow{\boldsymbol{\alpha}}=\alpha \hat{\boldsymbol{k}}\) we find:
\[\overrightarrow{\boldsymbol{a}}_{B, \text { ang }, 2 \mathrm{D}}=\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\boldsymbol{r}}_{B / A}=\alpha \hat{\boldsymbol{k}} \times\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\rho}}_{A}=\alpha\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A} \tag{9.23} \label{9.23}\]
The vector \(\overrightarrow{\boldsymbol{\omega}}\) is always perpendicular to the \(x y\)-plane, such that \(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A}=\) \(\omega\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A}\) and:
\[\overrightarrow{\boldsymbol{a}}_{B, \mathrm{cptl}, 2 \mathrm{D}}=\overrightarrow{\boldsymbol{\omega}} \times\left(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A}\right)=-\omega^{2} \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.24} \label{9.24}\]
This shows that the centripetal component of acceleration always points towards point \(A\), the centre of rotation. Note that a similar result for the centripetal acceleration term was obtained in Equation 5.81. Combining the three terms we obtain for the planar kinematics of a rigid body the following equation:
\[\overrightarrow{\boldsymbol{a}}_{B, 2 D}=\overrightarrow{\boldsymbol{a}}_{A}+\alpha\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}-\omega^{2} \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.25} \label{9.25}\]
For completeness we repeat the most important equations for analysing the kinematics of a rigid body:
\[\begin{align} \overrightarrow{\boldsymbol{v}}_{B} & =\overrightarrow{\boldsymbol{v}}_{A}+\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.26} \label{9.26}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B} & =\overrightarrow{\boldsymbol{a}}_{A}+\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\boldsymbol{r}}_{B / A}+\overrightarrow{\boldsymbol{\omega}} \times\left(\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\boldsymbol{r}}_{B / A}\right) \tag{9.27} \label{9.27}\\[4pt] \overrightarrow{\boldsymbol{v}}_{B, 2 D} & =\overrightarrow{\boldsymbol{v}}_{A}+\omega\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A} \tag{9.28} \label{9.28}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B, 2 D} & =\overrightarrow{\boldsymbol{a}}_{A}+\alpha\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \hat{\boldsymbol{\phi}}_{A}-\omega^{2} \overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.29} \label{9.29}\end{align}\]