6.10: Obtaining the equations of motion
After thorough preparation, we are now in the position to fully evaluate Newton’s second law to obtain the equations of motion using the following steps.
- Add the force components from the FBD along each of the axes.
- Apply \(\overrightarrow{\boldsymbol{F}}=m \overrightarrow{\boldsymbol{a}}\) along each of the axes.
- Express the components of the acceleration vector in terms of the position coordinates and their time derivatives using kinematics (see Sec. 5.9).
- Simplify the EoM by using the constraint equations and combining the scalar equations along the axes directions.
As an example of this procedure we apply it to obtain the EoM of the block \(A\) in Figure 6.4 using natural coordinates, and the projected forces from the previous section. We obtain three scalar equations of motion.
\[\begin{align} & \sum F_{A, t}=m_{A} g \sin \alpha-\mu_{k} F_{N}=m_{A} a_{A, t}=m_{A} \ddot{A}_{A} \tag{6.16} \label{6.16}\\[4pt] & \sum F_{A, n}=F_{N}-m_{A} g \cos \alpha=m_{A} a_{A, n} \underset{\text { Using Equation 6.10 }}{=} 0 \tag{6.17} \label{6.17}\\[4pt] & \sum F_{A, b}=0=m_{A} a_{A, b}=m_{A} \ddot{z}_{A} \tag{6.18} \label{6.18}\end{align}\]
where we used Equation 5.85 and Equation 5.86 to obtain the acceleration components in \(t, n, b\) coordinates. We see that the sum of forces in the \(\hat{\boldsymbol{b}}\) direction is zero, such that \(\ddot{z}_{A}\) is zero. We uses the constraint equation Equation 6.10 to find that \(a_{A, n}=0\), which implies that the sum of the forces in that direction is also zero. From that condition we find the equation for the normal force \({ }^{a}\) to be \(F_{N}=m_{A} g \cos \alpha\). By substituting this equation into Equation 6.16 we obtain the EoM along the \(\hat{\boldsymbol{t}}\) direction:
\[\begin{align} m_{A} \ddot{s}_{A} & =m_{A} g \sin \alpha-\mu_{k} F_{N} \tag{6.19} \label{6.19}\\[4pt] m_{A} \ddot{s}_{A} & =m_{A} g \sin \alpha-\mu_{k} m_{A} g \cos \alpha \tag{6.20} \label{6.20}\\[4pt] \ddot{s}_{A} & =g\left(\sin \alpha-\mu_{k} \cos \alpha\right) \equiv a_{0} \tag{6.21} \label{6.21}\end{align}\]
where we define \(a_{0}\) as the constant acceleration along the path coordinate. The same result could have been obtained by using Cartesian coordinates, but this would have been more complicated mathematically.