Good Night everyone:

I have one problem here that I KNOW how to solve using Lagragian Dynamics. But, I really want to know how to solve using Vector decomposition, Newton's Laws, first-year physics and so on..... I really apreciate tips and hints, both mathematical and physical. I DON'T WANT A SOLUTION OR STEP-BY-STEP.

Thank you.

(*) The "motivation" for my question is that we often hear that Lagrangian Dynamics is more General and powerful than Newton's Approach. It's true. But, I want to see for myself that it's true. On this particular problem, which is more difficult than basic Newtonian problems, the solution is hard (?) but still "possible".

(**): The concepts of forced,damped,simple and coupled oscillator are quite clear to me and basic ordinary differential equations as well. enter image description here


A good start is to make a free body diagram of all parts. Mark known and unknown forces. Remember that if you have a force $\mathbf{F}$ on one part at the contact point with another part, on the contacting part you have a force $-\mathbf{F}$. Then set up a differential equation for the motion given the total forces on the part(s) of interest.

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You have done the first step, and that is to recognize the degrees of freedom of the system ($x$ and $\theta$). Lets call the block body [1] and the sphere body [2] and the rod length $\ell$.

  • Kinematics
    • Express the position of the joints and the centers of mass as a function of the degrees of freedom. For example $$ {\bf r}_{2} = ( x + \ell \sin \theta, -\ell \cos \theta,0)$$
    • Express the rotational velocities in a similar manner $$ {\boldsymbol \omega}_2 = \dot{\theta} {\bf \hat{z}} $$
    • Take the total derivative of position to find the velocity and then the acceleration of the centers of mass. For example $${\bf v}_2 = (\dot{x} + \dot{\theta} \ell \cos\theta, \dot{\theta} \ell \sin \theta,0)$$ and $${\bf a}_2 = (\ddot{x} + \ddot{\theta} \ell \cos\theta - \ell \dot{\theta}^2 \sin \theta, \ddot{\theta} \ell \sin \theta + \ell \dot{\theta}^2 \cos \theta,0)$$
    • Take the total derivative of rotational velocity for rotational acceleration $${\boldsymbol \alpha}_2 = \ddot{\theta}{\bf \hat{z}} $$
  • Dynamics
    • Do a free body diagram for each body and sum up the forces on each body and torques about each center of mass on each body. For each joint apply appropriate reaction forces on the next body, and equal and opposite ones on the previous one. For example (considering tension $T$, friction $f_1$ and normal force $n_1$) $$\begin{align} \Sigma {\bf F}_1 & = {\bf T}_{12} + {\bf W}_1 & \Sigma {\bf F}_2 & = -{\bf T}_{12} + {\bf W}_2 \\ & = \begin{pmatrix} T \sin\theta+f_1 \\ -T \cos\theta-m_1 g+n_1 \\ 0 \end{pmatrix} & & = \begin{pmatrix} -T \sin\theta \\ T \cos\theta-m_2 g \\ 0 \end{pmatrix} \end{align} $$ ${\bf W}$ designates applied forces and ${\bf T}$ internal joint forces
    • Equations of motion $$ \begin{align} \Sigma {\bf F}_1 & = m_1 {\bf a}_1 &\Sigma {\bf F}_2 & = m_2 {\bf a}_2 \\ \Sigma {\bf M}_1 & = I_1 {\boldsymbol \alpha}_1 + {\boldsymbol \omega}_1 \times I_1 {\boldsymbol \omega}_1 & \Sigma {\bf M}_2 & = I_2 {\boldsymbol \alpha}_2 + {\boldsymbol \omega}_2 \times I_2 {\boldsymbol \omega}_2 \end{align} $$ Care must be taken on 3D problems to express the mass moment of inertia matrix along the world coordinates and not the body coordinates. $I = {\rm R} I_{body} {\rm R}^\top$
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