❓What is the total kinetic energy of the coupled mass system in terms of velocities of the generalize

What is the total kinetic energy of the coupled mass system in terms of velocities of the generalized coordinates?

The expression for the total kinetic energy, $T = \frac{1}{2}(m_1 + m_2)\dot{r}^2 + \frac{1}{2}(m_1 r^2)\dot{\varphi}^2$, is the sum of the kinetic energies of the two masses ($T = T_1 + T_2$), after grouping the terms related to the generalized velocities ($\dot{r}$ and $\dot{\varphi}$).

The formula reveals the effective inertia of the coupled system for each type of motion:

1. Radial Motion Term ($\dot{r}^2$):

   $\frac{1}{2}\underbrace{(m_1 + m_2)}{M{rr}} \dot{r}^2$

   • This term represents the kinetic energy associated with the radial velocity ($\dot{r}$), which is the speed at which the masses move toward or away from the central hole.

   • The coefficient of $\frac{1}{2}\dot{r}^2$ is $M_{rr} = m_1 + m_2$. This is the total mass of the system.

   • This makes physical sense because both mass $m_1$ (on the plane) and mass $m_2$ (hanging vertically) move with the same radial speed $\dot{r}$.

2. Angular Motion Term ($\dot{\varphi}^2$):

   $\frac{1}{2}\underbrace{(m_1 r^2)}{M{\varphi\varphi}} \dot{\varphi}^2$

   • This term represents the kinetic energy associated with the angular velocity ($\dot{\varphi}$), which is the speed of rotation.

   • The coefficient of $\frac{1}{2}\dot{\varphi}^2$ is $M_{\varphi\varphi} = m_1 r^2$. This is the moment of inertia of mass $m_1$.

   • This makes physical sense because only mass $m_1$ rotates in the horizontal plane; mass $m_2$ hangs vertically and does not contribute to the angular rotation about the $z$-axis.

This consolidated form is crucial because the coefficients of the squared velocity terms ($\frac{1}{2}\dot{r}^2$ and $\frac{1}{2}\dot{\varphi}^2$) are directly identified as the diagonal components of the Generalized Inertia Tensor ($\mathbf{M}$).

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