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:
-
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 $\dot{r}$.
-
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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What is the total kinetic energy of the coupled mass system in terms of velocities of the generalized coordinates-L.mp4