The expression, $M_{\varphi\varphi} = m_1 r^2$, represents the angular inertia component of the generalized inertia tensor for the coupled mass system.
This component is the moment of inertia of mass $m_1$ alone because:
Definition of Generalized Inertia Tensor Component: The component $M_{\varphi\varphi}$ is the coefficient of the squared generalized angular velocity term, $\dot{\varphi}^2$, in the total kinetic energy expression $T$.
$$ T = \frac{1}{2}(m_1 + m_2)\dot{r}^2 + \frac{1}{2}(\mathbf{m_1 r^2})\dot{\varphi}^2 $$
Kinetic Energy of $m_1$ (Mass on the plane): Mass $m_1$ moves in the horizontal plane using polar coordinates ($r, \varphi$). Its kinetic energy is $T_1 = \frac{1}{2} m_1 (\dot{r}^2 + r^2 \dot{\varphi}^2)$. The term $m_1 r^2 \dot{\varphi}^2$ is the rotational kinetic energy of $m_1$ about the vertical axis passing through the hole.
Kinetic Energy of $m_2$ (Hanging mass): Mass $m_2$ moves only vertically ($z_2 = r - L$), and its velocity is purely radial ($\dot{z}_2 = \dot{r}$). Its kinetic energy is $T_2 = \frac{1}{2} m_2 \dot{r}^2$. It has no angular velocity component ($\dot{\varphi}$) since it only moves straight up and down, and therefore contributes nothing to the $\dot{\varphi}^2$ term in the total kinetic energy.
Consequently, when calculating the total kinetic energy, only $m_1$ contributes to the $\dot{\varphi}^2$ term, making the angular inertia $M_{\varphi\varphi} = m_1 r^2$. This is the standard moment of inertia for a point mass $m_1$ moving a distance $r$ from an axis of rotation.
What is the component (angular inertia) of the generalized inertia tensor-L.mp4