❓What is the kinetic energy of mass moving in the horizontal plane?

The formula $T_1 = \frac{1}{2}m_1 (\dot{r}^2 + r^2 \dot{\varphi}^2)$ is the expression for the kinetic energy ($T_1$) of mass $m_1$ as it moves in the horizontal plane, expressed in polar coordinates ($r$ and $\varphi$).

Here is a breakdown of the components:

1. General Kinetic Energy Formula: Kinetic energy is generally given by $T = \frac{1}{2}m v^2$

2. Velocity Squared in Polar Coordinates ( $v^2$ ): For a mass moving in a plane, its velocity squared ($v^2$ or $\dot{r}_1^2$) in polar coordinates is the sum of the squares of its two orthogonal velocity components: the radial velocity and the angular (tangential) velocity.

   $v^2 = \dot{r}^2 + (r\dot{\varphi})^2 = \dot{r}^2 + r^2 \dot{\varphi}^2$

   • $\dot{r}^2$ (Radial Component): This term accounts for the velocity of the mass moving radially outward or inward along the string. The term $\dot{r}$ (pronounced "r-dot") is the rate of change of the radial distance $r$.

   • $r^2 \dot{\varphi}^2$ (Angular/Tangential Component): This term accounts for the velocity of the mass moving tangentially in a circle around the hole.

     • The tangential speed is $v_{\text{tan}} = r \dot{\varphi}$.

     • $\dot{\varphi}$ (pronounced "phi-dot") is the angular speed (rate of change of the angle $\varphi$).

     • The squared tangential speed is $v_{\text{tan}}^2 = (r\dot{\varphi})^2 = r^2 \dot{\varphi}^2$.

3. Final Expression: Combining the parts gives the total kinetic energy for $m_1$:

   $T_1 = \frac{1}{2}m_1 \left( \underbrace{\dot{r}^2}{\text{radial motion}} + \underbrace{r^2 \dot{\varphi}^2}{\text{angular motion}} \right)$

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