Generalized Velocity: Lagrangian

[WWCY]

Homework Statement

[/B]
In this example, I know that I can define the horizontal contribution of kinetic energy to the ball as $\frac{1}{2}m(\dot{x} + \dot{X})^2$.

In the following example,

Mass $M_{x1}$'s horizontal contribution to KE is defined as $\frac{1}{2}m(\dot{X} - \dot{x_1})^2$. Why is this? I have a hunch that it is due to the "origin" ($X$ line) $x_1$ and $x_2$ originate from, though I can't exactly put my finger on the exact reason.

Assistance is greatly appreciated!

2. Homework Equations

The Attempt at a Solution

 

[kuruman]

It has to do with the way the generalized coordinates are defined. In the top drawing $\dot{x}$ increases to the right (although not clear from the double arrow) so the velocity relative to the ground is $\dot{X}+\dot{x}$. In the second drawing, $x_1$ increases to the left while $X$ increases to the right, so the relative velocity would be $\dot{X}-\dot{x_1}$.

 

[WWCY]

It has to do with the way the generalized coordinates are defined. In the top drawing $\dot{x}$ increases to the right (although not clear from the double arrow) so the velocity relative to the ground is $\dot{X}+\dot{x}$. In the second drawing, $x_1$ increases to the left while $X$ increases to the right, so the relative velocity would be $\dot{X}-\dot{x_1}$.

Thanks for the response,

Would I be right to say that this is also the result of vector addition? Edit: With $x_1$ being defined as positive vector

 

[kuruman]

You could say that considering that it involves defining cartesian coordinates relative to origin $O_1$ and then adding a cartesian coordinate to define origin $O_1$ relative to new origin $O_2$.