Simple dynamics equation

zoom1

Didn't get this equation which is written for the pendulum part of the cart.
Must be easy but couldn't get it.

AlephZero

It is just "force = mass x acceleration" in the x direction, for the center of mass of the rod. (The length of rod isn't shown on your diagram, but if the equation is correct it is 2l, not l).

The ##\ddot\theta## term is from the tangential acceleration of the rod.

The ##\dot\theta^2## term is the from the radial (centripetal) accleration.

zoom1

It's a little clearer now, but didn't understand the centripetal force component.

Isn't it

AlephZero

Yes.

You also know ##v = \omega r## and ##\omega = \dot\theta##

The centripetal force is radial (along the length of the rod), but the equation is for the component of the force in the X direction.