Understanding the Pendulum Dynamics Equation for a Cart

[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 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.

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.

 

[Nickie]

I understand that equations can often be challenging to comprehend, especially if you are not familiar with the specific topic or field. In order to understand the dynamics equation for the pendulum part of a cart, it is important to have a basic understanding of pendulum motion and how it relates to the movement and stability of a cart.

A pendulum is a weight suspended from a pivot point that can swing back and forth. The motion of a pendulum is governed by several factors, including the length of the string, the mass of the weight, and the force of gravity. When a pendulum is attached to a cart, the dynamics equation takes into account the motion of the cart as well.

The dynamics equation for a pendulum on a cart takes the form of a differential equation, which describes the relationship between the position, velocity, and acceleration of the pendulum and the cart. This equation can be used to model and predict the behavior of the pendulum and cart system, such as how the cart will move when the pendulum swings.

To better understand this equation, it may be helpful to break it down into smaller components and study each part individually. Additionally, seeking out additional resources or consulting with an expert in the field may also aid in understanding the equation.

Overall, the dynamics equation for a pendulum on a cart is a fundamental tool for understanding the complex motion and stability of this system. With further study and practice, I am confident that you will be able to grasp this equation and its implications.