# Equation of motion for a simple mechanical system

• I
• davidwinth
In summary, the system has the following properties:The rod has a length of ##L## and a mass of ##m_b##.The disk has a radius of ##R## and a mass of ##m_d##.The collar is free to slide without friction on a vertical and rigid pole.The disk rotates without slipping on the floor.The translational and rotational velocities of the center of mass of the collar are strictly vertical.The total kinetic energy for the system is the sum of all these contributions:$$\frac{1}{2}m_{c}(Lcos(\theta)\dot{\theta})^2 + davidwinth TL;DR Summary I want to find an equation of motion for the following system through the use of the Lagrangian approach. The system is shown below. It consists of a rod of length ##L## and mass ##m_b## connecting a disk of radius ##R## and mass ##m_d## to a collar of mass ##m_c## which is in turn free to slide without friction on a vertical and rigid pole. The disk rolls without slipping on the floor. The ends of the rod are attached such that they rotate without friction. I choose the angle ##\theta## between the horizontal and the rod as the generalized coordinate.The velocity of the center of mass of the collar is strictly vertical, and is given by: ##V{_c} = Lcos(\theta)\dot{\theta}##. The velocity of the center of mass of the disk is strictly horizontal, and is given by ##V{_d}=-Lsin(\theta)\dot{\theta}##. The translational velocity of the center of mass of the rod is obtained from its components as: ##V_{b} = \sqrt{\dot{x}_{b}^2 + \dot{y}_{b}^2} = \frac{L\dot{\theta}}{2}##. The rotational velocity of the disk about its center of mass is given as: ##\omega_d = \frac{-Lsin(\theta)\dot{\theta}}{R}##. The rotational velocity of the rod about its center of mass is given as: ##\omega_b = \dot{\theta}##. Thus, the total kinetic energy for this system is the sum of all these contributions:$$\frac{1}{2}m_{c}(Lcos(\theta)\dot{\theta})^2 + \frac{1}{2}m_{b}\left(\frac{L\dot{\theta}}{2}\right)^2 + \frac{1}{2}m_{d}\left(Lsin(\theta)\dot{\theta}\right)^2 + \frac{1}{2}I_{d}\left(\frac{Lsin(\theta)\dot{\theta}}{R}\right)^2 + \frac{1}{2}I_{b}(\dot{\theta})^2

Where the ##I_d## and ##I_b## are the moment of inertia about the centroid of the disk and rod, respectively.

My doubt is whether I have correctly accounted for the motion of the rod by decomposing it this way. Is there something I have neglected in considering the motion of the rod?

vanhees71
davidwinth said:
My doubt is whether I have correctly accounted for the motion of the rod by decomposing it this way. Is there something I have neglected in considering the motion of the rod?
So far, everything looks good to me.

davidwinth
Thank you!

## 1. What is the equation of motion for a simple mechanical system?

The equation of motion for a simple mechanical system is a mathematical representation of the relationship between the position, velocity, and acceleration of an object. It is typically written as F = ma, where F represents the net force acting on the object, m is the mass of the object, and a is the acceleration of the object.

## 2. How is the equation of motion derived?

The equation of motion is derived from Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration. By rearranging this equation, we can get the familiar F = ma form.

## 3. What are the units of the equation of motion?

The units of the equation of motion depend on the units used for force, mass, and acceleration. In the SI system, the units are kilograms for mass, meters per second squared for acceleration, and newtons for force.

## 4. Can the equation of motion be used for any type of motion?

Yes, the equation of motion can be used for any type of motion as long as the system is simple and the net force acting on the object can be determined. It is commonly used in classical mechanics to analyze the motion of objects in a variety of situations.

## 5. Are there any limitations to the equation of motion?

While the equation of motion is a useful tool for analyzing simple mechanical systems, it does have some limitations. It assumes that the system is in a vacuum and that all forces are acting in one direction. It also does not take into account factors such as air resistance or friction, which can affect the motion of an object in the real world.

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