Rigid rod moving in potential field

In summary, at each point in the two-dimensional domain, the rod experiences a force -∇P, where P is the potential. The force is an attraction of the rod towards lower potentials.
  • #1
kirzoaktrt
4
0
Hi,

I'd like to write a program that "simulates" a rigid rod of length L moving in a potential field. The problem is in two dimensions. Friction is assumed to be negligible. The potential field and its gradient is known at every point of the two-dimensional domain.

The "mass" of the rod and the magnitude of the potential gradient can be arbitrarily set. The rod has uniform density.

I understand that the force acting upon a particle in a potential field is -∇P, where P is the potential. This is an attraction of the particle towards lower potentials. What is the force acting on the rod?

Since this is a computer simulation, time and space are discretized. The simulation would proceed in discrete timesteps Δt. How do I compute the position of the rod in each timestep? How would you generally approach this problem?

Your help is greatly appreciated.
 
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  • #2
How does the rod couple to the field?

For example, in an electric potential field, the units are V = J/C, i.e. energy per unit charge. What is the equivalent "charge" of the rod?
 
  • #3
The force is -∇P at every point in the two-dimensional domain, while

P=qψ.

In this context, P is the potential energy, which is known and ψ is the potential. q would be a constant, which could be the mass, charge or whatever. I can set it to be anything, but it is a real constant. I set Δt and q such that the problem can be handled numerically, but they don't have any significance besides this.

Answering your question, it does not matter whether you treat this as a gravity or electric problem. Units are unimportant.
 
  • #4
All you need is to figure the total force and total torque acting on the rod. These would be given by integrals, but in your computer program they will be calculated as sums. Break down the rod into N parts and for each part calculate the force as if it was a particle. calculate the torque using the center of mass as pivot. Add all the forces to find the total force and add all the torques to find the total torque. Use those to figure out the acceleration and angular acceleration of the rod. use those to update one time step to find what's the updated values for angular velocity and velocity. Use those to update the position of the center of mass and orientation (angle) of the rod. Rinse and repeat.
 
  • #5
Thanks!

Two questions.

1, Without losing generality, can the rod be treated as two particles connected by a massless rod?
2, I assume that it does not matter whether I rotate or translate the rod first at each iteration, yes?
 
  • #6
Answers:

1, No
2, Yes

Many thanks for the replies.
 

1. What is a rigid rod moving in a potential field?

A rigid rod moving in a potential field refers to a physical system in which a straight rod of fixed length and mass is able to move freely in a region where there is a conservative force acting on it. The motion of the rod is restricted to translations and rotations, and the potential field represents the energy stored in the system due to the conservative force.

2. What is the equation of motion for a rigid rod moving in a potential field?

The equation of motion for a rigid rod moving in a potential field is given by Newton's second law of motion, which states that the sum of all forces acting on the rod is equal to its mass multiplied by its acceleration. This can be written as F = ma, where F is the net force on the rod, m is its mass, and a is its acceleration.

3. How does the potential energy affect the motion of a rigid rod?

The potential energy in a potential field affects the motion of a rigid rod by determining the magnitude and direction of the conservative force acting on the rod. As the rod moves in the potential field, the conservative force will cause it to accelerate and change its velocity and position. The potential energy also determines the stability of equilibrium points for the rod's motion.

4. What is the difference between a conservative and non-conservative force in a potential field?

A conservative force in a potential field is one that does not depend on the path taken by the rigid rod, but only on the starting and ending points of its motion. This means that the work done by the force on the rod is independent of the path it takes. On the other hand, a non-conservative force is one that does depend on the path and can result in a change in the total energy of the system.

5. How can the motion of a rigid rod in a potential field be analyzed?

The motion of a rigid rod in a potential field can be analyzed using principles of classical mechanics, such as Newton's laws of motion and conservation of energy. The equation of motion can be solved to determine the position, velocity, and acceleration of the rod at any given time. Additionally, graphical and numerical methods can be used to visualize and analyze the motion of the rod in the potential field.

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