Solving 3 Body Problem in 1D: Tips for Calculating Motion of Mass C

[macrosshkli]

Homework Statement

if 3 identical objects with mass m are conneted by springs of spring constant k, moving in 1D. At t=0, the masses are at rest at their equilibrium. position. Mass A is then subjected to an external force.

A -> B -> C

Where A, B, C are the masses and -> represents the springs.

Calculate the motion of mass C.

Homework Equations

$$F(t) = F_{0}cos \omega t$$ (t > 0)

The Attempt at a Solution

The two body problem is easy to solve. (i.e. For A and B). But, I'm confused on how to incorporate mass C into the motion from A and B. Can anyone give me some tips on how to tackle this?

 

[Jhero]

I would first start by defining the problem and identifying any known variables. In this case, we have three identical masses (A, B, and C) connected by springs of spring constant k. The masses are initially at rest at their equilibrium position, and mass A is then subjected to an external force.

Next, I would draw a diagram to visualize the system and the forces acting on each mass. From the problem statement, we know that the external force on mass A is given by F(t) = F_{0}cos \omega t (t > 0), where F_{0} is the amplitude of the force and \omega is the angular frequency. We also know that the springs will exert a restorative force on each mass, given by Hooke's Law: F = -kx, where k is the spring constant and x is the displacement from equilibrium.

Now, to incorporate mass C into the motion from A and B, we can use Newton's Second Law, F = ma, to write equations of motion for each mass. Since the masses are connected, we can also use the fact that the acceleration of one mass will be equal to the acceleration of the other two masses. This will allow us to solve for the motion of mass C in terms of the motion of mass A.

To solve the equations of motion, we can use techniques such as differential equations or numerical methods. We can also use energy conservation principles to analyze the system and make predictions about its behavior.

In summary, to tackle this problem as a scientist, I would define the problem, draw a diagram, identify known variables, and use fundamental principles such as Newton's Laws and energy conservation to solve for the motion of mass C in terms of the motion of mass A. I would also use appropriate mathematical techniques to solve the equations of motion and make predictions about the system's behavior.