2 Masses Connected by a Spring

[orange300]

I was just wondering about this problem (this is not a homework problem or anything like that), and I'm not sure that I know how to solve it:

Homework Statement

2 masses, M1 and M2, are connected in the middle by a spring (spring constant k). How would I describe the force exerted by the spring on each mass so that the force is zero at some pre-determined equilibrium, outward when the spring is compressed, and inward when the spring is stretched?

Homework Equations

F=-kx (Hooke's Law)

The Attempt at a Solution

I have thought about putting the center (zero) of my coordinate system at the center of the spring, but I am not sure if this is correct. In other words, the displacement x is defined by the distance of either end of the spring from the center of the spring (at x=0). I set the equilibrium displacement at x=-3 and x=3 (3 units to the left of the center and 3 units to the right, respectively). Thus, when the spring is compressed (x>-3 on the left side of the center position, and x<3 on the right side), the spring exerts an outward force on each mass (left (negative) on M1, right (positive) on M2), so the Hooke's Law expression for the force on M1 would be F1=-k(x+3) and the expression for the force on M2 would be F2=-k(x-3). Similarly, the spring would exert an inward force when stretched (x<-3 on the left side, x>3 on the right). When the ends of the spring are at x=-3 on the left and x=3 on the right, the force exerted by the spring on either mass is zero

Ultimately, I would like to apply a force to one of the masses (a leftward force to the left-most mass, M1), add friction to the surface upon which the masses slide, and describe the equations of motion for the system, but I would like to be sure if I'm on the right track. I suppose my biggest hang-up involves the assumption that each end of the spring (on either side of x=0) is the same, which seems like an oversimplification, and the fact that the forces exerted by the spring on each mass do not balance.

I'm sure it is just something simple that I'm overlooking, but any help that could be offered would be appreciated.

 

[Shooting Star]

Hello, welcome to PF.

The centre of mass of the system experiences zero force. It would be convenient to take that as the origin. All the other points on the spring experiences tensional or compressional force. After simplification, you'll end up with something called a reduced mass, where you have to solve a single differential equation, instead of two DEs for two masses.

I suppose my biggest hang-up involves the assumption that each end of the spring (on either side of x=0) is the same, which seems like an oversimplification, and the fact that the forces exerted by the spring on each mass do not balance.

You are right. This is a very well discussed and elementary problem. It is heartening to note that you have already found the problems in the way you have tried to formulate it. Keep it up. Read up on this. For example, http://farside.ph.utexas.edu/teaching/336k/lectures/node63.html and http://math.fullerton.edu/mathews/n2003/SpringMassMod.html .

 

[nlightner]

I would first clarify that this is a classic example of a simple harmonic oscillator, where the forces are balanced at equilibrium and the motion follows a sinusoidal pattern. To describe the force exerted by the spring on each mass, we can use Hooke's Law, which states that the force is directly proportional to the displacement from equilibrium and in the opposite direction. In this case, the displacement is measured from the center of the spring, as you have correctly identified.

To ensure that the forces are balanced at equilibrium, we can set the equilibrium position at x=0 and the equilibrium force to be zero. This means that at this position, the force exerted by the spring on each mass is zero. As the spring is compressed (x<0), the force will be in the positive direction, while when the spring is stretched (x>0), the force will be in the negative direction. This is in line with your description of the problem.

To apply a force to one of the masses and add friction, we can use Newton's Second Law to describe the equations of motion for the system. The force applied to the mass will be balanced by the force exerted by the spring, and the friction force will act in the opposite direction of motion. This will result in a damped harmonic motion, where the amplitude of the oscillation decreases over time.

In summary, your approach to solving this problem is on the right track. Just make sure to clarify the assumptions and use the appropriate equations to describe the motion of the system.