Solve Coupled Oscillation Homework Problem

[Eric_meyers]

Homework Statement

"A particle of mass m is attached to a rigid support by means of a spring of spring constant k. At equilibrium, the spring hangs vertically downward. An identical oscillator is added to this system, the spring of the former being attached to the mass of the latter. Calculate the normal frequencies for one-dimensional vertical oscillations, and describe the associated normal modes."

The Attempt at a Solution

So I drew the picture as a spring from a ceiling with a mass at the end of it coupled to another spring with a mass hanging down from that second spring.

I define x1 to be the displacement of the first block (nearest to ceiling) in the downward direction

I define x2 to be the displacement of the second block (farthest down) in the downward direction.

m x1'' = -kx1 - k(x1 + x2) + mg

m x2'' = -k(x2-x1) + mg

x1'' = -2w^2x1 - w x2 + g

x2'' = -w (x2 - x1) + g

Solving these equations by setting them up as a matrix...

( -2w^2, -w ) (x1) = (g)
( w , -w ) (x2) = (g)

I don't think I set it up right because I can't really solve this...

 

[lippyka]

I think the normal frequencies should be w = sqrt(k/m) but I'm not sure how to get there.Can someone help me out?

 

[tomcue]

I would first clarify the problem statement and make sure all the variables and equations are properly defined. For example, it is not clear if the masses are initially at rest or if they are given an initial displacement before starting the oscillations. It is also important to specify the units used for the variables (e.g. is x1 in meters or centimeters?).

Once the problem is properly defined, I would approach it using the principles of coupled oscillations and linear algebra. I would start by writing the equations of motion for each mass, using the spring constants and masses to form a matrix equation as shown in the attempt at a solution. However, I would also include the initial conditions (e.g. initial displacements and velocities) to fully solve the system.

Next, I would use techniques such as eigenvalue analysis to solve for the normal frequencies and normal modes of the system. This would involve finding the eigenvalues and eigenvectors of the matrix equation, which would correspond to the normal frequencies and normal modes, respectively.

Finally, I would interpret the results and describe the physical meaning of the normal modes. In this case, there would be two normal modes, one in which both masses oscillate in phase (i.e. moving in the same direction at the same time) and one in which they oscillate out of phase (i.e. moving in opposite directions at the same time). The normal frequencies would depend on the masses and spring constants, with the in-phase mode having a lower frequency than the out-of-phase mode.

In conclusion, as a scientist, I would approach this problem by carefully defining the variables and equations, using principles of coupled oscillations and linear algebra to solve the system, and interpreting the results in a physical context.