Coupled oscillator mass on spring question?

In summary, the conversation discussed the concept of normal modes in a coupled system of two masses connected by springs. The equations of motion for the system were derived and the term "normal mode" was defined as a mode where all components oscillate with the same frequency. The frequencies of the normal modes were not known and the time-dependence of x and y for normal and general motion was also not known. The conversation ended with a request for help in finding a general expression for the solution of a normal mode.
  • #1
coffeem
91
0

Homework Statement



An object of mass m and another of mass M = 2m are connected to 3 springs of spring constant horixontally. The displacement of the two masses are defined as x and y. When x = y = 0, the springs are unextended.


a) Write down the two coupled equations of motion.

b) Define the term normal mode.

c) Find the frequencies of the normal modes.

d) Write down the time-dependence of x and y for the case of normal modes of motion and for the case of general motion.

e) Using appropriate sketched, describe the motion of the two masses for each normal mode.


The Attempt at a Solution



a) Write down the two coupled equations of motion.

mx1(dot dot) = -kx1 - k(x1-x2)

2mx2(dot dot) = -kx2 - k(x2-x1)

b) Define the term normal mode.

Mode at which all of the components in a couples system oscillate with the same frequency.

c) Find the frequencies of the normal modes.

I don't know how to do this but believe it involves finding a determinent.

d) Write down the time-dependence of x and y for the case of normal modes of motion and for the case of general motion.

Again I do not know how to do this.

e) Using appropriate sketched, describe the motion of the two masses for each normal
mode.

? Not sure.


Any help given would be appretiated.
 
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  • #2
You said that the normal modes are motions where all parts vibrate with the same frequency. Can you write an general expression that represents a solution of this form?
 
  • #3
No. Sorry I don't know how to do this can you explain this to me? thanks
 
  • #4
This is a very straight-forward question, and you either know how to do it or you don't. If you've never seen a question like this before, just read a textbook.
 

1. What is a coupled oscillator mass on spring system?

A coupled oscillator mass on spring system is a physical system consisting of multiple masses connected by springs that are also connected to a fixed point. The masses oscillate back and forth due to the restoring force of the springs.

2. How does the mass on spring system behave when the springs are coupled?

When the springs are coupled, the masses on the spring system will exhibit a phenomenon known as coupled oscillations. This means that the masses will oscillate with a single frequency, but with varying amplitudes and phases.

3. What is the equation of motion for a coupled oscillator mass on spring system?

The equation of motion for a coupled oscillator mass on spring system can be written as m(d^2x/dt^2) + kx = -k'x' - k''x''. Here, m is the mass of the oscillating object, k and k' are the spring constants, and x, x', and x'' represent the displacements of the masses from their equilibrium positions.

4. How does the coupling constant affect the behavior of a coupled oscillator mass on spring system?

The coupling constant, represented by k'' in the equation of motion, determines the strength of the coupling between the masses. A higher coupling constant will result in a stronger interaction between the masses, leading to more pronounced coupled oscillations.

5. How is energy transferred in a coupled oscillator mass on spring system?

In a coupled oscillator mass on spring system, energy is transferred between the masses through the connecting springs. When one mass is at its maximum displacement, it will transfer its energy to the other mass, causing it to oscillate with a greater amplitude. This energy transfer continues back and forth between the masses, leading to the characteristic coupled oscillations.

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