Coupled Oscillators: Masses m and 2m in 3l_0 String

In summary, the conversation discusses a problem involving two masses attached to a string and oscillating between two rigid supports. The tension in each segment of the string is related to its extension. The solution for this particular case is provided, with the initial conditions being the displacement of the masses. The question remains of how to show that the oscillations of the masses are in phase. Additionally, a friend has a clock that is having an issue with the weight and pendulum being in sync, and it is questioned whether the above equation would be helpful in fixing the issue.
  • #1
trelek2
88
0
The problem is:
A mass m and a mass 2m are attached to a light string of unstretched length [tex] 3l _{0} [/tex], so as to divide it into 3 equal segments. The string is streched between rigid supports a distance [tex] 3l \textgreater 3l _{0} [/tex] apart and the masses are free to oscillate longitudinally. The oscillations are of small enough amplitude that the string is never slack. The tension in each segment is k times the extension. The masses are initially displaced slightly in the same direction so that mass m is held at a distance [tex] \sqrt{3} -1 [/tex] further from its equilibrium position than the mass 2m. They are released simultaneously from rest.

The task is to show that they oscillate in phase and explain why.

I have found the general solution and applied the initial conditions and found the solution for this particular case to be:
[tex]{x _{1} \choose x _{2} } = {-1 - \sqrt{3} \choose 1} ( \frac{- \sqrt{3} }{2}(a+1)+ \frac{1}{2})cos \omega _{1} t+ {1- \sqrt{3} \choose 1} (a+ \frac{ \sqrt{3} }{2} (a+1)- \frac{1}{2} )cos \omega _{2}t[/tex]
where I set a to be the initial displacement of mass 2m.
Is my answer correct and how do I show that the oscillate in phase?
 
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  • #2
What are the length(s) of the two vertical string(s) between the masses and the horizintal string?

A friend of mine spent many many hours building a very tall grandfather clock, It had a long pendulum and a weight hanging on a chain to supply mechanical power. He put in in a prominent place in his living room, on a high-pile wall-to-wall carpet. He told me that he was having a problem with the clock. Whenever the weight on the chain dropped to a point where it was level with the pendulum, it slowly began swinging back and forth, in sync with the pendulum. He asked me what was wrong with the clock and how to fix it. He was an engineer. Should I show him the above equation?
 
  • #3
There are no vertical strings in this case. This just one horizontal string with two masses attached to it.
 

What are coupled oscillators?

Coupled oscillators refer to a system of two or more oscillators that are connected or interact with each other in some way. In the case of "masses m and 2m in 3l_0 string", the two masses are connected by a string and their motion is dependent on each other.

What is the equation of motion for coupled oscillators?

The equation of motion for coupled oscillators can be derived using Newton's Second Law of Motion and Hooke's Law. In the case of "masses m and 2m in 3l_0 string", the equation of motion would be: m(d^2x1/dt^2) = -kx1 + k(x2-x1) and (2m)(d^2x2/dt^2) = -2k(x2-x1), where x1 and x2 are the displacements of the two masses and k is the spring constant of the string.

What is the natural frequency of coupled oscillators?

The natural frequency of coupled oscillators can be determined by solving the equation of motion. In the case of "masses m and 2m in 3l_0 string", the natural frequency would be: ω = √(k/m) for the first mass and ω = √(2k/2m) = √(k/m) for the second mass. This means that both masses have the same natural frequency.

How do the masses behave in coupled oscillators?

The behavior of the masses in coupled oscillators depends on their initial conditions and the parameters of the system (e.g. mass, spring constant). In the case of "masses m and 2m in 3l_0 string", the two masses will oscillate in opposite directions with the same frequency and amplitude. This is known as anti-phase motion.

What are the applications of coupled oscillators in science and engineering?

Coupled oscillators have many applications in various fields. In physics, they are used to model molecular vibrations and atomic interactions. In engineering, they are used in designing and analyzing systems such as bridges, buildings, and electrical circuits. They also have applications in fields such as biology, chemistry, and economics.

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