So the question goes if I has a spring with spring constant $k$ and two masses attached to this spring (one on either side) what is the resonant frequency of the system in terms of $m$ and $k$?
Diagram of system:
[m]-////-[m]
Now the real problem I'm having is trying to decide what the forces are acting on the system in order to come up with my differential equation? I know that for a horizontal spring say attached to wall we can take the differential equation
$$m\frac{d^2 x}{d t^2}+kx= 0$$
And then use the equation $Asin(\omega t^2+\phi)$ as a solution and say this is true when $\omega= \sqrt{\frac{k}{m}}$
But I was thinking maybe I could just use the differential equation
$$m\frac{d^2 x}{d t^2}+2kx= 0$$ but I feel like that may be too simple? Is there something I'm missing? Any help would be appreciated! 🙂
Note: None of this system is undergoing any damping
Best Answer
Assuming the spring is "ideal" (massless) you actually have 2 masses. You can describe your problem as the motion of the center of mass, and either of the masses. And if no external force is exerted on your system, you are only left with the motion of 1 mass relative to the center of the mass of the system.
Let's say your masses are m1, m2, and the spring constant is k. Denote the spring length (at rest) d.
The distance of m1 to center of mass is d * m2 / (m1+m2). Note that if m1 is displaced by x from the center of mass, then the overall spring displacement is given by x * (m1+m2) / m2. In other words, the effective spring constant for the motion of m1 is k * (m1+m2)/m2
So that the oscillation frequency of such a system is given by:
ω = [ k * (m1+m2) / (m1*m2) ] ^ (1/2)
Taking m2 to infinity reduces the formula to ω = [ k/m1 ] ^ (1/2), which is reasonable.