Two springs connected by a spring

[compliant]

Homework Statement

Two masses, m1 and m2, are connected to each other by a spring with a spring constant k. The system moves freely on a horizontal frictionless plane. Find the natural frequency of oscillation.

Homework Equations

F = -kx
F = ma

The Attempt at a Solution

Let m1 be the mass on the left, and let m2 be the mass on the right.
Let positive be in the direction of m2.

Let x1 be the displacement of m1 onto the spring, and x2 the displacement of m2 onto the spring.

$$-m_{1}\ddot{x_{1}}=k{x_{1}}$$

$$m_{2}\ddot{x_{2}}=-k{x_{2}}$$

$$m_{2}\ddot{x_{2}}-m_{1}\ddot{x_{1}}=-k(x_{2}-x_{1})$$

$$m_{2}\ddot{x_{2}}-m_{1}\ddot{x_{1}}=-k(\Delta{x})$$

I tried manipulating this into an ODE, but got nowhere.

 

[Inferior89]

What if you define a new variable, say, $$q = x_2 - x_1$$?

 

[compliant]

Well, then I would have

$$m_{2}\ddot{x_{2}}-m_{1}\ddot{x_{1}} = -kq$$

...which, from what I see, doesn't do a whole lot because I still can't factor the left side to do anything.

 

[Inferior89]

Well, firstly, the force on the first mass depends on the location of both itself AND the other mass. Think for example if one mass is very far away. Then there would be a big force on the second one.

So your equations of motion are incorrect (total forces should add up to zero).

 

[rwisz]

I can provide a different approach to solving this problem. First, we can consider the system as a single mass with an effective mass, meff, given by the sum of the two masses connected by the spring: meff = m1 + m2. This is because the two masses are connected and will move together as one unit.

Next, using the equation F = -kx, we can write the equation of motion for the system as:

-meff * d^2x/dt^2 = kx

This is a simple harmonic oscillator equation with a natural frequency, ω0, given by:

ω0 = √(k/meff)

Substituting the value of meff, we get:

ω0 = √(k/(m1 + m2))

So, the natural frequency of oscillation of the system is directly dependent on the spring constant and the total mass of the system. This means that if we increase the spring constant or the total mass, the natural frequency will increase as well. Conversely, decreasing the spring constant or the total mass will result in a decrease in the natural frequency.

I hope this helps in solving the problem. As a scientist, it is important to consider different approaches and perspectives in problem-solving to find the most accurate and efficient solution.