# Two springs connected by a spring

• compliant
In summary, to find the natural frequency of oscillation for a system of two masses connected by a spring on a frictionless plane, the equations of motion must be correctly set up. The equations given in the conversation are incorrect and cannot be manipulated into an ODE. Instead, a new variable, q, can be defined to properly account for the forces on both masses, leading to the correct equation of motion.

## 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.

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.

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

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.

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).

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.

## What is a system of two springs connected by a spring?

A system of two springs connected by a spring is a physical arrangement where two individual springs are connected to each other by a third spring. This results in a complex spring system that can exhibit unique behaviors and properties.

## How does the stiffness of the connecting spring affect the overall stiffness of the system?

The stiffness of the connecting spring plays a significant role in determining the overall stiffness of the system. If the connecting spring is stiffer than the individual springs, the overall stiffness will be higher. Conversely, if the connecting spring is less stiff, the overall stiffness will be lower.

## What happens to the frequency of oscillation when two springs are connected by a spring?

The frequency of oscillation in a system of two springs connected by a spring depends on the stiffness of the individual springs and the connecting spring. In general, the frequency will be higher than that of a single spring, but the exact value will depend on the specific parameters of the system.

## Can a system of two springs connected by a spring exhibit resonance?

Yes, a system of two springs connected by a spring can exhibit resonance. This occurs when the frequency of the external driving force matches the natural frequency of the system, resulting in a large amplitude of oscillation. The presence of the connecting spring can alter the natural frequency of the system and potentially lead to different resonance behaviors.

## What are some real-life applications of a system of two springs connected by a spring?

A system of two springs connected by a spring can be found in various mechanical systems, such as car suspensions, shock absorbers, and even trampolines. It is also commonly used in engineering and physics experiments to study the effects of different spring configurations on the overall behavior of a system.