# Mechanical point

# Coupled oscillators

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

Two masses and are attached by a spring of spring constant and of relaxed length .

They move freely on a horizontal axis (x'x) without friction.

When , the spring is relaxed, the masses and are at rest on and .

From , a horizontal constant force is exerted on .

Let and .

## Question

Find and .

### Hint

Apply the second law of Newton on the two masses.

### Solution

We apply the second law of Newton on each of the two masses, we project it on the axis (Ox) :

We sum the equations :

We integrate and take the initial values into account :

We can write the equations as :

And :

By subtracting these equations :

We note :

Then :

The solution is :

Using the initial values :

From which we deduce and , since we know their sum and their difference.