Chapter 6

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 .

Coupled oscillators

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.

SimulationJAVA animations by de JJ.Rousseau (Université du Mans)

  • Deux oscillateurs harmoniques couplés : click HERE

  • Trois oscillateurs harmoniques couplés : click HERE

  • Oscillations d'une chaîne linéaire : click HERE

  • Oscillations d'une chaîne linéaire avec deux masses alternées : click HERE

  • Pendules élastiques superposés : click HERE

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