Chapter 6

Mechanical point

Kinematics : changing the frame of reference

RemarqueRelativity on motion

Trajectories change when the reference frame does.

The movement of a point is not the same when it is studied in another frame of reference : it is relative to the frame.

Problem :

We know the movement of a material point in a reference frame (R).

How is a point moving when it is studied in a reference frame (R') which is moving relative to (R) ?

For instance :

What trajectory does the moon describe relatively to the sun ?

What trajectory does the moon describe relatively to the sun ?

SimulationA JAVA animation by Jean-Jacques Rousseau (Université du Mans)

  • Motion of the Moon : click HERE

FondamentalPure translation of reference frames

A rigid body is in translation if all of its points have at all times the same velocity vector.

For exemple, a Ferris wheel is in circular translation relative to its axis, and the Earth is in circular translation relative to the sun.

The next figure explicits the chosen notations.

The reference frame, which we describe by the Cartesian coordinates is moving in translation relatively to the frame, at the speed of the point in this frame.

We can also note that the basis vectors of ( ) keep fixed directions.

Pure translation of reference frames

Composition of velocity :

We can decompose the vector :

Let us calculate the velocity in :

The index means that velocity is calculated in the referential frame, in which the basis vectors are constant.

Most of the time, the basis vectors of evolve through time.

When the movement is a translation, these vectors are constant, thus we can simplify the relation :

Finally : (composition of velocity)

Composition of acceleration :

In the same way :

We can see that for a uniform translation :

FondamentalPure rotation of reference frames

The frame of reference is in rotation around the axis of the "fixed" frame (see figure) :

We write :

We write :

is the angular velocity vector of the frame relative to .

We will use : (see the kinematics courses on the derivatives of polar vectors)

Pure rotation of reference frames

Composition of velocity :

Let us express the velocity vector :

We find :

We know :

From where we can derive the composition of velocity :

Sidenote : temporal derivative and change of reference frame

We can generalize this formula for any vector :

This is the relation we will use to determine the composition of acceleration vectors.

Composition of acceleration :

Let us use the former relation :

Then :

From which we extract :

And finally : (Composition of acceleration)

Composition of acceleration

The Coriolis acceleration is equal to zero if the point is at rest in .

For a uniform rotation (ω= cste) :

We can evaluate the term :

A useful mathematic reminder :

If we apply this :

Let H be the orthogonal projection of M on the axis :

We can see :

Finally :

We find the acceleration of a uniform circular motion.

SimulationJAVA animations by Jean-Jacques Rousseau (Université du Mans)

  • Mouvements relatifs (oscillateur harmonique) : click HERE

  • Mouvements relatifs (manège) : click HERE

  • Mouvements relatifs et manège (bis) : click HERE

  • Mouvements relatifs (grande roue) : click HERE

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Kinematics in a frame of reference
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Phase portrait