Chapter 6

Mechanical point

Inertia force, Coriolis force

FondamentalFrom a mathematical point of view, why do we use inertia forces ?

(R) is an inertial frame of reference, (R') is a frame which as a motion with respect to (R).

Let M (masse ) be a material point, subject to « real » forces, the resultant of which we note .

Let and be the velocity and acceleration vectors of M in (R).

Let and be the velocity and acceleration vectors of M in (R').

In general : (see the lesson "Kinematics : changing the frame of reference")

The second law of motion in the inertial frame (R) gives us :

If we compose the acceleration vectors :

We note :

  •  : the inertia force

  •  : the Coriolis force

Then :

We can finally apply the second law of motion in a non-inertial reference frame, but only if we add to the « real » forces the inertia force and the Coriolis force.

ExempleElevator and apparent weight :

Here, the frame of reference (R') is in pure translation relative to the frame (R), so there is no Coriolis force.

The equation of the equilibrium in the frame of the elevator is :

If we rewrite it using the notation of the « apparent weight » ,  :

Note :

is an absolute value ( ), whereas can be positive or negative.

When the elevator goes from the ground floor to the 6th floor for instance, is positive : the apparent weight is greater than the real weight. That is what we feel when we are in an elevator that goes up.

We can also note that the apparent weight is zero when the elevator is in free fall !

Elevator and apparent weight :

Note :

If (R') is in a rectilinear uniform translation relative to (R), the inertial acceleration is zero.

Then :

The second law of motion in (R') is :

The second law of motion is thus still valid in (R').

We finally show that a frame in rectilinear uniform translation relative to an inertial frame of reference is also inertial.

FondamentalCentrifugal force

The motion of (R') relative to (R) is a rotation around a fixed axis.

The angular velocity is constant.

The inertia force is :

It's the well known centrifugal force !

Centrifugal force
Une centrifugeuse à la foire du trône

FondamentalCoriolis force

is the velocity vector of a material point in the frame (R').

The Coriolis force at that point is :

A few effects of the Coriolis force in terrestrial mechanics are reminded in the courses on non inertial frames of reference.

A video about Coriolis force :

A few examples of Coriolis force

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

  • Forces de Coriolis et centrifuge : click HERE

  • Chute libre sur un manège : click HERE

  • Régulateur de Watt (centrifuge) : click HERE

The atmosphere braking a satellite
Non-inertial frame of reference :