# What is the Coriolis force

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

Coriolis force is a non-physical force, appearing, like centrifugal force, only in rotating frames of reference.

It is an inertial force, like centrifugal force and gravity, meaning that it affects all matter, proportionately to its mass (inertia), but independently of any other characteristic (such as charge).

It is perpendicular to the velocity of each body.

Equations

Coriolis force is minus mass times twice the cross-product of angular momentum of the frame, and the velocity of the object relative to the frame:

$$-2m\,\mathbf{\Omega} \times \mathbf{v}_{rel}$$

(By comparison, centrifugal force depends on position rather than velocity, and is $m\,\mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r})\,=\,m\,\mathbf{\Omega} \times \mathbf{v}_{rot}$ where $\mathbf{v}_{rot}$ is the velocity of rotation.

So centrifugal force is usually much larger than Coriolis force:

$$\frac{|centrifugal|}{|Coriolis|}\,=\,\frac{v_{rot}}{2v_{rel}}$$

Extended explanation

In a car:

Coriolis force depends on speed relative to the frame, and, for example, it is zero on a car moving uniformly in a circle and being observed in the frame of the driver of the car.

But that is almost the only circumstance in which the driver can ignore it.

In particular, it is twice the centrifugal force (and opposite to it) for stationary objects (like a house) being observed in the frame of the driver of that car:

The house has tangential velocity $-\,\Omega\,r$, and so experiences:
centrifugal force $m\,\Omega^2\,r$ outward;
and Coriolis force $2m\,\Omega^2\,r$ inward;
net force: $m\,\Omega^2\,r$ inward, forcing the house to move in a circle round the driver!

Weather:

Because of the rotation of the Earth, Coriolis force tends to make the atmosphere rotate in circles proportional to windspeed and to the sine of the latitude.

It also tends to produce circular currents in the oceans. These have a much smaller size, since waterspeed is much slower than windspeed.

Euler force:

Centrifugal force and Coriolis force appear in all rotating frames.

In non-uniformly rotating frames, a third non-physical force, the Euler force, appears:

$$-m\,\frac{d\mathbf{\Omega}}{dt} \times \mathbf{r}$$

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