# Coriolis and Centrifugal forces - explanation

Describe the Coriolis and Centrifugal force and how to derive them. Give examples of where their effects can be seen.

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CORIOLIS & CENTRIFUGAL FORCES

The Coriolis and Centrifugal forces are fictitious forces arising from considering motion in an accelerating frame of reference. They are proportional to mass and are given by: -2m(wÃ—v) and -mwÃ— (wÃ—r) respectively, where w is the angular frequency, m the mass, v the velocity, r the radius and x denotes the cross product.

They can be derived from the following argument. Consider a vector r0 in a frame S0 given by xex+yey+zez=r and a frame S rotating at an angular frequency of w. In S the actual velocity and acceleration are obtained by differentiating r0 with respect to t, giving v0= v + wÃ—r and a0=a +2m(wÃ—v) + mwÃ— (wÃ—r) respectively, where v is the apparent velocity and a the apparent acceleration in the rotating frame. Alternatively it can be obtained by considering an operator approach where [d/dt]s0 = [d/dt]s +wÃ—A. If the angular frequency also depends on time another term arises: -m((dw/dt)Ã—r), called the Euler force.

Examples of the effects of the centrifugal effect are the Earth's equatorial bulge. The radius to the equator is roughly 43km (or 1/300th) bigger than to the poles because of the centrifugal force which points outwards from the equator. Another example is a centrifuge which separates particles of different mass by rapidly rotating the mixture of particles and fluid. The heavier particles move to the outside because they feel a larger centrifugal force in their frame of reference. In the frame of reference of the centrifuge the outward force is described in terms of the centripetal acceleration.

The Coriolis force always points to the right in the Northern Hemisphere irrespective of what direction you are travelling in. Its effects can be seen on objects falling on Earth despite it being a very small effect, equivalent to around 8mm sideways over a 50m drop. The Foucault pendulum is another way of observing the effects of the Coriolis force. A large pendulum which is set in motion with the smallest of sideways motion, usually by burning through a thin thread holding the bob in place, precesses at x*sin(l) where x is the angular frequency of the Earth's rotation and l the latitude. This is because of the Coriolis force acting to the right on each swing and hence slowly pushing it round. At the North and South pole it takes 24 hours to precess 360Â° and at the equator it doesn't precess at all, due to the result of the cross-product term. Coriolis force also has an effect on weather systems, although here there are many other influences which may swap this small effect.

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