Rotating Reference Frame: Centrifugal and Coriolis Forces
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An observer positioned in a rotating reference frame, rotating at a constant angular
speed of ω, will observe:
F = m.
a + 2m
v x
ω
- m.
ω x (
ω x
r)
where:
F : Force (N)
m : Mass (kg)
v : Speed (m / s)
ω : Angular velocity (rad / s)
r : position vector (m)
x : vector cross product
The component m.
a indicates that the Second law of Newton still holds in an rotating
reference frame. In fact, a reference frame rotating at a constant,
angular speed is also an inertia frame, in which the Newton laws apply.
The component (2m.
v x
ω) is called the Coriolis
force (after the name of the scientist who discovered it).
The component -m.
ω x (
ω x
r)
is called the centrifugal force.
Both forces are inertia forces, as their magnitudes depend on the mass, m. The centrifugal
force has a direction that is orthogonal to the axis of rotation, and is
always directed away from this axis. This Coriolis force is orthogonal to both v
and
ω. When v=0, the Coriolis force is zero.
Explanation
In a rotating reference frame, an observer experiences a centrifugal force. One can imagine the existence of this force by realizing that in order for a mass to
make a circular motion, a centripedal force of mω2r
is needed. When the observer in the rotating reference frame holds a mass m in place,
it must, therefore, exercise
a force of the same magnitude and in the same direction
in order to hold the object in place and at rest (in the moving reference frame).
The observer experiences this as a force directed away from the axis of rotation. This force is called the centrifugal force.
Note that we can only speak of a centrifugal force in a rotating reference frame.
The centripedal and centrifugal forces are not reaction forces in the sense of the
Third Law of Newton, although they have the same magnitude and are opposite in direction.
They are forces that exist in different reference frames.
The Coriolis force can be explained as follows. The observer throws a stone away
at speed v in the direction of the axis of rotation (see figure x). Because
the observer is rotating, and the rock has no real forces working on it, it will
not reach point A but point B, because the systems rotate at angular speed ω. So, the
observer sees the rock bending away (in the opposite direction of rotation). Since the motion of the stone is
curved and hence accelerated, the
observer experiences a force which is orthogonal to the speed v of the object. This
is called Coriolis force, and will only exist if v>0. The Coriolis force also
only exists in a rotating reference frame.
Rotating reference frames play an important role because they simplify reasoning and
calculations
when looking at rotating systems.
Example
We look again at the example of the rotor that is turning at 500 rpm. We make
the assumption that the blades do not have any mass. Instead, all the mass is concentrated
at the end of the blades, at one point. This point of mass equals 5 kg.
How great
is the centrifugal force working on the mass?
Fcf = - m. ω x (ω x r) = -5 . (500 2π / 60)
^{2} . 5 = -68,469 N
Note that the centrifugal forces in a rotor system are highly significant!