Environmental Engineering Reference
In-Depth Information
is out of the
plane of the page.
w
v
'
Figure 8.3
Trajectory of a ball rolled on a rotating disk.
Solution 8.2.1
The first thing to realise is that the centrifugal force always acts
radially outwards and hence it does not affect the general argument. Our attention
therefore is focused upon the Coriolis force. In case (a), for the anti-clockwise
rotation illustrated in the figure,
points out of the plane of the page. The velocity in
the rotating frame points radially outwards and hence the vector
ω
v
pushes the
particle as illustrated. In case (b), the velocity vector points in the opposite direction
and so the particle is pushed in the opposite direction. This result might at first seem
counter intuitive, especially the result of part (b). There is however a simple way
to understand what is happening. Viewed from an inertial frame, when the ball is
released from the rim of the turntable it has both a radial and tangential component
to its velocity. The tangential component is equal to ωR where R is the radius of the
turntable. Now, after it has moved inwards slightly it is at a distance smaller than
R from the centre but its tangential component of velocity is still (approximately)
equal to ωR and this speed is faster than is needed to keep the particle travelling on
a radius vector. Hence the particle moves in the direction of the rotation. Crudely
stated, it is as if the particle has been thrown in the direction of motion with a speed
equal to the speed of the rim of the turntable. The opposite is true for the case
where the particle is rolled from the centre: it never has enough tangential speed to
keep up with the rotating disk and hence it moves in the opposite direction to the
rotation.
−
ω
×
8.2.1 Motion on the Earth
Motion in the vicinity of some region on the Earth's surface is most conveniently
described by employing a system of co-ordinates fixed to the Earth. Such a system
is illustrated in Figure 8.4 and this choice of basis vectors is most convenient for
describing physics in the vicinity of a point at latitude
λ
on the Earth's surface (in
principle they can of course be used to describe any physics anywhere else in the
Universe). Looking at Figure 8.4, it should be clear that (for
π/
2
<λ<π/
2)
the basis vector
e
2
points to the North,
e
1
points East (i.e. into the plane of the
page) and
e
3
points upwards (i.e. radially outwards from the centre of the Earth).
−
