Page 63 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Tidal Energy Chapter | 3 55

3.6 CORIOLIS
The Earth rotates around its own axis from west to east. Therefore, a reference
frame attached to a fixed position on the Earth rotates around the Earth axis.
This leads to a complication when we try to apply Newton’s law of motion
on a coordinate system that is attached to the Earth, specially when studying
large-scale ocean circulation and tidal dynamics. An object (or here, our
reference frame), which rotates around an axis, has acceleration (i.e. centripetal
acceleration), because its velocity is changing with time and Newton’s law of
motion is not valid in a frame, which has an acceleration. Whilst the speed
(scaler) of an object rotating around an axis may be constant, its velocity (vector)
changes due to the change in direction. To further clarify this concept, consider
an object on the surface of the Earth which rotates around the Earth with an
angular velocity of Ω (Fig. 3.10). The speed of the object will be u = R E Ω,
where R E is the radius of the Earth. The velocity of the object with respect to a
nonrotating frame at the centre of the Earth is given by
u =−u sin θ ˆ i + u cos θ ˆ j = R E Ω[− sin θ ˆ i + cos θ ˆ j] (3.9)
Therefore, we can calculate the acceleration by taking the derivative of velocity
as follows
d u dθ dθ dθ
a = = R E Ω − cos θ ˆ i − sin θ ˆ j =−R E Ω [cos θ ˆ i+sin θ ˆ j] (3.10)
dt dt dt dt
dθ
Because Ω = , then
dt
2
a =−R E Ω [cos θ ˆ i + sin θ ˆ j] (3.11)

FIG. 3.10 Centrifugal acceleration for a rotating body.