Mathematics of Physics and Engineering

Uniform Rotation of Frames 55

Newton's Second Law implies mr = F. By (2.1.24),

mf\ = F - 2mu> xfi- mu> x (u> x n). (2.1.30)

The force F is the sum of the Earth's gravitational force Fa =
— {Km/\r\\^2 )ri and the net "propulsive" force Fp(t) that ensures the
motion of the point. If we prescribe the trajectory r(t) of m in the frame
0\, then the force Fp necessary to produce this trajectory is given by

FP(t) = mri + 2mu>u x ri + rnu x(wx n) — Fa- (2.1.31)

Conversely, if the force Fp — Fp(t) is specified, then the resulting trajec-
tory is determined by solving (2.1.30) with the corresponding initial condi-
tions ri(0), ri(0) and with F = FG + FP.
Note that (2.1.30) can be written as

mri = F — m acor — mac = FQ + Fp + Fcor + Fc.

Both forces FQ and Fc act in the meridian plane (NOP). Indeed, by the
Law of Universal Gravitation, the gravitational force FQ acts along the
line OP, where P is the current location of the point. The centrifugal force
Fc = mijj x (u> x n) acts in the direction of the vector b, as follows from
the properties of the cross product; see Figure 2.1.5.
To analyze the effects of the Coriolis force Fcor = —macor = -2mu x
fi on the motion of the point mass, we again assume that the point is in
the Northern Hemisphere and moves north along a meridian with constant
angular speed. According to (2.1.26), the force Fcor is perpendicular to
the meridian plane and is acting in the eastward direction. The magnitude
of the force is proportional to sin#, with the angle 6 measured from the
equator; see Figure 2.1.5. In particular, the force is the strongest on the
North pole, and the force is zero on the equator. By (2.1.25) and (2.1.31), to
maintain the motion along a meridian, the force Fp must have a westward
component to balance Fcor. Thus, to move due North, the mass must be
subject to a propulsive force Fp having a westward component. The other
component of Fp is in the meridian plane.

The following exercise analyzes the Coriolis force when the motion is
parallel to the equator.

EXERCISE 2.1.9. Let ii, be the northward vector along the axis through
the North and South poles. We assume that the Earth rotates around this
axis, and denote by u> k the corresponding rotation vector. Let O be the
fixed inertial frame and 0\, the frame rotating with the Earth; the origins