17.6. CORIOLIS FORCE AND CENTRIPETAL FORCE 363

i

kj

Denote the first as i(t), the second as j (t) , and the third as k (t). If you are standing onthe earth you will consider these vectors as fixed, but of course they are not. As the earthturns, they change direction and so each is in reality a function of t. What is the descriptionof the angular velocity vector in this situation?

Let i∗,j∗,k∗, be the usual basis vectors fixed in space with k∗ pointing in the directionof the north pole from the center of the earth and let i(t) ,j (t) ,k (t) be the unit vectorsdescribed earlier with i(t) pointing South, j (t) pointing East, and k (t) pointing awayfrom the center of the earth at some point of the rotating earth’s surface p(t). (This meansthat the components of p(t) are constant with respect to the vectors fixed with the earth. )Letting R(t) be the position vector of the point p(t) , from the center of the earth, observethat this is a typical vector having coordinates constant with respect to i(t) ,j (t) ,k (t) .Also, since the earth rotates from West to East and the speed of a point on the surface ofthe earth relative to an observer fixed in space is ω |R|sinφ where ω is the angular speedof the earth about an axis through the poles and φ is the polar angle measured from thepositive z axis down as in spherical coordinates. It follows from the geometric definition ofthe cross product that

R′ (t) = ωk∗×R(t)

Therefore, the vector of Theorem 17.4.2 is Ω(t) = ωk∗ because it acts like it should forvectors having components constant with respect to the vectors fixed with the earth. Asmentioned, you could let θ ,ρ,φ each be a function of t and use the formula above alongwith the chain rule to verify analytically that the angular velocity vector is what is claimedabove. That is, you would have θ (t) = ωt and the other spherical coordinates constant.See Problem 12 on Page 369 below for a more analytical explanation.

17.6 Coriolis Force and Centripetal ForceLet p(t) be a point which has constant components relative to the moving coordinate sys-tem described above {i(t) ,j (t) ,k (t)}. For example, it could be a single point on therotating earth or more generally simply a generic moving coordinate system. Let i∗,j∗,k∗

be a typical rectangular coordinate system fixed in space and let R(t) be the position vectorof p(t) from the origin fixed in space. In the case of the earth, think of the origin as the cen-ter of the earth. Thus the components of R(t) with respect to the moving coordinate systemare constants. A general observation is this. If w (t)=w1 (t)i(t)+w2 (t)j (t)+w3 (t)k (t) ,let w′

B (t) bew′

B (t) = w′1 (t)i(t)+w′

2 (t)j (t)+w′3 (t)k (t)

A dot will indicate the total derivative. Thus

ẇ (t) ≡ w′1 (t)i(t)+w′

2 (t)j (t)+w′3 (t)k (t)

+w1 (t)i′ (t)+w2 (t)j ′ (t)+w3 (t)k′ (t)

≡w′B (t)+Ω(t)×w (t)