Mathematics of Physics and Engineering

50 Kinematics and Dynamics of a Point Mass

the design of water-wheels. More familiar effects of the Coriolis force, such
as rotation of the swing plane of the Poucault pendulum and the special
directions of atmospheric winds, were discovered in the 1850s and will be
discussed in the next section.
Recall that in our example r\ = 0. From (2.1.16) we conclude that
F + Fc + Fcor = 0. The real (as opposite to inertial) force F must balance
the effects of the inertial forces to ensure the required motion of the object in
the rotating frame. For a passenger sliding outward with constant velocity
VQ r in a turning car, this real force is the reaction of the seat in the form
of friction and forward pressure of the back of the seat..

EXERCISE 2.1.3? Find the vector function describing the trajectory ofm in

the O frame. What is the shape of this trajectory? Verify your conclusion

using a computer algebra system.

EXERCISE 2.1.4.A Suppose a point mass m is fixed at a point P in the O
frame, that is, m remains at P in the O frame for all times. Find the vector
function describing the trajectory ofm in the 0\ frame. What is the shape
of this trajectory? Verify your conclusion using a computer algebra system.
Hint. This is the path of m relative to the car seen by a passenger riding in the
car. Show that O is fixed in Ox.

Coming back to Figure 2.1.3, note that the coordinate vectors in the
frame 0\ spin around 0\ with constant angular speed WQ, while the origin
0\ rotates around O with the same angular speed WQ. This observation
leads to further generalization by allowing different speeds of spinning and
rotation.
EXERCISE 2.I.5.'^4 Suppose the origin 0% rotates around the point O with
angular speed U>Q, while the coordinate vectors (r, 9) spin around 0\ with
constant angular speed 2WQ. Suppose a point mass is fixed at a point P in
the 0 frame. Find the vector function describing the trajectory of m in
the 0\ frame. What is the shape of this trajectory? Verify your conclusion
using a computer algebra system.

Our motivational example with the car illustrated some of the main
effects that arise in rotating frames. The example was two-dimensional in
nature, and now we move on to uniform rotations in space. There are many
different ways to describe rotations in R^3. We present an approach using
vectors and linear algebra.
We start with a simple problem. Consider a point P moving around a