Previous:  Rotation and Angular Velocity

In the previous section, we derived a formula that relates the velocity of an object in a fixed coordinate system to the velocity of the particle is observed in a coordinate system that is rotating relative to the fixed system.  The expression is

where the primed quantities refer to the rotating coordinate system.  This expression can be rewritten in terms of the derivatives of the position vectors r and r’ of the particle:

This expression relates the time derivative of the vector r in the fixed frame to the time derivative of the vector r’ in the rotating coordinate system.

In fact, this expression can be written slightly differently, as follows:

That is, the operation of taking a time derivative in the fixed system (the left-hand side of the above equation) is equivalent to taking a time derivative plus a vector product in the rotating system (the right-hand side of the above equation).  We can therefore write the following:

The compound operator is easily expanded, and the right-hand side of the above equation is just

The second term in this expression is just

In terms of more conventional symbols for velocity and acceleration, and assuming that the angular velocity does not change with time (so that the first term in the above equation is zero) the last few equations can be combined to give

where of course the a’s represent accelerations, and the v’s represent velocities.

Now, in the fixed coordinate system, Newton’s second law takes the form

where F represents the physical force acting on a particle of mass m.  In the rotating system, using the two equations above, Newton’s second law takes the form

That is, the acceleration a’ in the rotating system is determined on the basis of the physical force F, and also two additional “forces” that arise because of the rotation of the system.  On the right-hand side of the above equation, the second term is called the “Coriolis force” and the third term is called the “centrifugal force”.

Finally, if the rotating coordinate system is also undergoing translational motion then, as shown in a previous article, an additional inertial term

Is required on the right-hand side of the above equation, where A is the translational acceleration of the rotating system.

It is important to note that these forces do not arise from any physical agency – they arise solely because of the rotation of the coordinate system.  Thus, if the angular velocity ω is reduced to zero, then both the Coriolis and the centrifugal forces vanish.  For this reason, the Coriolis and centrifugal forces are referred to as “fictitious” or “inertial” forces.

One of the most obvious rotating systems is the earth, and in the next two sections, we will take a look at some of the consequences that arise from studying dynamics in such a rotating system, in particular projectile motion.

Next:  Projectile Motion