The (vector) position R of the CM with respect to the ground is just the mass weighted average over all the parts of the car:

 (1)

The external forces on the car are also vectors: they have X components and Y components. So, we write the sum of all the forces on the car with a bold F. Similarly, the velocity of the CM is a vector. It is the change in R over a small time, dt, divided by the time. This is written

 (2)

The d/dt notation is called a derivative. In turn, the acceleration is a small change in the velocity divided by the time:

 (3)

The d2/dt2 notation is called a second derivative and results from two derivatives in succession.

Newton's Second Law for the CM of the car is then

 (4)

where M is the total mass of all parts in the car. Simple, eh? This is a differential equation, and theoretical physics is overwhelmingly concerned with the solutions of such things. In this case, a solution is finding R given M and F. We can also simplify the writing of the equations in general by replacing time-derivative notations with dots: one dot for one time derivative and two dots for two derivatives. We get

 (5)

Now, we consider the parts of the car separately as they yaw (and pitch and roll) about the CM while remaining firmly attached to the car. Let's write all position variables measured with respect to the coordinate grid fixed in the car with overbars, so the vector position of the i-th mass in our list is ri.

However, we don't need to use vectors (in two dimensions), because in pure yawing motion about the CM of the car, the radial distance of each car part from the CM remains fixed and each part has the same yaw angle as the whole car.

Let the yaw angle of the car and its coordinate grid measured against the ground-based, inertial coordinates be . As each part is affected by forces, it moves in a yaw-arc around the CM. A small amount of yaw is written d . Each part moves perpendicularly to a line drawn from the part to the CM of the car, and the distance it moves is equal to its radial distance from the CM, ri (non-bold: a number, not a vector), times the little amount of yaw d . Divide by the little time over which the motions are measured, and you have the velocity of each car part:

 (6)

Now, it's easy to apply Newton's second law. Equate the force on the i-th part Fi, to the mass of the part times the acceleration of the part:

 (7)

We're almost done with the math, so hang in there. If we multiply equation (7) by ri on both sides, the left-hand side becomes the torque of the forces on the i-th part about the CM:

 (8)

Now, if we sum this equation up over all the parts in our list, we can drop the i subscript:

 (9)

remembering that all parts have the same . The reason for doing this is that resulting equation looks like Newton's Second Law, equation (5). If you replace  with a symbol, I, the equation is identical in form:

 (10)

Physicists like to find formal equivalences amongst equations because they can use the same mathematical techniques to solve all of them. The equivalences also hints at deeper insights into similarities in the Universe.

OK, if you haven't already guessed it,  is the polar moment of inertia. To compute it for a given car, we take all the parts in the car, measure their masses and their distances from the CM, square, multiply and add. In practice, this is very difficult. I doubt if PIMs are measured very often, but when they are, it is probably done experimentally: by subjecting the car to known torques and measuring how quickly yaw angle accumulates.

We can also see that, for a given rotational torque, the acceleration of yaw angle is inversely proportional to I. Thus, we have backed up, from first principles, our statement that cars with low PMI respond more quickly, by yawing, to transient cornering forces than do cars with large PMI. A car with a low PMI is designed so that the heavy parts - primarily the engine - are as close to the CM as possible. Moving the engine even a couple of inches closer to the CM can dramatically decrease the PMI because it varies as the square of the distance of parts from the CM. Since equation (10) is formally equivalent to Newton's Second Law, an analogous insight applies to that law. A car with low mass responds more quickly to forces with straight-line changes in motion just as a car with low PMI responds more quickly to torques with rotational changes in motion.

Why would one design a car with a high PMI? Only to get a big, powerful engine into it that might have to be placed in the front or the rear, far from the CM. So, take your pick. Choose a car with a low PMI that yaws very quickly and give up on some engine power. Or, choose a car with colossal engine and give up on some handling quickness.