Why do vehicles with a higher center of mass roll more easily considering that they have a higher moment of inertia?

It is a well known fact that cars with a higher center of mass roll over more easily, but why is this true considering that higher center of mass = higher moment of inertia?

I understand that the higher center of mass = higher torque being applied as the car turns, but this increase in torque is linearly related to the distance from the point of rotation (T = FD), with F being the force applied by the mass of the vehicle as it turns. Moment of inertia, however, is quadratic to the distance of the mass from the point of rotation (I = MR^2), so it seems that moment of inertia increases more than torque does as you move the mass further from the point of rotation, which should always be the tires.

Here's a couple of equations to further show what I am talking about

CoM = 5 meters from point of rotation

Mass = 10 Kg

Centripetal force = 15 N (I know that these aren't realistic values for a car, but let's use them for simplicity)

Moment of inertia = 10*5^2 = 250KgM^2

Torque = 15*5 = 75NM

With a higher CoM

CoM = 20 meters

Mass = 10 Kg (same as before)

Centripetal force = 15 N (same as before since mass in unchanged)

Moment of inertia = 10*20^2 = 4,000KgM^2

Torque = 10*20 = 200 NM

As you can see the torque increased by much less than the moment of inertia.

• Interesting question. Outline of an answer (hopefully to flesh out later): consider two cars both shaped like books, with the same moment of inertia for rotations about the same edge. But one has the wheels so that the back cover of the book faces the road, and the other has the wheels so that the spine of the book faces the road. It's a little clearer in this case why the upright book is more likely to tip over. – rob Aug 22 '18 at 3:52

The moment of inertia tells you rate of rotation given a net torque. But for cars, we don't care how fast they tip over, we care if they tip at all. So the question isn't the moment of inertia, but the net torque. Is there something about a high center-of-mass vehicle that makes it more likely to enter a regime where the torques can't be countered?

Lets imagine a car is turning to the left and in the car's rotating frame of reference, it is experiencing a fictitious (centrifugal) force apparently to the right. The car will not tip if the torque from $F$ does not exceed the torque from the center of mass. So the tip limit is when

$$F y = mgx$$

If mass, width, and forces are kept constant, then increasing the height makes it easier to tip.

The situation is easiest to understand if we take the extreme case of a moving car where the mass is concentrated high up on the end of a vertical lever arm that is long in comparison to the car's track width. In this case, we initiate rolling with a steering command, say to the left.

the car will begin to roll over towards the right when the car's wheel platform has established a sideways velocity component which moves the center of support out from under the center of mass. The higher the center of mass is situated, the less cornering force is required to accomplish this and the more easily the car will begin to roll.

In the limit of a very high center of mass and a very narrow track width, the system behaves more like a motorcycle and the vehicle's moment of inertia about its roll axis is small compared to the frictional force generated at the steered wheel's contact point with the road surface. In this case, a "steer left" movement of the front wheel immediately initiates a "roll right" response of the motorcycle as a whole.

In the limit of a very low center of mass and a wide vehicle track, the opposite condition occurs: the wheel adhesion with the pavement fails before sufficient cornering moment can be generated to flip the car, and instead of rolling over, it slides sideways.

There are three (at least three) issues with a vehicle with a high center of mass that makes them considerably more subject to rollover than other vehicles.

1. Two of the main culprits in vehicle rollover is the vehicle hitting a curb or sliding off the road onto a rough shoulder. The torque that initiates the rollover results from the interaction between the tires and the surface. The torque is given by $\vec r \times \vec F$, where $\vec r$ is the vector from the center of mass to the tire that just hit a curb or went off the road. A higher center of mass exacerbates this torque. The torque is rather low in vehicles with very low centers of mass.

2. It's game over when the center of mass rotates to being above the tires that triggered the rollover. A huge roll angle is needed in the case of a squat F1 race car, but only a tiny roll angle for one of those death trap vans in which even a basketball player can stand upright.

3. Cargo shift. The above concern is exacerbated when passengers or cargo aren't tightly strapped in. A good fraction of fatal passenger van rollovers occur because the passengers weren't wearing seat belts. Momentum makes those unbelted passengers fly in a way that increases the chance of a rollover. They give an additional torque, in exactly the wrong way, when they hit the vehicle wall, and where they hit the wall increases the odds the center of mass is outside the axles of the vehicle.