Fraught Freight

Now that I'm actually going into an office every day, I'm once again a car commuter, and I'm filled with transportation-related questions! Our route mostly involves the highway, and I was curious about the rollover signs on off-ramps:

rxrsignals.com

This shows a truck tilting at a 30° angle, while going 25 mph. The tilt comes from the fact that the truck is going around a curve, which requires applying a centripetal force. Since this force is applied to the tires, there's a torque produced, which will act to tip the truck toward the outside of the curve:

The red dot marks the truck's center of mass, with m the total mass of the truck, and a the acceleration required to make the turn. This is given by

where R is the radius of the turn. To find this, we can use Google Maps' measuring tool on the ramp by our home:

Given this R, along with the speed and dimensions of the truck, we can find the height of the center of mass that would produce the given tilt – The diagram above is too high, and the truck would tip over, but to balance g and a, we would need

where θ is the tilt angle. Using our 30° case, and the turn shown above, we can check the heights for several different speeds:

In most cases, we should expect the truck to be loaded such that the center of mass is in the center of the truck, which means the posted limit for our turn of 25 mph is just about right!