Banked Curves, Rotating Frames and Artificial Gravity

The normal force always acts perpendicular to the surface. On a flat road that means straight up, so it cannot help with turning. Tilt the road by an angle $\theta$ and the normal force tilts by the same angle, toward the centre of the curve. Now it has two jobs. Its vertical component $F_N\cos \theta$ holds the car up against gravity, and its horizontal component $F_N\sin \theta$ points toward the centre and supplies the centripetal force. At exactly the right speed, called the design speed, these two components do everything and friction is not needed at all. Dividing the horizontal equation by the vertical one cancels both $F_N$ and $m$.

Real roads have friction, and that widens the range of safe speeds. Go faster than the design speed and the car needs more inward force than the normal force alone provides, so it tends to slide up and out; static friction then acts down the slope, adding to the inward force. Go slower and the normal force provides too much inward push, so the car tends to slide down toward the inside; friction then acts up the slope. Setting friction to its maximum, ${\mu }_s{F}_N$, in each case gives the two limits.

Connected Objects and Frames of Reference showed that an accelerating frame needs apparent forces to make Newton's laws work. A rotating frame needs two. The centrifugal effect, $m{\omega }^{2}r$ directed outward, is what you feel pressing you into the outside door of a turning car. The Coriolis effect acts only on objects that move relative to the rotating frame: it seems to push them sideways, at right angles to their velocity. Nothing actually pushes. The ground simply rotates underneath while the object travels in a straight line. Earth is a slowly rotating frame, so in the Northern Hemisphere moving objects appear to veer to the right, and in the Southern Hemisphere to the left. Air rushing toward a low-pressure centre veers right, so it ends up circling counterclockwise, which is why hurricanes and Ontario's large storm systems spin that way in the north and the opposite way south of the equator.

A spinning space station can imitate gravity. The floor is the outer wall, and it must push each astronaut inward to keep them moving in a circle. That normal force feels exactly like standing on a floor on Earth. To feel $g$, the centripetal acceleration must equal $9.81{\text{m/s}}^2$, so ${\omega }^{2}r=g$ and the period is $T=2\pi \sqrt{r/g}$. A small station must spin fast, and fast spinning has costs: the 'gravity' changes noticeably between head and feet, and turning your head in a fast-spinning frame triggers Coriolis effects in the inner ear that cause motion sickness. Studies suggest many people tolerate only a few revolutions per minute, which pushes designs toward large radii. Amusement-park rotor rides use the same physics with friction added: the wall supplies the centripetal force, and the friction it produces holds riders up when the floor drops away.