Theory of General Relativity
First, don’t confuse this with the theory of special relativity. Special relativity is all about one simple thing: No matter how fast different people are moving, they all see the speed of light as the same constant value (3 x 108 m/s). Why is it special? Well, before Einstein, everyone thought reference frames were relative. If you are moving toward me, that is the same as me moving toward you. Yes, there are tons of cool things that come from special relativity—but we are going to talk about general relativity.
The theory of general relativity says several things. But the main idea is that gravity is a result of curved space-time. If space-time is curved, then a light beam should be bent by a massive object—and yes, this actually happens. However, there is another effect of the theory of general relativity: you can’t tell the difference between acceleration and gravity.
Suppose you have two rockets (with no windows). One of the rockets is at rest (and not accelerating) on the surface of Earth where the gravitational field is 9.8 N/kg. The other rocket is in deep space and accelerating at 9.8 m/s2. In both rockets, a person throws a ball.
What does acceleration and gravity have to do with hoverboards?
An Accelerating Hoverboard
There you are, standing still on your new hoverboard. Just standing there isn’t much fun. You want to go somewhere. Since you are at rest, this means you must accelerate. Here’s what happens when you accelerate on a hoverboard.
Yes, you have to rotate your ankles to turn on the electric motors in hoverboard, but you must also lean forward. If you don’t lean forward, you fall backward. Trust me on this and don’t try it yourself. Let’s consider a moment during this brief hoverboard acceleration.
Three forces act on the rider in this diagram. One is the force of the floor pushing up on the board. This force has an upward component to keep the rider from falling and a horizontal friction component that accelerates the board.
Another is the gravitational force pulling down. I have labeled this mg. The third is another gravitational force labeled mg2. This is the equivalent gravitational force due to the accelerated motion of the rider. The sum of these two gravitational force vectors would be directly along the line of his body such that in his reference frame he would feel like he is standing vertically.
The greater the acceleration of the hoverboard, the greater the lean angle. Since the vector sum of these two gravitational forces must be along the direction of the human body, there is a relationship between the lean angle and the acceleration.
You should be able to measure both the lean angle and the acceleration of the hoverboard to show this is true. That would make a nice homework assignment for you. Oh, and the same thing happens when a hoverboard slows down.
What about moving at a constant speed? In that case, you aren’t accelerating so there is no additional equivalent gravitational force. You just stand up straight. What about turning? Turning also is an acceleration that would require another gravitational force (from a general relativity viewpoint). However, turning is different in that you have two feet side by side. You can actually turn and push more with one foot than the other to keep yourself from falling over since your feet are probably spread apart.
You can look at an accelerating hoverboard from a stationary frame and there is only a vertical downward gravitational force or you can view it from the perspective of the person with an extra gravitational force. Either way, it’s still physics.