A roller coaster has a loop, r = 20m, how fast should it travel so that riders don't fall out?

For the riders to not fall out two forces acting on them must be equal, those forces being the gravity pulling them down towards Earth's surface and the centripetal force acting away from the centre of rotation. By equating these 2 terms the velocity can be calculated.

The force due to gravity = mg and the centripetal force = m*(v²/r). Equating the two; mg=m*(v²/r), as m is a on both sides we can simplify to mg=m(v²/r). Now we can re-arrange to find v, our equation is now g=v²/r, so firstly we can multiply both sides by r to get gr=v² . Next we want to get rid of the squared term, to do this we quare root both sides (this removes the squared term as square rooting is essentially raising a number to a half, e.g. x1/2 ) , so the equation becomes (gr)^1/2 = v²1/2 which simplifies to (gr)^1/2 = v. Finally we can sub in numbers to get (9.18*20)^1/2 = 14.0 ms^-1.