A motorcycle of mass M moves on a bowl-shaped surface specified by its height h(r) where r=x2+y2 is the radius in cylindrical polar coordinates (r,ϕ,z). The torque exerted by the motorcycle engine on the rear wheel results in a force F(t) pushing the motorcycle forward. Assuming F(t) is directed along the motorcycle's velocity and that the motorcycle's vertical velocity and acceleration are small, show that the motion is described by
where dots denote time derivatives, F(t)=∣F(t)∣ and g is the acceleration due to gravity.
The motorcycle rider can adjust F(t) to produce the desired trajectory. If the rider wants to move on a curve r(ϕ), show that ϕ(t) must obey
ϕ˙2=gdrdh/(r+r2(dϕdr)2−dϕ2d2r)
Now assume that h(r)=r2/ℓ, with ℓ a constant, and r(ϕ)=ϵϕ with ϵ a positive constant, and 0⩽ϕ<∞ so that the desired trajectory is a spiral curve. Assuming that ϕ(t) tends to infinity as t tends to infinity, show that ϕ˙(t) tends to 2g/ℓ and F(t) tends to 4ϵMg/ℓ as t tends to infinity.