Engr Help

Euler’s First Law

Example: A biker is speeding around the inside of a cylindrical track as shown in the figure

below. The outer radius of the track is R. The center of mass of the biker/motorcycle is

a distance d from the outer radius of the track’s wall. The coefficient of friction between the

tires and the wall of the track is μ. Find the minimum speed of the biker without sliding

down the wall of the track.

Strategy: The first step in applying Euler’s first law is to draw a free body diagram to

identify all forces acting on the body, in this case the motorcycle/rider. Here N is the

normal force on the tires, F is the friction force on the tires, mg is the combined weight of

the motorcycle and rider, e_n is the unit normal vector, and k is the unit vector in the z-direction.

Next apply Euler’s first law F = m a_C which results in

(F – mg) k + (N) e_n = m [ 0 k + v²/ρ e_n ] so F = mg and N = m v²/ρ .

Here ρ is the radius of curvature; in this case ρ = R – d.

Now for minimum speed, v, the friction force, F, must also equal μN or μ m v²/ρ

So mg = μ m v²/ρ and solving for v gives v = √ [ g(R-d)/μ ] (result)