“Hidden” Momentum in an Unbalanced Tire

1 Problem Discuss the motion of an unbalanced tire that rolls without slipping on a horizontal surface, subject to conservation of energy. In what sense(s) can this system be said to contain " hidden momentum? " 2 Solution 2.1 The Motion We consider the tire to be a hoop of radius a and mass M that is unbalanced due to an additional mass m at some fixed point on the rim, as shown in the figure below. Taking the origin of an x-y coordinate system to be at a point of contact of the tire with the road when mass m is at its lowest position, when the point of contact is at x the geometric center of the tire has horizontal velocity v and the center of mass of the unbalanced tire has rotated through angle θ = x/a. The angular velocity ω of the tire is ω = ˙ θ = v a. (1) The mass m has coordinates x m = x − a sin θ, y m = a − a cos θ, (2) and velocity ˙ x m = v − a ω cos θ = (1 − cos θ)v, ˙ y m = a ω sin θ = sin θv, v 2 m = 2(1 − cos θ)v 2. (3) The conserved energy E of the system is E = Mga + mga(1 − cos θ) + Mv 2 + mv 2 m 2 = Mga + mga(1 − cos θ) + [M + (1 − cos θ)m]v 2. (4)