A truck with four identical wheels runs down a slope freewheeling, i.e. with the motor disengaged, without braking or slipping. Each wheel is considered to be a homogeneous disc of radius r and mass m. The rest of the truck, i.e. wheels not included, has a mass M. To a point A of the truck is suspended an object of non-negligible mass m′, and of moment of inertia I with respect to a horizontal axis perpendicular to the velocity of A and passing through A. All considered movements are frictionless and parallel to the plane of the Figure.
(1) Give the Lagrange equations of the system.
(2) Give a first integral of the system.
(3) Show that the system can take, for appropriate initial conditions, a movement in which the inclination of the object of mass m′ and the truck acceleration remain both constant.
(4) Calculate this angle of inclination in case 3 and show that it is not zero or equal to a right angle if the road followed by the truck is neither horizontal nor vertical.
(5) Show that, if the wheel mass can be neglected, the angle of inclination in case 3 can be equal to the slope inclination.