Motion of a rod pushed at one point in a weightless environment in space

Abstract

We analyze the motion of a rod floating in a weightless environment in space when a force is applied at some point on the rod in a direction perpendicular to its length. If the force applied is at the centre of mass, then the rod gets a linear motion perpendicular to its length. However, if the same force is applied at a point other than the centre of mass, say, near one end of the rod, thereby giving rise to a torque, then there will also be a rotation of the rod about its centre of mass, in addition to the motion of the centre of mass itself. If the force applied is for a very short duration, but imparting nevertheless a finite impulse, like in a sudden (quick) hit at one end of the rod, then the centre of mass will move with a constant linear speed and superimposed on it will be a rotation of the rod with constant angular speed about the centre of mass. However, if force is applied continuously, say by strapping a tiny rocket at one end of the rod, then the rod will spin faster and faster about the centre of mass, with angular speed increasing linearly with time. As the direction of the applied force, as seen by an external (inertial) observer, will be changing continuously with the rotation of the rod, the acceleration of the centre of mass would also be not in one fixed direction. However, it turns out that the locus of the velocity vector of the centre of mass will describe a Cornu spiral, with the velocity vector reaching a final constant value with time. The mean motion of the centre of mass will be in a straight line, with superposed initial oscillations that soon die down.