Describe the possible motion of a solid object that is suspended so it is free to rotate about its center of?

Describe the possible motion of a solid object that is suspended so it is free to rotate about its center of gravity. Is it a physical pendulum? (14-15)



please help thanks!Describe the possible motion of a solid object that is suspended so it is free to rotate about its center of?WHO is asking YOU this question?! What level of dynamics are you studying?



This is a classical problem that was originally solved by Euler in the 18th century, and clarified by a remarkable geometric representation and interpretation in the 19th century.



The general motion of an arbitrarily shaped body is FAR more complicated than that of a simple physical pendulum. It can be appreciated however if you have good visualization skills.



It was shown that the general motion consists of a certain kind of rolling, but not necessarily periodic, motion. It can however be periodic if the body has certain dynamical symmetries, such as being some kind of axially symmetrical "top." (Initially the name of the person who first thought of this interpretation escaped me, but on reflection the celebrated French scientist "Poinsot" comes to mind.)



Any 3-D body can be imagined to have a so-called "momental ellipsoid" --- in general having three principle axes of different lengths --- attached firmly to it. [This involves a generalization of the simpler concept of the moment(s) of inertia of a plane body.] Now imagine also a fixed plane (the "invariable plane") perpendicular to the body's angular momentum vector.



Then the general motion of the body is given by the following motion: the rigidly attached momental ellipsoid, while keeping its centre of gravity in the same place, rolls around on the invariable plane while maintaining non-slipping contact with the invariable plane. Although the body's angular momentum and kinetic energy ARE of course conserved, its angular velocity ISN'T. In fact the variable vector from the centroid (a term preferred to "centre of gravity") to the point of contact between the momental ellipsoid and the invariable plane is a proportional representation of the angular velocity vector.



This general motion can in fact be quite mesmerizing. In the early days of the space program, astronauts would frequently set something freely spinning before the ever present TV cameras. If the chosen body had 3 quite different principle axes of inertia, the body would seemingly slow down and speed up its tumbling motion for no apparent reason. Despite its apparent peculiarities, the concept of the rolling, fully 3-D momental ellipsoid helps to bring some order to what is taking place.



Words that crop up in this treatment (words that might help you in a web search) are "polhode" and "herpolhode" --- these are certain curves traced out by the imagined point of contact of the momental ellipsoid and the invariable plane.



Live long and prosper.