20 Motion of Rigid Body III

Ashok Goyal

1. Introduction

The rigid body motion in the absence of any external force or torque is described by the Euler’s equations. The centre of mass in such a situation can be taken to be at rest without any loss of generality because it may atmost be moving with a uniform velocity in the absence of an external force. Rotation of the earth thus can be considered as a case in point. The earth is symmetrical about the polar axis and is slightly flattened at the poles. It can thus be considered as a symmetrical top. The application of Euler’s equations would result for an observer to find that the axis preceses in a circle about the north pole once roughly every ten months. The motion of a symmetrical body in a gravitational field with one point fixed in space has been of wide interest in the motion of a variety of physical systems like the motion of a gyroscope to the motion of a top and of a child’s toy like the ‘tippie top’. The motion is complicated and full of surprises. We will discuss a few cases in this unit.

2. Force-Free motion of a symmetrical Top

and the magnitude of the angular velocity does not change and we find that the angular velocity  revolves or preccesses about the body symmetric axis  with a constant angular frequency

which depends on the level of asymmetry that is the difference between the moment of inertia about the symmetry axis and the other principal axis which lie in a plane perpendicular to the symmetry axis.

Thus an observer in the body coordinate system would observe the angular velocity of rotation of the body to trace out a cone about the body symmetry axis.

The angular momentum  of the system in the inertial frame does not change in time since there are no external forces or torques. The kinetic energy of the system is another constant of motion, since the C.M. of the system is taken to be at rest, the rotational kinetic energy is also conserved i.e.

day. This is attributed to the fact that earth is not perfectly rigid and is neither has the shape of an orbit spheroid.

3. Motion of Symmetry Top with one Point Fixed

We will consider now the motion of a symmetrical top in the gravitational field with one point fixed. We take the origin of the body fixed axis to coincide with the centre of mass of the body so that the translational kinetic energy of the top vanishes. The symmetry axis of the top is one of the principal-axis and can be chosen as the -axis of the body-fixed coordinate system. Since the tip of the top is assumed to be stationary, we are left with three degrees of freedom to describe the orientation of the top. These three generalized coordinates are chosen to be the three Euler angles  and  where  gives the inclination of the  axis with the vertical chosen to be the z-axis of the inertial coordinate system.  is the angle made by the body fixed  axis with the line of nodes and  is the angle between the line of nodes and -axis of the inertial system as shown

We can solve the problem using these constants of motion without taking recourse to Lagrange’s equations of motion.

4. The stability of Rigid Body Rotation

When a rigid body is in force-free rotation about one of its principal axis and a small perturbation is applied, if the system reverts back to its motion or performs small oscillations about it,its motion is said to be stable. Consider a rigid body with principal moments of inertia  and

and the motion is always stable.

5. Summary

•  For the case of force-free motion of a rigid body like earth’s rotation, an observer will find that the axis precesses in a circle about the North Pole. This follows from the application of Euler’s Equation to earths rotation.
• Motion of a symmetrical top exhibits the phenomenon of precession, Nutation and sleeping depending on the initial conditions.