Classical Mechanics Rigid Body Dynamics
Dipan Kumar Ghosh UM-DAE Centre for Excellence in Basic Sciences Kalina, Mumbai 400098 November 14, 2016
1 Introduction
A rigid body is defined as a collection of particles such that the relative distance between any pair of particles in the body remains fixed. The position of the rigid body gets fixed if we specify the positions of three non-colinear points in the body with respect to a fixed (external to the body) coordinate system OXYZ. Such a space-fixed coordinate system will also be called as the inertial coordinate system. Since three particles have a total of 9 degrees of freedom, the rigid body itself has six degrees of freedom because the distance between three possible pair of particles are fixed by three holonomic constraints. (Adding another particle would introduce another three degrees of freedom but it will bring in three additional constraints as well!). It is convenient to define a second set of axes which is fixed with the body, i.e. it moves or rotates with the body. We denote this set of axes by Cxyz where C is the origin of this “body-fixed” system of coordinates. Frequently (but not necessarily) C is chosen to be the centre of mass of the body. The six degrees of freedom of the rigid body is specified by the position vector R~ of the origin C of the body fixed system (with respect to the origin O of the inertial frame) and three independent angles that Cxyz makes with OXYZ (e.g. we can fix Cx with resoect to OX two angles - polar and azimuthal; Cy can then be chosen by one more angle; Cz gets fixed since it is perpendicular to both Cx and Cy).
1 c D. K. Ghosh, IIT Bombay 2
Z
z r’ y
C
r x R
O Y
X Let us consider the situation where Cxyz axes rotate about the OZ axis where the points O and C coincide. Our interest will be to find the relationship between dynamical quantities in the rotating frame of reference with those in the fixed frame.(Note that what follows in the next section is true for a rotating frame which and is not just for the rigid body).
2 Velocity and Acceleration of a particle as seen from a Rotating Frame
Let us consider what happens to a vector when it is rotated about a fixed axis by an angle δϕ with an angular velocity Ω.~