A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.5 Centre of mass

X

Now that we have defined the vector product of two vectors, let us find a use

for this concept. Figure 71 shows a rigid body rotating with angular velocity ω.

For the sake of simplicity, the axis of rotation, which runs parallel to ω, is as-

sumed to pass through the origin O of our coordinate system. Point P, whose

position vector is r, represents a general point inside the body. What is the veloc-

ity of rotation v at point P? Well, the magnitude of this velocity is simply

v = σ ω = ω r sin θ, (8.15)

where σ is the perpendicular distance of point P from the axis of rotation, and
θ is the angle subtended between the directions of ω and r. The direction of
the velocity is into the page. Another way of saying this, is that the direction of

the velocity is mutually perpendicular to the directions of ω and r, in the sense
indicated by the right-hand grip rule when ω is rotated onto r (through an angle

less than 180 ◦). It follows that we can write

v = ω × r. (8.16)

Note, incidentally, that the direction of the angular velocity vector ω indicates

the orientation of the axis of rotation—however, nothing actually moves in this

direction; in fact, all of the motion is perpendicular to the direction of ω.

8.5 Centre of mass

The centre of mass—or centre of gravity—of an extended object is defined in much

the same manner as we earlier defined the centre of mass of a set of mutually

interacting point mass objects—see Sect. 6.3. To be more exact, the coordinates

of the centre of mass of an extended object are the mass weighted averages of

the coordinates of the elements which make up that object. Thus, if the object

has net mass M, and is composed of N elements, such that the ith element has
mass mi and position vector ri, then the position vector of the centre of mass is

given by

rcm

1

= mi

M

i=1,N

ri. (8.17)