14.5. PROOF OF THE DISTRIBUTIVE LAW 299

Definition 14.5.4 Let an object consist of p point masses m1, · · · ,mp with the po-sition of the kth of these at Rk. The center of mass of this object R0 is the point satisfying

p

∑k=1

(Rk −R0)×gmku= 0

for all unit vectors u.

The above definition indicates that no matter how the object is suspended, the totaltorque on it due to gravity is such that no rotation occurs. Using the properties of the crossproduct (

p

∑k=1

Rkgmk −R0

p

∑k=1

gmk

)×u= 0 (14.22)

for any choice of unit vector u. You should verify that if a×u= 0 for all u, then it mustbe the case that a= 0. Then the above formula requires that

p

∑k=1

Rkgmk −R0

p

∑k=1

gmk = 0.

dividing by g, and then by ∑pk=1 mk,

R0 =∑

pk=1Rkmk

∑pk=1 mk

. (14.23)

This is the formula for the center of mass of a collection of point masses. To considerthe center of mass of a solid consisting of continuously distributed masses, you need themethods of multi-variable calculus.

Example 14.5.5 Let m1 = 5,m2 = 6, and m3 = 3 where the masses are in kilograms. Sup-pose m1 is located at 2i+ 3j + k, m2 is located at i− 3j + 2k and m3 is located at2i−j+3k. Find the center of mass of these three masses.

Using 14.23

R0 =5(2i+3j+k)+6(i−3j+2k)+3(2i−j+3k)

5+6+3=

117i− 3

7j+

137k

14.5.3 Angular Velocity

Definition 14.5.6 In a rotating body, a vector Ω is called an angular velocity vec-tor if the velocity of a point having position vector u relative to the body is given by Ω×u.

The existence of an angular velocity vector is the key to understanding motion in amoving system of coordinates. It is used to explain the motion on the surface of the ro-tating earth. For example, have you ever wondered why low pressure areas rotate counterclockwise in the Northern hemisphere but clockwise in the Southern hemisphere? To quan-tify these things, you will need the concept of an angular velocity vector. Here is a simpleexample. Think of a coordinate system fixed in the rotating body. Thus if you were ridingon the rotating body, you would observe this coordinate system as fixed.