92 CHAPTER 6. VECTOR PRODUCTS

which reduces to (F3−F2) i+ (F1−F3) j+ (F2−F1) k=−27i−8j−8k. This amountsto solving the system of three equations in three unknowns, F1,F2, and F3,

F3−F2 =−27, F1−F3 =−8, F2−F1 =−8

However, there is no solution to these three equations. (Why?) Therefore no single forceacting at the point i+j+k will produce the given torque.

6.5.2 Center Of MassThe mass of an object is a measure of how much stuff there is in the object. An object hasmass equal to one kilogram, a unit of mass in the metric system, if it would exactly balancea known one kilogram object when placed on a balance. The known object is one kilogramby definition. The mass of an object does not depend on where the balance is used. Itwould be one kilogram on the moon as well as on the earth. The weight of an object issomething else. It is the force exerted on the object by gravity and has magnitude gmwhere g is a constant called the acceleration of gravity. Thus the weight of a one kilogramobject would be different on the moon which has much less gravity, smaller g, than on theearth. An important idea is that of the center of mass. This is the point at which an objectwill balance no matter how it is turned.

Definition 6.5.4 Let an object consist of p point masses m1, · · · ,mp with the position of thekth 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 (6.30)

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

. (6.31)

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 calculus.