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

Definition 14.5.4Let an object consist of p point masses m_{1},

⋅⋅⋅

,m_{p}with theposition of the k^{th}of these at R_{k}. The center ofmass of this object R_{0}is the pointsatisfying

∑p
(Rk − R0)× gmku = 0
k=1

for all unit vectors u.

The above definition indicates that no matter how the object is suspended, the total
torque on it due to gravity is such that no rotation occurs. Using the properties of the
cross product

(∑p ∑p )
Rkgmk − R0 gmk × u = 0 (14.22)
k=1 k=1

(14.22)

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

p p
∑ ∑
Rkgmk − R0 gmk= 0.
k=1 k=1

dividing by g, and then by ∑_{k=1}^{p}m_{k},

∑p
R0 = --k∑=p1Rkmk--. (14.23)
k=1 mk

(14.23)

This is the formula for the center of mass of a collection of point masses. To consider the
center of mass of a solid consisting of continuously distributed masses, you need the
methods of multivariable calculus.

Example 14.5.5Let m_{1} = 5,m_{2} = 6, and m_{3} = 3 where the masses are inkilograms. Suppose m_{1}is located at 2i + 3j + k,m_{2}is located at i− 3j + 2k and m_{3}is located at 2i − j + 3k.Find the center of mass of these three masses.