362
6.3. DENSITY, MASS, AND CENTER OF MASS
the same axes. Without doing any calculations, which bar do you expect to
have the greater center of mass? Why?
(f) Compute the exact center of mass of the bar described in (e) whose density
function is p(x) = 4e 0.020732x . Check the result against the prediction you
made in (e).
⊳
Summary
In this section, we encountered the following important ideas:
• For an object of constant density D, with volume V and mass m, we know that m = D·V .
• If an object with constant cross-sectional area (such as a thin bar) has its density
distributed along an axis according to the function ρ(x), then we can find the mass of
the object between x = a and x = b by
m =
b
a
ρ(x) dx.
• For a system of point-masses distributed along an axis, say m 1 , . . . , m n at locations
x 1 , . . . , x n , the center of mass, x, is given by the weighted average
x =
n
i=1 x i m i
n
i=1 m i
.
If instead we have mass continuously distributed along an axis, such as by a density
function ρ(x) for a thin bar of constant cross-sectional area, the center of mass of the
portion of the bar between x = a and x = b is given by
x =
b
a
x ρ(x) dx
b
a
ρ(x) dx
.
In each situation, x represents the balancing point of the system of masses or of the
portion of the bar.
Exercises
1. Let a thin rod of length a have density distribution function ρ(x) = 10e −0.1x , where x
is measured in cm and ρ in grams per centimeter.
(a) If the mass of the rod is 30 g, what is the value of a?