Greenwood press

152
VARIATION

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and the moment of inertia 
I of the cross section of the board. The variation for-
mula is 
D = k
W L
3
3EI
.
Ohm’s law is a direct variation statement 
V = IR, where V is voltage, I is
current, and 
R is the resistance in a particular conductor. R, which is measured
in ohms, is constant of variation for the particular conductor. Resistance is meas-
ured in ohms and will vary across different wires. For example, electrical resist-
ance
R in a wire varies directly as its length L, and inversely as its cross-sec-
tional area 
A: R =
ρL
A
, where 
ρ is the constant of variation. The constant of vari-
ation is called resistivity and has been computed for many materials: gold has a
resistivity of 
2.35 × 10
−8
; carbon, 
3.50 × 10
−5
; and wood, 
10
8
. If one assumes 
that the wire is round, then the variation is 
R =
kL
r
2
. The coefficient of variation, 
k, would be the resistivity divided by π. If resistance in a wire must be reduced,
there are two routes: you could shorten the wire, or you could use a wire with a
larger radius. The latter might have the most payoff, because the radius is squared
in the formula. The three-dimensional graph below shows the resistance (verti-
cal axis) of copper wire wrapped into a coil. The lower-left axis shows the radius
in meters of wire running from 2 mm up to 1 cm (0.01 meter). The axis on the
right shows how long the wire would be if it were unwrapped. The axis runs,
right to left, from 0 to 1,200 meters. The length does not appear to affect results.
However, radii under 5 mm send the resistance soaring.
Some laws appear in different forms of variation depending on the situation.
The simple form of Poiseuille’s law states that the speed 
S at which blood moves
through arteries and veins varies directly with the blood pressure 
P and the
fourth power of the radius 
r of the blood vessel: S = kP r
4
. This is derived from
Poiseuille’s law for the flow of fluids, which relates to flow 
F rather than speed 
(flow is speed times cross-section area of the tube): 
F =
k∆P r
4
nL
, where 
∆P is
the change in pressure from the beginning of a tube to the end, 
r is the radius of
the tube, 
n is a measure of viscosity of the fluid, L is the length of the tube, and
k is a constant of variation. In this version, which is used to determine the flow
of oil through pipes and also fluids through tubes in an automobile, flow is
directly related to the change in pressure and fourth power of tube radius, and is
inversely related to viscosity and the length of tube.
In general, solving variation problems involves two steps: first, solve for the
constant of variation in a known situation; and second, use that constant to