Linear Algebra

156 Chapter 2. Vector Spaces
Remark. An especially interesting application of the above formula occurs
when when the two bodies are a planet and the sun. The mass of the sun m1
is much larger than that of the planet m2. Thus the argument to ˆ f is approximately
0, and we can wonder if this part of the formula remains approximately
constant as m2 varies. One way to see that it does is this. The sun’s mass is
much larger than the planet’s mass and so the mutual rotation is approximately
about the sun’s center. If we vary the planet’s mass m2 by a factor of x then the
force of attraction is multiplied by x, andx times the force acting on x times
the mass results in the same acceleration, about the same center. Hence, the
orbit will be the same, and so its period will be the same, and thus the right side
of the above equation also remains unchanged (approximately). Therefore, for
m2’s much smaller than m1, the value of ˆ f(m2/m1) is approximately constant
as m2 varies. This result is Kepler’s Third Law: the square of the period of a
planet is proportional to the cube of the mean radius of its orbit about the sun.
In the final example, we will see that sometimes dimensional analysis alone
suffices to essentially determine the entire formula. One of the earliest applications
of the technique was to give the formula for the speed of a wave in deep
water. Lord Raleigh put these down as the relevant quantities.
Considering
quantity
dimensional
formula
velocity of the wave v L 1 M 0 T −1
density of the water d L −3 M 1 T 0
acceleration due to gravity g L 1 M 0 T −2
wavelength λ L 1 M 0 T 0
(L 1 M 0 T −1 ) p1 (L −3 M 1 T 0 ) p2 (L 1 M 0 T −2 ) p3 (L 1 M 0 T 0 ) p4 = L 0 M 0 T 0
gives this system
with this solution space
p1 − 3p2 + p3 + p4 =0
p2
=0
−p1 − 2p3 =0
⎛ ⎞
1
⎜
{ ⎜ 0 ⎟
⎝−1/2⎠
−1/2
p1
�
� p1 ∈ R}
(as in the pendulum example, one of the quantities d turns out not to be involved
in the relationship). There is thus one dimensionless product, Π1 =
vg −1/2 λ −1/2 , and we have that v is √ λg times a constant ( ˆ f is constant since
it is a function of no arguments).
As those three examples show, analysis of the relationships possible among
quantities of the given dimensional formulas can bring us far toward expressing