1 INTRODUCTION 1.7 Dimensional analysis
g
fqWe can now compare the exponents of [L], [M], and [T ] on either side of the
above expression: these exponents must all match in order for Eq. (1.5) to be
dimensionally consistent. Thus,
0 = y + z, (1.7)^
0 = x, (1.8)^1 = − 2 z. (^) (1.9)
It immediately follows that x = 0 , y = 1/2, and z = −1/2. Hence,
t = C
‚
., h
. (1.10)
Now, (^) √the actual tower of Pisa is approximately 100 m tall. It follows that since
tf ∝ h (g is the same for both the real and the model tower) then the 1 m
high model tower falls over a factor of 100/1 = 10 times faster than the real
tower. Thus, the film must be slowed down by a factor 10 in order to make it
look realistic.
Worked example 1.1: Conversion of units
Question: Farmer Jones has recently brought a 40 acre field and wishes to replace
the fence surrounding it. Given that the field is square, what length of fencing (in
meters) should Farmer Jones purchase? Incidentally, 1 acre equals 43,560 square
feet.
Answer: If 1 acre equals 43,560 ft^2 and 1 ft equals 0.3048 m (see Tab. 2 ) then
1 acre = 43560 × (0.3048)^2 = 4.047 × 103 m^2.
Thus, the area of the field in mks units is
A = 40 × 4.047 × 103 = 1.619 × 105 m^2.
Now, a square field wi√th sides of length l has an area A = l^2 and a circumference
D = 4l. Hence, D = 4 A. It follows that the length of the fence is
D = 4 ×
q
1.61 9 × 105 = 1.60 9 × 103 m.