THE SUN-EARTH SYSTEM
APPENDIX II
Units and Dimensional Analysis
Units
The principle of conservation of energy says
that the amount of energy (of all types) around
before an event takes place exactly equals the
amount after the event. Let us assume that the
amount of energy (called potential energy, PE) a
ball of mass m has when raised to a height h is
mgh, where g is the acceleration due to gravity
(PE = mgh). We know the ball has energy, because, for example, if we dropped it on a drumhead it would not only bounce back up some
distance but would also create sound waves, a
form of energy. Before we drop the ball, its
speed is zero and its height is h. We know that
it will have a certain speed, v, when it reaches
the drumhead after falling. Since at the drumhead its height will be zero (we are measuring
height from the drum), mgh will be zero. But
energy must be conserved, so the energy of the
ball right at the drumhead must depend on its
speed. It now has some speed, but no height, so
the potential energy must have been converted
into some other form of energy that has to do
with its speed, v. How does this “energy of
motion” (called kinetic energy, KE) depend on
the ball’s speed and possibly its mass?
Scientists use an International System of
Units (SI) to describe and measure various
quantities. The three basic units from which all
other quantities may be derived are length,
mass, and time. In the SI system, length is measured in meters and mass in kilograms. Time is
measured in seconds. In the SI system, the basic
quantities are written as follows:
• Length in meters, m (approximately
1.1 yard)
• Mass in kilograms, kg (approximately
2.2 pounds)
• Time in seconds, s
• Temperature in kelvins, K
All other quantities are derived from these
units. For example, we express energy in the SI
system as joules, J. In terms of the basic units a
joule is a kg m2s–2 (s–2 means per second
squared, or per second per second), and is
about equal to the amount of energy imparted
to the floor by dropping a 2.25 kg (5 pound) bag
of sugar a distance of 5 cm (2 inches). The familiar measure of radiant power, watt (W), is a
joule per second.
In science, the units on each side of an equation must be the same. This is another way of
saying that we must compare apples with
apples and not oranges. This property can help
scientists find a correct relationship between
quantities even when the theory behind the
relationship is not completely understood.
Dimensional Analysis
Dimensional analysis or “keeping the units
the same on both sides of the equation” can
help. We know that the kinetic energy must in
some way depend on the speed and mass,
maybe each raised to some power. Let us set up
an equation with the total energy at height h
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APPENDIX II
becomes LT–2. On the right side, v has dimensions of length per time, so v becomes LT–1.
Using these notations for mass, length, and
time, we may write mgh = mavb as
equal to the total energy at height zero, so that
energy is conserved.
Energy at height h + Energy with speed zero =
Energy at height zero + Energy with speed v
(1)
In symbols this may be written:
mgh + 0 = 0 + KE
mgh = KE
(M)(LT–2)(L) = (Ma)(LT–1)b
ML2T–2 = Ma(LT–1)b
ML2T–2 = MaLbT–b
(2)
(3)
Since the powers of like units on each side of
the equation must be the same, we see from
inspection that a = 1 and b = 2. Going back to
Equation 4, since v = LT–1, we may write KE as
As we reasoned earlier, KE must also equal
m times v, each raised to some power. We will
raise m to the power “a” and v to the power
“b,” realizing that these exponents could be
positive, negative, or zero.
KE = mavb
KE = mv2.
Actually, the correct expression is KE =
(1/2)mv2, a fact we could determine by measuring the mass of the ball and the velocity after it
had fallen a distance h. But the point is that the
dependence of KE on mass and speed are right,
and we obtained the relationship solely by
some intuition and dimensional analysis.
(4)
So, because they both equal KE, mgh = mavb.
In order to keep tabs on the basic units we will
write mass as M, length as L, and time as T.
Now, g is the acceleration due to gravity, and
its units are length per time squared, so g
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