The Physics of "s is one half
gt^{ }^{2}"
This song contains five important
equations from firstsemester physics. The title equation, s
= 1/2 gt^{ 2}, gives the distance
s fallen by an object in a time t, assuming the
object starts at rest and that air resistance is negligible; g
is the acceleration of gravity, which is about 9.8 m/s^{2}
on the surface of the Earth. (Note: for medium to small objects, the
force of air resistance becomes important after a drop of 10 meters
or less. You might be interested in my answer to the question, "What
would happen if you dropped a superball off the Empire State Building".)
The second equation in the song, PV=
nRT is the "ideal gas law". It gives the relationship
between the pressure P, the volume V, the number
of moles n, and the temperature (measured from absolute
zero) T of any gas, with the assumptions that the gas
molecules only interact with each other in very short collisions,
and that the volume of the gas molecules themselves is negligible
compared to the total volume occupied by the gas. The constant
R is the "ideal gas constant", which equals
8.31 J/mol K. The equation says that, if you start with a given
container filled with gas, you can increase the pressure either
by 1) adding more gas (increasing n), 2) heating the
gas (increasing T), or 3) squeezing the container to
a smaller volume (decreasing V). (Note: the "number
of moles" is equal to the number of gas molecules divided
by Avogadro's number, 6.02 x 10^{23}.) Perhaps the most
remarkable thing about this equation is that the gas pressure
doesn't depend on the type of gas  you would get the same pressure
for a given number of oxygen molecules as you would for the same
number of helium atoms.) Click the image at right to open a Java
applet from Ohio State University that nicely illustrates this
law. 

The third equation
gives the expression for kinetic energy: KE = 1/2 mv^{
2}, where m is the mass of a moving object and
v is its velocity. Although the idea of energy is so
central to our current understanding of physics, and indeed
the way we think about everyday life and geopolitics, it is
a relatively recent notion. The law of conservation of energy,
i.e. that energy can never be destroyed, but only transformed
from one form to another, was initially obscured by the common
observation that things that move usually eventually come to
rest, so that their energy of motion seems to disappear. We
now understand that, when this happens, the kinetic energy is
transformed into thermal energy through friction. However, it
was not until experiments performed by the brewer James P. Joule
in the early 1840s that this really became clear, and it took
many years after that for the idea of energy conservation to
be widely accepted in the scientific community. Read
more about the history of energy here.


The fourth equation
in the song gives the formula for the period (the time for a
complete swing back and forth) of a pendulum: .
Here L is the length of the pendulum, and g
is the acceleration of gravity, which is about 9.8 m/s^{2}
on the surface of the Earth. There are two astonishing things
about this equation: 1) The period of the pendulum doesn't depend
on the mass of the pendulum bob (the weight at the end of the
pendulum)! 2) The period doesn't depend on the amplitude of
swinging. In other words, the period is the same when the pendulum
is swinging back and forth only a tiny amount as when its swinging
more vigorously. (In fact, this equation only works well for
moderate or small amplitude swings, say less than 30 degrees
back and forth.)
There is a wonderful
legend (apocryphal, but so what) of how Galileo discovered the
lack of dependence on amplitude: "Galileo was bored. As
he listened to a Mass in the drafty cathedral of Pisa in 1581,
the 17yearold student noticed something interesting. A chandelier
high overhead was swaying in the breeze, sometimes barely moving
and other times swinging in a wide arc. His curiosity aroused,
he timed the swings with his pulse. To his surprise, it took
the same number of pulse beats for the chandelier to complete
one swing no matter how far it moved. The wider the swing, the
faster the motion, but always in the same amount of time. So
time could be measured by the swing of a pendulum  the basis
for the pendulum clock."^{1 }
The pendulum, of course,
was then used to construct the first accurate clocks, which
were eventually adapted for use at sea, allowing accurate navigation.

Galileo
15641642

The last equation in the song is
Newton's second law, "force is mass times acceleration",
or F = ma. This equation is really the definition
of force: a force is something that, when applied to a body
of mass m produces an acceleration a.
In one of the best parts of the song,
the professor goes on to say, "or more precisely the time
derivative of the linear momentum". In fact, this is the
way Newton originally stated the law  his wording was, "The
net force acting on a body is equal to the rate at which the
body's quantity of motion is changing." What he called
"quantity of motion" is what we now call "linear
momentum" (linear as opposed to angular momentum); it is
usually assigned the symbol p, and is given by p
= mv.
We can easily see that the two ways
of stating Newton's second law are equivalent. ("There
he goes again!") The "rate of change of " (equivalently
"time derivative of") the momentum is written ,
so that Newton's second law in his original version would be
.
Since, for most situations, the mass of the accelerating object
doesn't change, we can rewrite this as .
Finally, the acceleration
is the time derivative (i.e. the rate of change) of the velocity:,
so .
Usually, the familiar F =
ma form is more convenient. However, the original form,
is important for developing the idea of "impulse",
which is useful for analyzing collisions.
WFS 8/28/05


References: 1
 Prof. Roger Newton, author of Galileo's Pendulum (Harvard
University Press, 2004)
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