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/s² 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 17-year-old 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
1564-1642

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