Lecture Notes, Day 3: Gravity, Weight and Thermodynamics

Dealing with Euclidean Space

Vector addition in Euclidean space. Dealing with the combination of motions and a parallelogram of force.

One essential property with Euclidean space is flatness. As soon as the space is NOT flat the geometry goes non-Euclidean. For example, the surface of a sphere is non-Euclidean; a triangle on a sphere (suitably defined) will have angles that sum to something greater than 180 degrees. In fact, there is essentially only one Euclidean space of each dimension, while there are many non-Euclidean spaces of each dimension.

p.19 of the handout…when dealing with equations on the page that are written out in English, use symbols in the margins to work out the details. Get used to making sentences into equations when forces, matter, energy and distances are being discussed. Example: the surface area of a sphere is directly proportional to the square of its radius. Make sure to write this out on the side of the paragraph as: A α r²

Perihelion precession of Mercury (footnote on page 19 of handout) The exceptions to Newton’s laws are few. However, when we have them, they take new mathematics to solve the differences in gravitational effects and space-time warping. One such problem child was Mercury’s orbit. Some of the theories used to explain Mercury’s curious perturbation was that there was another planet closer to the Sun and it was even given a name, Vulcan. Thanks to the discoverer of Neptune (French mathematician Urbain Le Verrier). He thought the same method could be used for explaining the weird movements of Mercury. No such luck!

In Newtonian physics, a lone object orbiting a spherical mass would trace out an ellipse with the spherical mass at a focus. The point of closest approach, called the perihelion in the solar system, is fixed. There are a number of solar system effects that cause the perihelion of a planet to precess, or rotate around the sun. These are mainly because of the presence of other planets, which perturb orbits. Another effect is solar oblateness, which produces only a minor contribution.

The precession of the perihelion of Mercury was a longstanding problem in celestial mechanics. Careful observations of Mercury showed that the actual value of the precession disagreed with that calculated from Newton's theory by 43 seconds of arc per century, which was much larger than the experimental error at the time. However, the problem was resolved by Einstein's theory, which predicted exactly the observed amount of perihelion shift. This was a powerful factor motivating the adoption of Einstein's theory.

p. 21 of the handout explains stresses, strain and elasticity of matter. Can anyone find a few videos on the web that demonstrate the elasticity of matter as forces of stress are in operation? I’ve seen them used on PBS documentaries where you actually see a spherical baseball flatten itself against the bat and then “snap back” into its original form after it is hit by the bat. It totally rocks!

How to measure Gravitational Forces to assist Newton with his gravitational constant.
Robert Hooke created Hooke’s Law to explain: The strain is proportional to the stress. And this principle can be applied to twisting forces by means of a torsion balance. Thanks to Henry Cavendish for his delicate torsion balance that gave us a number to use! We now use an even more exact figure than Cavendish was able to measure; G=6.67 x 10^-11m³/kg-sec²

How Much Do You Weigh?

Centripetal Force - The force directed inward to a rotating body

Centrifugal Force – The force directed outward from a rotating body

These two forces acting together on the earth as it spins on its axis give it an oblate spherical shape.

This causes the measurement of weight to be different if one is located at the poles or at the equator. If you want to weigh less, move to Alaska! The way scientists have managed to negate the differences in gravitational effects is by calibrating double-pan balances with known weights.

Thermodynamics

The study of heat requires us to understand the three different scales of temperature that have been created. Fahrenheit, Celsius and Kelvin.

F = 9/5C + 32

C = 5/9(F-32)

K = C + 273 more commonly written as T = t + 273

When building the St. Louis Arch the builders had a dilemma putting in the last block due to temperature changes and steel expansion at mid-day.

This is the coefficient of linear expansion at work.

To get a handle on the coefficient of cubic expansion one can look at the Chernobyl accident of 1986 to determine what happens when huge amounts of liquid are instantaneously turned to steam due to high levels of heat.

The First Law of Thermodynamics:

Whenever heat is added to a system, it transforms to an equal amount of some other form of energy. Or in any process, the total energy of the universe remains constant.

Parallelepiped - is a three-dimensional figure formed by six rectangles.

The Second Law of Thermodynamics:

Heat will never itself flow from a cold object to a hot object. Or it can be stated, Natural systems tend to proceed toward a state of greater disorder.

The Third Law of Thermodynamics:

As temperature approaches absolute zero, the entropy of a system approaches a constant.

Entropy – the measure of the amount of disorder

Here is the diagram I used in class to discuss supercritical fluid