Introduction to Physics and Constants
Constants are characteristics of the physical and mathematical world that can be “taken for granted.” They don’t change, at least not within an ordinary human lifetime, unless certain other factors change too.
Math Versus Physics
In pure mathematics, constants are usually presented all by themselves as plain numbers without any units associated. These are called dimensionless constants and include π, the circumference-to-diameter ratio of a circle, and e , the natural logarithm base. In physics, there is almost always a unit equivalent attached to a constant. An example is c , the speed of light in free space, expressed in meters per second.
Table 6-2 is a list of constants you’ll encounter in physics. This is by no means a complete list. Do you not know what most of the constants in this table mean? Are they unfamiliar or even arcane to you? Don’t worry about this now. As you keep on learning physics, you’ll learn about most of them. This table can serve as a reference long after you’re done with this course.
Table 6-2 Some Physical Constants
Here are a few examples of constants from the table and how they relate to the physical universe and the physicist’s modes of thought.
Mass Of The Sun
It should come as no surprise to you that the Sun is a massive object. But just how massive, really, is it? How can we express the mass of the Sun in terms that can be comprehended? Scientific notation is generally used; we come up with the figure 1.989 × 10 ^{30} kg if we go to four significant figures. This is just a little less than 2 nonillion kilograms or 2 octillion metric tons. (This doesn’t help much, does it?)
How big is 2 octillion? It’s represented numerically as a 2 with 27 zeros after it. In scientific notation it’s 2 × 10 ^{27} . We can split this up into 2 × 10 _{9} × 10 ^{9} × 10 ^{9} . Now imagine a huge box 2,000 kilometers (km) tall by 1,000 km wide by 1,000 km deep. [A thousand kilometers is about 620 miles (mi); 2000 km is about 1240 mi.] Suppose now that you are called on to stack this box neatly full of little cubes measuring 1 millimeter (1 mm) on an edge. These cubes are comparable in size to grains of coarse sand.
You begin stacking these little cubes with the help of tweezers and a magnifying glass. You gaze up at the box towering high above the Earth’s atmosphere and spanning several states or provinces (or even whole countries) over the Earth’s surface. You can imagine it might take you quite a while to finish this job. If you could live long enough to complete the task, you would have stacked up 2 octillion little cubes, which is the number of metric tons in the mass of our Sun. A metric ton is slightly more than an English ton.
The Sun is obviously a massive chunk of matter. But it is small as stars go. There are plenty of stars that are many times larger than our Sun.
Mass Of The Earth
The Earth, too, is massive, but it is a mere speck compared with the Sun. Expressed to four significant figures, the Earth masses 5.974 × 10 ^{24} kg. This works out to approximately 6 hexillion metric tons.
How large a number is 6 hexillion? Let’s use a similar three-dimensional analogy. Suppose that you have a cubical box measuring 2.45 × 10 ^{5} meters, or 245 kilometers, on an edge. This is a cube about 152 mi tall by 152 mi wide by 152 mi deep. Now imagine an endless supply of little cubes measuring 1 centimeter (1 cm) on an edge. This is about the size of a gambling die or a sugar cube. Now suppose that you are given the task of—you guessed it—stacking up all the little cubes in the huge box. When you are finished, you will have placed approximately 6 hexillion little cubes in the box. This is the number of metric tons in the mass of our planet Earth.
Speed Of Electromagnetic (em) Field Propagation
The so-called speed of light is about 2.99792 × 10 ^{8} m/s. This works out to approximately 186,282 miles per second (mi/s). Radio waves, infrared, visible light, ultraviolet, x-rays, and gamma rays all propagate at this speed, which Albert Einstein postulated to be the same no matter from what point of view it is measured.
How fast, exactly, is this? One way to grasp this is to calculate how long it would take a ray of light to travel from home plate to the center-field fence in a major league baseball stadium. Most ballparks are about 122 m deep to center field; this is pretty close to 400 feet (ft). To calculate the time t it takes a ray of light to travel that far, we must divide 122 m by 2.99792 × 10 ^{8} m/s:
t = 122/(2.99792 × 10 ^{8} )
= 4.07 × 10 ^{−7}
That is just a little more than four-tenths of a microsecond (0.4 μs), an imperceptibly short interval of time.
Two things should be noted at this time. First, remember the principles of significant figures. We are justified in going to only three significant figures in our answer here. Second, the units must be consistent with each other to get a meaningful answer. Mixing units is a no-no in any calculation. It almost always leads to trouble.
If we were to take the preceding problem and calculate in terms of units without using any numbers at all, this is what we would get:
seconds = meters/(meters per second)
s = m/(m/s) = m × s/m
In this calculation, meters cancel out, leaving only seconds. Suppose, however, that we were to try to make this calculation using feet as the figure for the distance from home plate to the center-field fence? We would then obtain some value in undefined units; call them fubars (fb):
fubars = feet/(meters per second)
fb = ft/(m/s) = ft × s/m
Feet do not cancel out meters. Thus we have invented a new unit, the fubar, that is equivalent to a foot-second per meter. This unit is basically useless, as would be our numerical answer. (As an aside, fubar is an acronym for “fouled up beyond all recognition.”)
Always remember to be consistent with units when making calculations! When in doubt, reduce all the “givens” in a problem to SI units before starting to make calculations. You will then be certain to get an answer in derived SI units and not fubars or gobbledygooks or any other nonsensical things.
Gravitational Acceleration At Sea Level
The term acceleration can be somewhat confusing to the uninitiated when it is used in reference to gravitation. Isn’t gravity just a force that pulls on things? The answer to this question is both yes and no.
Obviously, gravitation pulls things downward toward the center of the Earth. If you were on another planet, you would find gravitation there, too, but it would not pull on you with the same amount of force. If you weigh 150 pounds here on Earth, for example, you would weigh only about 56 pounds on Mars. (Your mass, 68 kilograms, would be the same on Mars as on Earth.) Physicists measure the intensity of a gravitational field according to the rate at which an object accelerates when it is dropped in a vacuum so that there is no atmospheric resistance. On the surface of the Earth, this rate of acceleration is approximately 9.8067 meters per second per second, or 9.8067 m/s ^{2} . This means that if you drop something, say, a brick, from a great height, it will be falling at a speed of 9.8067 m/s after 1 s, (9.8067 × 2) m/s after 2 s, (9.8067 × 3) m/s after 3 s, and so on. The speed becomes 9.8067 m/s greater with every second of time that passes. On Mars, this rate of increase would be less. On Jupiter, if Jupiter had a definable surface, it would be more. On the surface of a hugely dense object such as a neutron star, it would be many times more than it is on the surface of the Earth.
The rate of gravitational acceleration does not depend on the mass of the object being “pulled on” by gravity. You might think that heavier objects fall faster than slower ones. This is sometimes true in a practical sense if you drop, say, a Ping-Pong ball next to a golf ball. However, the reason the golf ball falls faster is that its greater density lets it overcome air resistance more effectively than the Ping-Pong ball can. If both were dropped in a vacuum, they would fall at the same speed. Astronomer and physicist Galileo Galilei is said to have proven this fact several centuries ago by dropping two heavy objects, one more massive than the other, from the Leaning Tower of Pisa in Italy. He let go of the objects at the same time, and they hit the ground at the same time. This upset people who believed that heavier objects fall faster than lighter ones. Galileo appeared to have shown that an ancient law of physics, which had become ingrained as an article of religious faith, was false. People had a word for folks of that sort: heretic . In those days, being branded as a heretic was like being accused of a felony.
Practice problems of these concepts can be found at: Units And Constants Practice Test
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From Physics Demystified: A Self-Teaching Guide. Copyright © 2002 by The McGraw-Hill Companies, Inc. All Rights Reserved.