There are quite a few fundamental constants. These are things like the speed of light (c) the charge on an electron (e), and the Planck constant (h). These constants are determined with some type of interesting experiment. The first values of these constants were often difficult to find—the speed of light, for example, was calculated by tracking the moons of Jupiter. Of course, now we have much better methods to get a very precise value for the speed of light. We don"t need to resort to moons anymore.
Perhaps the most difficult constant to measure is the gravitational constant (G). This gravitational constant is used to give the value of the force between two objects with mass. It is used in the following gravitational model.
In this expression, the gravitational force depends on both the masses of the two interacting objects as well as the distance between them (the r) in the expression. I apologize for the other weird notation (the "hat" on the r and the other vector stuff)—but that"s the vector expression for the gravitational force. The last point to mention is the value of G. It"s about 6.67 x 10-11 N*m2/kg2. This means that two 1 kilogram masses a distance of 1 meter apart would have a gravitational force of a super tiny value. Gravity is very weak.
But how do you find the value of G? The are multiple methods now, but I want to flash back in time to perhaps the first method of finding this constant—using a mountain. Let me start off with a simpler experiment. Suppose I hold a mass on a string over a perfectly symmetrical Earth. It might look like this (not to scale).
There are two forces on that mass. First, the string pulls up and the gravitational force pulls down (where "down" means "towards the center of the Earth"). These upward and downward forces must have the same magnitude so that the total force is zero and the mass stays at rest. It wouldn"t be too difficult to measure the upward pulling force—you could use a spring scale or something like that. Then this upward pulling force would give you the magnitude of the downward, gravitational force.
Once you have the gravitational force, you just need to know two things (other than the value of the mass in kilograms). You need to know the radius of the Earth and the mass of the Earth. The radius of the Earth isn"t too difficult—the Greeks made a pretty good approximation of its size. Oh, you need the radius of the Earth because this is the value for the "distance" between the two masses in the gravitational force calculations. But what about the mass of the Earth? Yup, no one knew what that was. There"s your problem.
What you really need is some other object for which you know the mass. But it has to be a pretty big object because the force would otherwise be super small and difficult to measure. What about a mountain? Those have large masses. So that"s exactly what they used—a mountain. Here"s how this would work. You once again take a mass and suspend it from a string just like in my previous example. However, you put this mass near a mountain. Now the suspended mass will have two gravitational forces—the gravitational force from the Earth pulling "down" and the gravitational force from the mountain. Here is a diagram to help you picture this.
Since the two gravitational forces from the mountain is sideways (relative to "down"), the force from the string must be diagonal. Now you just need to know the mass and distance to the mountain. Assuming both gravitational forces depend on the same G constant, the tilted angle of the string would give you the relation between the mass of the mountain and the mass of the Earth (the rest of the Earth). Boom. Use that mass of the Earth to then calculate G.
Of course, there are some problems with this method. Let me go over some of them.
How do you find the mass of a mountain?
If this was my job, I would just assume the mountain is a sphere and a assume a constant density. Since I know the volume of a sphere, I could use the density to calculate the mass. Not too difficult. However, there is a big problem—the deflection of a hanging mass would be so tiny that the difference in a spherical calculated mass vs. actual mass would be significant. Honestly, I would still do this calculation. Why? Because it at least allows me to calculate an approximate expected deflection of the mass—so I would have an idea of how precise to build my measurements.
A better way to find the mass of the mountain is to actually measure it. You can get the height with a barometer, but what about the other dimensions? The answer: counter lines. Yes, by mapping lines of constant altitude around the mountain, the mass could be calculated in horizontal slices. It seems that this mountain problem was the source of the rediscovery of counter lines in the 18th century.
But wait! It"s not just the mass of the mountain that you need, it"s also the total gravitational force. Part of the mountain is closer to the hanging mass and will have more effect than parts that are farther away. In essence, you have to do a volume integral over the mountain to find its total gravitational pull.
How do you measure "down"?
Suppose you hang a mass and stand near a super massive mountain—which way does the mass hang? The answer is straight down. Humans define up and down based on the direction of the gravitational field. So, even though a massive mountain would result in a gravitational field that doesn"t point towards the center of the Earth, we wouldn"t be able to tell—at least not with a hanging mass (which we also call a plumb bob).
Instead, there needs to be an alternative method to find "up" and "down." The answer is the stars. By measuring the location of a star vs. its predicted location, you can get a value for up and down based on the stars. Oh, it"s not easy, but you can do it. No one ever said that science was easy.
No comments:
Post a Comment