Newton's law of gravitation says that the gravitational force F exerted on one object (call it m1) by another object (call it m2) is given by the equation:
F = Gm1m2/(d2)
The force is equal to the gravitational constant G multiplied by the product of the two objects' masses and then divided by the square of the distance between the two objects.
When we measure distances in meters and masses in kilograms, then the force F turns out to be measured in units called Newtons. What is a Newton? If we have a force of one Newton, and apply it to an object that masses 1 kg for one second, then, if the object were stationary at the start of the second, at the end of the second it would have a velocity of 1 m/s--one meter per second--in the direction the force pushed it. (In more familiar units, 9.8 Newtons = 2.2 pounds: 4.55 Newtons = 1 pound; 0.224 pounds = 1 Newton.)
Through many careful and ingenious experiments, scientists have determined the value of the gravitational constant G. In this metric system of measurement, G = 6.67 x 10-11.
This gravitational force can be very weak. Consider the gravitational force exerted on a 1 kg object by another 1 kg object located 1 meter away is. Using Newton's graviational equation and substituting in "1" for all the masses an distances, we get:
F = G x 1 x 1 / (12) = G = 6.67 x 10-11 m kg/(s2)
How strong is this force? If we apply such a force to a one-kilogram object for one second, it will give the object a speed of 6.67 x 10-11 m/s. Recall that there are 86,400 seconds in a day, and 365 days in a year. Multiplying, we see that a speed of 6.67 x 10-11 m/s is a speed of 0.002 meters/year--or two millimeters a year.
Suppose we take an 88 pound boy--a boy that masses 40 kg, and weighs 392 Newtons. By "weighs 392 Newtons," we mean that 392 Newtons is the strength of the earth's gravitational pull on the boy. How large is the earth to exert such a pull?
We know that the boy stands 6,400 km from the center of the earth, that the earth is roughly a sphere, and that when we calculate the gravitational attraction by a spherical mass we can do our calculations as if all the mass of the sphere were concentrated at its center. (This is one of the many things that Sir Isaac Newton discovered).
So we take our equation:
F = Gm1m2/(d2)
and we substitute in the things that we know:
So our equation, after substitution, is:
392 = (6.67 x 10-11)(40)(m2)/((6.4 x 106)2)
Let's use the commutative law to move the thing we don't know--the mass of the earth, m2--our in front of the rest of the right-hand-side:
392 = (m2)(6.67 x 10-11)(40)/((6.4 x 106)2)
Let's multiply together the (6.67 x 10-11) and the (40):
392 = (m2)(2.67 x 10-9)/((6.4 x 106)2)
Let's square the distance in the denominator:
392 = (m2)(2.67 x 10-9)/(4.10 x 1013)
Let's do the division:
392 = (m2)(6.51 x 10-23)
And now if we divide both sides by (6.51 x 10-23), we get m2 by itself on the right-hand-side and we have our answer:
6.02 x 1024
6,020,000,000,000,000,000,000,000 kilograms is the mass of the earth. That's 6,020,000,000,000,000,000,000 tons. If we remember that there are six billion people alive on the earth, that is about 1,000,000,000,000,000--one quadrillion--tons of earth-stuff per person.Posted by DeLong at December 12, 2002 11:09 AM | Trackback