One last estimate of the geometry of the whole Earth, and its implications for the composition of the planet.

Transcription

1 One last estimate of the geometry of the whole Earth, and its implications for the composition of the planet. Unless otherwise noted the artwork and photographs in this slide show are original and by Burt Carter. Permission is granted to use them for non-commercial, non-profit educational purposes provided that credit is given for their origin. Permission is not granted for any commercial or for-profit use, including use at for-profit educational facilities. Other copyrighted material is used under the fair use clause of the copyright law of the United States.

2 Remember that the current estimate for the Earth s meridional (through the poles) circumference is 40,008km, close to what we estimated. The equatorial circumference is 40,074km. The average circumference is approximately 40,060km. We will use this value. The circumference of a circle (or sphere) is its diameter x, so Earth s diameter is ~12,752km. The radius of a circle (or sphere) is half the diameter, so Earth s radius is 6,376km.

3 The volume of a sphere is 4/3 r 3. Our goal in this exercise is to estimate Earth s density, which is usually expressed as g/cm3. Therefore we need to convert the km to cm. This makes for doing arithmetic with very big numbers now, but makes our final step MUCH easier. 1km = 1000m, and 1m = 100cm, so our conversion requires us to multiply the radius in km by (1000x100) This gives us a radius of about x 10 8 cm Plugging this value into the sphere-volume equation above tells us that Earth s volume is about x cm 3. This gives us the volume we need for a density calculation, but we still need a mass a number of grams.

4 Obviously there are some options that are not open to us. Isaac Newton realized that he could determine the mass of Earth by seeing how it interacted gravitationally with other bodies. This is presented on the next 2 pages if you are interested, but the math is a bit more involved than what we ve done before, so if you prefer to skip it, then just take professor Newton s word for it. Xg

5 Newton didn t really discover gravity. The first guy to drop a big rock on his toe discovered gravity. What Newton discovered was a couple of very important details about how gravity works. At least at big scales. The first is what we call the gravitational constant (usually signified by capital G). This is a number that enters into every calculation of the gravitational attraction between objects and lets us calculate what the strength of their attraction is. The current estimate of the value of that constant is: G = [(6.673x10-11 m 3 )/(kg*s 2 )]. The force of a gravitational attraction (F) is [(G x the mass of one object x the mass of the other object)/(the distance between them 2 ), or F=[G(m1*m2)]/d 2. Newton also had the great insight that gravity is an acceleration of objects. Things fall with an ever-increasing speed, at least in a vacuum. It was well known even before Newton that a force generated during motion is equal to the mass of the moving object x its acceleration. The acceleration of gravity is usually represented by a small g, and mass by a small m. On Earth, g = ~9.8m/s 2. So the force of gravity affecting a ball, for example, as it falls to Earth is F=mg. Notice that we now have two equations for the value of F. If we substitute mg for F in the first equation, we get: mg = [G(m1*m2)]/d 2. Either m1 or m2 is the mass of our hypothetical ball [call it m (b) ], the other is the mass of the earth [call it m (e) ] to which the ball is falling. So if we re-label, we get m (b) g = [G(m (b) *m (e) )]/d 2. We are interested in the value of m (e), so we can rearrange and get m (e) =[(m (b) gr 2 )/(G m (b) )]. Cancelling we find (to our relief) that the other mass (m (b) ) cancels and we get: m (e) = gr 2 /G. We already know what g and G are. Newton took this distance to be between the centers of mass of the two objects. This is not perfectly accurate and we will see how to use that inaccuracy in the second geology course, but for now this will give us a good enough value for the distance we need. Remember that the distance between a falling ball (just before it hits the ground) and the center of the Earth is, on average, about 6,376km the radius of the Earth.

6 Newton s work lets us derive a formula for the mass of the Earth, if we know a few things. First, we need Newton s gravitational constant: G = [(6.673x10-11 m 3 )/(kg*s 2 )]. Second, we need to know the local force of gravity, which is g = ~9.8m/s 2 Finally we need to know the radius of Earth, which we ve already calculated to be ~6,376km (which we must convert to meters to match the units in the other values). 6,376km = 6,376,000m or x 10 6 m. The equation we need is: m (e) = gr 2 /G. (m (e) is read mass of earth.) Substituting values for variables gives us: m (e) = [(9.8m/s 2 )*(6.376 x 10 6 m) 2 ]/[(6.673x10-11 m 3 )/(kg*s 2 )] = [(9.8m/s 2 )* x m 2 *(kg*s 2 )]/ (6.673x10-11 m 3 ) (Notice that we can cancel the s 2 terms. Soon we ll get rid of all the m terms too.) = ( m 3 kg)/ (6.673x10-11 m 3 ) (As promised, we now have m 3 terms in numerator and denominator, and can cancel them.) = ~5.97 x kg. This is about what you ll find if you look it up in a book.

7 If you are just rejoining us, the short version of Newton s insight is that he can mathematically determine the mass of earth to be about 5.97 x kg. Density is a material s mass divided by a unit volume, generally expressed as g/cm 3 so we need to multiply by 1000 to convert kg to g. This is easy with scientific notation: the Earth s mass is 5.97 x g. Now recall that we determined a few pages back that the volume of Earth is about x cm 3. Notice that both the mass and the volume have tacked onto the end. Great news -- these will cancel! Earth s overall density is therefore roughly : (5.97 x g)/( x cm 3 ) = ~5.5 g/cm 3, a value you can check in an astronomy book. Now we have a problem. What is it? (HINT: What is Egypt made of?)

8 Egypt is made, for the most part, of granite, sedimentary rocks (sandstone, limestone, and mudstone), loose sediment (sand and mud), and a little basalt. GRANITE: SANDSTONE: LIMESTONE: BASALT: Very hard Very hard and brittle Soft Very hard Light color Light color Light color Dark color Coarse Crystals Medium Crystals Crystals variable Very Fine Crystals Uncommon or inaccessible. Common Common in most areas. Rare. Moderately dense: ~ g/cc Lower density: ~ g/cc Lower density: ~ g/cc Very dense: ~3.0 g/cc Used in Egypt for construction/decoration of important buildings and statues. Used in Egypt for construction when limestone was not available. Used in Egypt for construction of buildings and for statues. Used in Egypt for decoration of important buildings and for statues. Density of Whole Earth = ~5.5 g/cm 3. Absolutely nothing in Egypt is as dense as that!

9 How can the Earth be denser than any of the rocks we know?