AMATH 383 Spring 2016 Homework 1 Due Thurs Apr 7th in class Show work for full credit! The grader will subtract points for poor presentation. 1. Getting a feel for Fermi problems: Come up with an estimate of the answer for the following problems. For each problem, clearly state the assumptions you make, where/how you obtained estimates for numbers necessary for your calculations, and show the calculations you use to obtain your estimate. Remember, the goal is not to get an accurate answer, just an estimate. While you want your nal estimate to be roughly the correct order of magnitude, showing your thought process here is far more important. a) How many days would it take to walk from Seattle to Chicago, assuming you could walk continuously at roughly the same pace the whole way? (i.e., pretend you don't need to stop to eat, sleep, go to the bathroom, take seles, etc.) After you have obtained your estimate, compare it with the estimate that Google Maps gives. Note that Google Maps's estimate is still the output of a model it's just a far more sophisticated model. b) How many raindrops fell on the UW Campus in 2015? How might you check the accuracy of your estimate? 2. Why doesn't anything have units of sin(cm2 )? (Dimensional analysis) In this problem you will prove the claim I made in class, that the dimensions of a quantity x can only be of power-law form, i.e., [x] = M L T in the M LT class of units. We'll solve a simplied case here, in which we are considering a system with only length units L (e.g., geometry). Extension to the full M LT class (or any system of independent units) is a straightforward extension of the results of this problem. Suppose we want to measure some quantity equal to x in some choice of standard units. Using a dierent choice of units L1 , the numerical value of x in this unit system is X1 = x(L1 ). (L) is an unknown function that we will prove is equal to (L) = [x] = L , for some value of . In some other unit system L2 , the numerical value of x is X2 = x(L2 ). Hence, (L1 ) X1 = X2 (L2 ) Now, we must also be able to convert from units L1 to L2 (or vice versa). We can do so by decreasing the units L1 by a factor of L2 . X1 is then related to X2 by L1 X1 = X2 . L2 Putting everything together, this implies (L1 ) = (L2 ) L1 L2 . (1) This kind of equation is called a functional equation. It relates the value of the unknown function at one set of inputs to the value of the unknown function at a dierent set of inputs. To solve this equation, (a) dierentiate both sides of the equation with respect to L1 . You will need to use the chain rule on the right-hand-side. After you have taken the derivative, set L1 = L2 = L. The quantity (1) should appear in your result; this is just a constant and we can dene it to be (1). (b) The result should be a dierential equation for (L). Solve this dierential equation and hence show that (L) = CL , where C is an arbitrary constant. (c) Use Equation (1) above to show that C = 1 and hence establish the desired result, [x] = L . 1 3. Proving Pythagoras using the Pi theorem Let's prove Pythagoras' theorem using the Buckingham Pi Theorem. As all of you (hopefully!) remember, the Pythagorean theorem relates the lengths of the sides of a right-angled triangle: a2 + b2 = c2 . Consider Fig. 1, below. Figure 1: Right-angled triangle with hypotenuse c and smaller acute angle . The center line divides the triangle into two smaller geometrically similar right-angled triangles with hypotenuses a and b. The area, Ac , of the largest right-angled triangle is determined by its hypotenuse c, and, for deniteness, the smaller of its acute angles . The other lengths and angles can be related to these two quantities, so we must be able to write the area in the form Ac = f (c, ), for some function f (that you do not need to determine!). a) Using the quantities Ac , c, and , Dene dimensionless parameters , 1 , . . . , n , for however many n as necessary (including the possibility that n = 1), and recast Ac = f (c, ) as a relationship between the dimensionless quantities, = c (1 , . . . , n ). The perpendicular to the hypotenuse of the large triangle in Fig. 1 divides it into two geometrically similar right angle triangles, one with hypotenuse b and one with hypotenuse a. b) For each of these triangles, dene dimensionless parameters and write down the relationship between them, similar to the relationship in a). (The relationships for each triangle are separate). c) The areas of the smaller triangles add to give the area of the larger triangle: Ac = Aa + Ab . Plug your dimensionless relationships into this equation for the areas and thus deduce Pythagoras' theorem. (Hint: The key insight here is that the three triangles are geometrically similar i.e., just scaled up versions of each other. What does this imply about the functions a , b , and c in the dimensionless relationships you dened? ) d) Pythagoras' theorem only holds true in \"Euclidean geometry\" geometry on at surfaces. For example, the sides of a triangle plastered to the surface of a globe do not obey Pythagoras' theorem. (The interior angles do not even add up to 180 !). Curved surfaces have an intrinsic length scale that describes the radius of curvature of the surface. Explain how this fact spoils our proof of Pythagoras' theorem for triangles on curved surfaces. 2 4. Scaling laws and espionage (This question is borrowed from Nigel Goldenfeld's course on phase transitions at the University of Illinois. It has been modied slightly.) It is 1947 and you are a spy for superpower R. You notice in Life magazine a series of time lapse photographs of the early stages of the rst test of an atomic bomb, at Trinity, New Mexico. They are reproduced at http://guava.physics. uiuc.edu/~nigel/courses/563/Trinity/ The photographs show the expansion of the shock wave caused by the blast at successive times in ms. Assuming that the motion of the shock is unaected by the presence of the ground, and that the motion is determined only by the energy released in the blast E and the density of the undisturbed air into which the shock is propagating, , derive a scaling law for the radius R of the reball as a function of time t. (Hint: in the MLT class of units, [E] = M L2 T 2 and [] = M L3 .) Extract data from the photographs to test your scaling law and hence deduce the yield of the blast. A scale bar is given on the photographs use this to convert from your measurements of the radius in the photographs to the actual radius. You must test your scaling law by plotting a graph. (Hint: It will be useful to plot log10 (R) versus log10 t). The density of air is = 1.225 kg/m3 . You should assume that all numerical factors are of order 1. This information will not be declassied for another 3 years, so you may reasonably expect promotion and other rewards for your eorts. 3