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92.450 Homework / 550 Mathematical Modeling
Due June 1
1.
Use common sense, dimensions and special cases to explain why each of the following statements is incorrect. Do not refer to the correct formula. (a) The circumference of an ellipse with semimajor and semiminor axes of lengths + and , is #1 (b) The surface area on the earth (radius V) visible from a satellite at height 2 is given by 1V # 2 . V2 +# , # . #
(c) A cylinder of radius < and height 2 runs through the center of a sphere of radius V and the entire body has mass 7. The moment of inertia of this body about an axis through the center of the sphere and perpendicular to the cylinder is 7V # <2# . (d) The force due to air friction on a sphere of mass 7 and radius < as it moves through the air with speed @ at 7@# 2 altitude 2 is given by the formula . <#
2.
A block of ice is melting. If the entire surface of the block is exposed, then it is reasonable to assume its volume Z decreases at a rate proportional to its surface area W , i.e. .Z 5W .> for some positive constant 5 . (a) Determine the units of 5 . What factors do you think influence 5 ? (b) For a given body, the quantity 5 W Z #$ is called shape the factor. Find the shape factor for a sphere and cube. What does 5 say about the size and shape of a body? What bodies do you think have the smallest 5? The largest 5 ? Find two differently shaped bodies with the same shape factor. (c) Assuming constant 5, use dimensional analysis to argue that the time X for a block of ice of initial volume "$ Z! to melt is given by a formula of the form X Z! 5 J 5 for some dimensionless function J . (d) Use your physical intuition to sketch a qualitative graph of the function J . Be careful with the domain. (e) From (b), W 5Z #$ . Using this in (1) gives the separable equation .Z .> 55Z #$ . Solve this subject to the initial condition to determine X and J...