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Due: Thu Mar 7 2013 MTH 437 Homework 2 Problem 1 (10 points) Spherical coordinates are dened by x = r sin cos y = r sin sin z = r cos 1. Draw a small sketch to interpret the geometry. 2. Compute the kinetic energy T in spherical coordinates T = m2 ??x_ 2 + y_2 + z_2Problem 2 (10 points) Consider the damped harmonic oscillator given by x + x_ + x = 0; x(0) = 0; x_ (0) = 1 1. Redo the calculations in class to nd the analytical solution to the above problem 2. Write a Matlab code that solve the above ODE using a 4-th order Runge-Kutta. Find the maximum error of the numerical solution (compared to the analytical solution) for dt = 0:1 in [0; 8]. Problem 3 (10 points) Free fall in a uniform gravitational eld can be modelled by the following equation for the fall velocity mv_ = 12CDAv2 ?? mg where is the air density, CD the drag coecient and A the cross-sectional area. Verify the following formulas: 1. assuming an object falling from rest, we have v(t) = ??v1 tanhgt v1; v1 = r 2mg CDA 2. and for the vertical position y, we ndy = y0 ?? v21 g ln coshgt v1