Download Sample Problems for Final Exam - Physics Computations I | PHY 102 and more Exams Physics in PDF only on Docsity! Sample One Hour Final Exam: PHY102 Due Friday April 18th, 6pm Use Mathematica to solve all parts of the three problems Vectors, Lists and Matrices Problem 1. Consider the circuit shown in the figure. The equations for the currents I1, I2, I3 in a circuit are given by, V1 − I1R1 − I2R2 = 0 V1 − V2 − I1R1 − I3R3 = 0 I1 − I2 − I3 = 0 where V1 and V2 are applied voltages and R1, R2, R3 are resistances. Write these three equations as a matrix equation and use Mathematica to find the currents I1, I2, I3 as a function of the applied voltages and the resistances. Check that your code is correct by setting V2 = 0, R1 = 0, R2 = R3 and solving the equations by hand. What are the currents for the particular case V1 = 1, V2 = 4, R1 = 1, R2 = 10, R3 = 4? Ordinary differential equations and plotting Problem 2. A mass(m = 1kg), spring(k = 1N/m) system hangs verti- cally at equilibrium in Earth’s gravity. It is driven vertically by a periodic force acos(ωt) where a = .25 and it experiences a damping of −b~v, where b = 0.1. Consider a small amplitude initial displacement of 0.25 and ini- tial velocity of zero. Solve the linear differential equation for this problem. (x′′(t) + bx′(t) + x(t) = asin(ωt)). Plot x(t) for ω = 0.1, ω = 1, ω = 10. This is a damped driven oscillator which has a surprisingly rich behavior and is used to model many dynamical systems. Partial differential equations and Plotting Problem 3. Confirm that the travelling soliton ρ(x, t) = 3cSech2[c1/2(x− ct)/2] solves the kdV equation given by, ∂ρ(x, t) ∂t + ∂3ρ(x, t) ∂x3 + ρ(x, t) ∂ρ(x, t) ∂x = 0, provided that c > 0. For c = 1, t = 0 plot ρ(x, t) as a function of x for the x-range −10 < x < 10. What is the velocity of the soliton? Solitons play an important role in studies of non-linear process in physics, engineering and biology. For example solitons in the kdV equation provide a simple model for a Tsunami. 1
