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MSU PHY 102 - worksheet04

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Worksheet #4 - PHY102 (Spr. 2006)Solving equationsSolving equations in MathematicaLook up how to solve algebraic equations exactly(Solve) and numeri-cally(NSolve). If you have a transcendental equation (e.b. x = sin(x)) youneed to use “FindRoot”.In simple kinematics and simple applications of Newton’s second law,the physics is often described by a second order linear differential equa-tion. This may be solved analytically using DSolve, or numerically usingNDSolve. We shall consider initial value problems in which it is necessaryto specify the initial conditions. In Newton’s second law, this is the initialposition and velocity. An example is: “DSolve[{x00[t] + 0.05x0[t] + x[t] ==1, x0[0] == 0, x[0] == 2}, x[t], t]”. Note the double equals (“==”) occurs inall of the “Solve, DSolve ...” functions. It is Mathematica’s way of expressinga “Truth” statement. Use the Mathematica help index to loop up DSolveand see some other examples.Extracting what you wantThis is a pretty confusing, but essential, part of Mathematica syntax. Thesolutions are given as a list of substitution rules. First you have to choose theelement of the output list that you want. Then you have to correctly use thesubstitution rule. Look at the two examples in the notebook worksheet00.nbof how to do this.1Problems - Due Thursday 9th February 9pmProblem 1.(i) Find and print the real root of the equation:x3+ 2x2+ x = 1 (1)(ii) Plot the two functions, x, and, 2tanh(x), on the same graph (usePlot). Then find and print the largest real root of the equation.x = 2tanh(x) (2)Problem 2.Set up the differential equation for the displacement x(t) of a simple har-monic oscillator with mass m = 1 and angular frequency ω = 2. ProgramMathematica to solve this differential equation (DSolve) to find x(t). Plot itskinetic energy as a function of time, given x(0) = 5, v(0) = x0[t] = 0. Nowadd damping to the equation, in the form 0.05x0(t). Repeat your calculationwith this damping term. Plot over a time which includes at least 10 periodsof the motion. Is this underdamped or underdamped motion. Put your an-swer in a text cellProblem 3.A projectile is thrown (from earth) with initial velocity speed u, at anangle θ to the horizontal.(i) Program Mathematica functions describing its equations of motionalong the x and the y directions as a function of time.(ii) Program Mathematica to “Solve” for the “range” of this projectilemotion using these functions. At what angle to the horizontal is the


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