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Unformatted text preview: TA's Name:____________________ Section: ____ Your Name: _________________________________ Physics 2214 Assignment 1 Concepts: oscillations complex numbers complex exponentials traveling waves wave equation Reading: Lecture Notes Lec. 1-2.; Y&F, Vol. 1, Chapter 13 Assignment: Due in homework boxes opposite 125 Clark Hall before 430 pm on Tuesday, Feb. 1. Please turn in this sheet stapled to the top of your work. This first assignment reviews some of the math we’ll use in the first half of the course. 1. Consider the function x(t) = A cos (ωt + φ). (a) Evaluate (i) dx/dt; (ii) d2x/dt2 and (iii) d2x/dt2 / x (b) Show by direct subsitution that x(t) can be a solution to the second order differential equation a d2x/dt2 + cx = 0 where a and c are constants. Show that it is a solution only if ω satisfies ω2 = c/a Explain your reasoning. 2. Consider the complex function z(t) = A ei(ωt + φ) (a) What are the real and imaginary parts of z(t), x(t) = Re [z(t)] and y(t) = Im [z(t)]? (b) Evaluate (i) dz/dt; (ii) d2z/dt2; (iii) dx/dt = d/dt {Re[z]}; (iv) Re [dz/dt]; (v) d2x/dt2 = d2/dt2 {Re[z]}; and (vi) Re [d2z/dt2]. Do the operations of taking the time derivative d/dt and taking the real part Re [ ] commute (i.e., can they be performed in either order and yield the same result?). (c) Consider a second order differential equation of the form a d2x/dt2 + b dx/dt + cx = 0 where a, b and c are constants. Using your result from (b), show that if x(t) = Re[z(t)], then z(t) must satisfy a d2z/dt2 + b dz/dt + cz = 0. (d) Substitute for z(t) in (c). Show by explicit calculation that z(t) is only a solution if ω satisfies - aω2 + ibω + c =0. Physics 2214, Spring 2011 1 Cornell University (e) Solve this equation for ω. Under what conditions is ω real? (f) If the condition of part (e) is not satisfied, then we can write ω = ω0 ± i /τ . Write an expression for x(t) = Re[z(t)] in this case. 3. If x(t) = A exp(-t/τ), show that x2 can be written as x2(t) = B exp (-t/τ’). What is the ratio τ’/τ ? 4. Now consider the differential equation a d2x/dt2 + b dx/dt + cx = F cos ωDt. (a) Substitute x(t) = Re [z(t)] where z(t) = A ei(ωt + φ) to obtain a d2z/dt2 + b dz/dt + cz = B . In this case, a solution requires that ω = ωD. Why? (b) Substitute for z(t) and obtain an expression that must be satisfied by A. 5. (a) Determine the values of A, ω, f, T, and φ from the graph of x(t) = A cos (ωt + φ) shown below. x￿nm￿ 4 2 ￿ 0.5 0.0 0.5 ￿2 1.0 t￿ Μs￿ ￿4 (b) Suppose that, instead of being a constant, φ increased linearly with time, i.e., φ = φ0+ Ct, where C is constant. How would the graph shown in (b) change? Would the curve move in the +t or -t direction? Assume C>0. 6. Consider y(x,t) = A cos (kx - ωt), which is a function of both position x and time t. Here x and t are the independent variables, and you can think of y(x,t) as a vertical displacement (for example of a string) as a function of time and the horizontal coordinate x along the string. (a) Sketch y(t) for x =0. What is the period of the wave in t? (b) Sketch y(x) for t=0. What is the period of the wave in x? (c) What must be the relation between x and t in order that y(x,t) = A? (d) Show that y(x,t) as given above can be a solution to the partial differential equation Physics 2214, Spring 2011 2 Cornell University . What must be the relation between k, ω and v in order for y(x,t) to satisfy this equation? Physics 2214, Spring 2011 3 Cornell University ...
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