Phys 3750 Midterm Exam 1 Name____________________ 1 You may use a pen/pencil and a 3"×5", handwritten note card on the exam. Answer all questions as completely as possible; show all of your work. If you run out of space on a problem, continue on the back of that sheet of paper. Good Luck! Q1. (5 points each) Short and Simple. (a) Write the complex number πie3 in the form iyx+. (b) What is special about a normal-mode coordinate ()tQi? (c) A solution to the 1D wave equation is ()()tkxAωsinsin. What kind of wave is this? (d) Describe, in words, what a dispersion relation is. Q2. (10 points) Consider the potential energy function ()()2−+=−qqeeAqV . Find the harmonic approximation to this potential-energy function near 0=q . Identify the effective spring constant k.Phys 3750 Midterm Exam 1 Name____________________ 2 Q3. (10 points) An harmonic oscillator with mass m and spring constant k has the following initial conditions: ()aq =0 , ()00 =q&. Write down the time-dependent motion ()tq of this oscillator. Q4. (10 points) Consider the equation ()()()−+−+=321321121202121000αααqqq. Find 3α in terms of the initial displacements ()0iq . Q5. (5 points) The dispersion relation for the coupled oscillator system can be written as ()= kdk2sin~2ωω. Find the long wavelength limit of this expression and thus identify the parameter 2c in the wave equation.Phys 3750 Midterm Exam 1 Name____________________ 3 Q6. Consider the system illustrated in the following picture. The two objects have different masses, but all three springs are the same. Assume that the objects are constrained to move horizontally. (a) (10 points) Write down the equation of motion for each object. (b) (10 points) Find the characteristic equation that determines the normal-mode frequencies for this system. (Do not solve this equation.) mM k k kPhys 3750 Midterm Exam 1 Name____________________ 4 Q7. (10 points) Find the necessary condition for the function ()()()txAtxqβαcoscos,= to be a solution to the wave equation. Q7. For two coupled oscillators the normal-mode transformation is given by −=111121M . (a) (5 points) Find 2M. (b) (10 points) Thus find 1−M, the inverse of