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Unformatted text preview: 18.03 Problem Set 5 Spring 2009 Due in Room 2-106 at 12:55 pm, Friday, March 13 Part A 0 points 1. (Lec 14, Fri Mar 6: Inhomogeneous equations: Particular solutions: Expo- nential response formula) Read: Notes S, O.1, 2, 4 , EP 2.6 2. (Lec 15, Mon Mar 9: Undetermined coefficients) Read: EP 2.5 (pp 148–157); 3. (Lec 16, Wed Mar 11: Frequency response) Read: EP 2.7 Part B 60 points 0. (at due date) Write the names of any person, web site, or materials you consulted. Write “No C” (no consultation) on your paper if you consulted no outside materials/people. 1. Lec 13 Wed Mar 4; Conservation and dissipation of energy (14 pts = 2 + 2 + 4 + 1 + 2 + 3) Consider the equation mx 00 + bx + kx = 0 a) Show that the general solution to the undamped equation ( b = 0, m > 0, k > 0) has the form x = A cos( ω n t- φ ) and find ω n , the natural frequency. b) The kinetic energy is given by E K = 1 2 mv 2 = 1 2 m ( x ) 2 For the undamped system, find the kinetic energy of the mass starting at x (0) = x and x (0) = 0 at the time the solution reaches the equilibrium x = 0. (Answer: 1 2 kx 2 ) We have just shown that if we stretch a spring so that the mass is at position x and then release the mass at zero velocity, then when the mass returns to equilibrium it will have kinetic energy equal to 1 2 kx 2 . For this reason we define this as the potential energy of a mass at position x . Equivalently, this number 1 2 kx 2 is the work we would have to perform to move the mass from the equilibrium position to the position x . In general, the potential energy of the mass at position x is defined to be E P = 1 2 kx 2 c) Show that the sum of kinetic and potential energy E = E K + E P = 1 2 m ( x ) 2 + 1 2 kx 2 is constant in time for any solution to the undamped equation. (Don’t plug in a general solution, although this works. Instead, just show that E = 0.) This is known as conservation 1 of energy. Go to the applet Damped Vibrations (Log Zoom) and try various values of m > 0, k >...
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Spring '09 term at MIT.
- Spring '09