#40 a-f B-A. (B+A ". Beats slation Recall the identity cos A-cos Be2-2A)sin(-2A) a. Show that 0-10,a, . 9 and (ii)o_10,us2toverify the identity. In which case do you see Gaph the functions on...
B-A. (B+A ". Beats slation Recall the identity cos A-cos Be2-2A)sin(-2A) a. Show that 0-10,a, . 9 and (ii)o_10,us2toverify the identity. In which case do you see Gaph the functions on both sides of the equation in part (a) with (i) beats? b. 40 Analysis of the forced damped oscillation equation Consider the equation my"+ey'+ky Fo cos wof, which oscillator. Assume all the parameters in the equation are positive. a. Explain why the solutions of the homogeneous equation decay in time. b. Show that the particular solution is yp -A sinwt + B cos ot, where coFo and B- and tan e. Using the amplitade phase form of a solution, show that y, Asinr B cs utC'sin ), where whe Fo Show that C.- d. PTE e. What is the relationship between the forcing frequepcy v and the natural frequency wo that produces the largest amplitude C"7 Explain why this result is analogous to resonance in the case of forced undamped motion. Let m-c-F0-1 and co-3. Graph the amplitude C. as a function ofu. Describe how C. varies with respect to ω. t. 41. Impedance Use the result of Exercise 40d and wirite the amplitude in terms of circuit parameters; that is, replace m by Le by R by C and Fo by o Express the amplitude of the steady-state current as 100% Similarly (15) and (16) come from (6) and (7)
B-A. (B+A ". Beats slation Recall the identity cos A-cos Be2-2A)sin(-2A) a. Show that 0-10,a, . 9 and (ii)o_10,us2toverify the identity. In which case do you see Gaph the functions on both sides of the equation in part (a) with (i) beats? b. 40 Analysis of the forced damped oscillation equation Consider the equation my"+ey'+ky Fo cos wof, which oscillator. Assume all the parameters in the equation are positive. a. Explain why the solutions of the homogeneous equation decay in time. b. Show that the particular solution is yp -A sinwt + B cos ot, where coFo and B- and tan e. Using the amplitade phase form of a solution, show that y, Asinr B cs utC'sin ), where whe Fo Show that C.- d. PTE e. What is the relationship between the forcing frequepcy v and the natural frequency wo that produces the largest amplitude C"7 Explain why this result is analogous to resonance in the case of forced undamped motion. Let m-c-F0-1 and co-3. Graph the amplitude C. as a function ofu. Describe how C. varies with respect to ω. t. 41. Impedance Use the result of Exercise 40d and wirite the amplitude in terms of circuit parameters; that is, replace m by Le by R by C and Fo by o Express the amplitude of the steady-state current as 100% Similarly (15) and (16) come from (6) and (7)