SECTION 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves 591 In Problems 47–52, the function d models the distance (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds). The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at = t 0? (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? 47. π ( ) ( ) = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − d t e t 20 cos 2 5 0.49 1600 t 0.7 40 2 48. π ( ) ( ) = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − d t e t 20 cos 2 5 0.64 1600 t 0.8 40 2 49. π ( ) ( ) = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − d t e t 30 cos 2 7 0.36 6400 t 0.6 80 2 50. π( ) ( ) = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − d t e t 30 cos 2 0.25 4900 t 0.5 70 2 51. π( ) ( ) = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − d t e t 15 cos 3 0.81 900 t 0.9 30 2 52. π ( ) ( ) = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − d t e t 10 cos 2 3 0.64 2500 t 0.8 50 2 53. Loudspeaker A loudspeaker diaphragm is oscillating in simple harmonic motion described by the function ω ( ) ( ) = d t a t cos with a frequency of 520 hertz (cycles per second) and a maximum displacement of 0.80 millimeter. Find ω and then find a function that describes the movement of the diaphragm. 54. Orlando Eye Opened in 2015 at Icon Park on International Drive, the Orlando Eye is a giant Ferris wheel that provides scenic views of the surrounding area. Its diameter is 390 feet; it takes 18 minutes to complete one rotation; and the bottom of the wheel is 10 feet above the ground. Find a function that gives a rider’s height h above the ground at time t. Assume the passenger begins the ride at the bottom of the wheel. 55. Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the function ω ( ) ( ) = d t a t sin . If a tuning fork for the note A above middle C on an eventempered scale (A ,4 the tone by which an orchestra tunes itself) has a frequency of 440 hertz (cycles per second), find ω. If the maximum displacement of the end of the tuning fork is 0.01 millimeter, find a function that describes the movement of the tuning fork. Source: David Lapp. Physics of Music and Musical Instruments. Medford, MA: Tufts University, 2003. 56. Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the function ω ( ) ( ) = d t a t sin . If a tuning fork for the note E above middle C on an eventempered scale ( ) E4 has a frequency of approximately 329.63 hertz (cycles per second), find ω. If the maximum displacement of the end of the tuning fork is 0.025 millimeter, find a function that describes the movement of the tuning fork. Source: David Lapp. Physics of Music and Musical Instruments. Medford, MA: Tufts University, 2003. 57. Charging a Capacitor See the figure (top, right). If a charged capacitor is connected to a coil by closing a switch, energy is transferred to the coil and then back to the capacitor in an oscillatory motion. The voltage V (in volts) across the capacitor will gradually diminish to 0 with time t (in seconds). (a) Graph the function relating V and t: π ( ) ( ) = ≤ ≤ − V t e t t cos 0 3 t 3 (b) At what times t does the graph of V touch the graph of = − y e ? t 3 When does the graph of V touch the graph of = − − y e ? t 3 (c) When is the voltage V between −0.4 and 0.4 volt? Capacitor Switch Coil + – 58. The Sawtooth Curve An oscilloscope often displays a sawtooth curve.This curve can be approximated by sinusoidal curves of varying periods and amplitudes. (a) Use a graphing utility to graph the following function, which can be used to approximate the sawtooth curve. f x x x x 1 2 sin 2 1 4 sin 4 0 4 π π ( ) ( ) ( ) = + ≤ ≤ (b) A better approximation to the sawtooth curve is given by π π π ( ) ( ) ( ) ( ) = + + f x x x x 1 2 sin 2 1 4 sin 4 1 8 sin 8 Use a graphing utility to graph this function for ≤ ≤ x 0 4 and compare the result to the graph obtained in part (a). (c) A third and even better approximation to the sawtooth curve is given by π π π π ( ) ( ) ( ) ( ) ( ) = + + + f x x x x x 1 2 sin 2 1 4 sin 4 1 8 sin 8 1 16 sin 16 Use a graphing utility to graph this function for x 0 4 ≤ ≤ and compare the result to the graphs obtained in parts (a) and (b). (d) What do you think the next approximation to the sawtooth curve is? 50mv V1 2B. Gm.V Trig TVline OH1 Obase1
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