Question

Ae-kt sin út or f(t)-Ae-kt oos ωt des crites the position (10 pts) An equation of the form f(t) of an object in damped harmonic motion, with the following characteristics: A is the initial amplitude k is the damping constant -is the period The frequency of the motion is simply the reciprocal of the period, ie, fA common unit for frequency is 2e the hertz (Hz), which represents one cycle per second. Suppose the G-string on a violin is plucked from an initial distance of 0.5 cm above its rest position, then allowed to vibrate. The damping constant k is determined to be 1.4 (depends on the size of the string and the material its made of). The note produced is a pure G (frequency- 200 Hz) a. Write an equation of the form f(t)-Aekt cost that de scribes the motion of the point at which the string was plucked, at time b. Find the vertical position of the string at the following times (round to 6 decimal places, and put the answers to iv-vi in scientific notation. You may use a program such as Microsoft Excel if you wish. visit Math Lab if necessary) t 0.0007 seconds t-0.0025 seconds Lt03 seconds v.t6.286 seconds v. t6.29 seconds vi. t 6.58 seconds C. Note that the answers for () and (i) go from positive to negative. What conclusion can be drawn about the motion of the string between 0.0007 and 0.0025 seconds? d. Find the value of t at which the string first passes back through its initial resting position. [You must set your equation from part (a) equal to 0and solve for . Hint: Look at the Unit Circle. When is the first time that the cosine value equals zero?1 e. Is your answer from part (d) above consistent with your results from part (c)? Explain

Answer 1

(a)

A = 0.5, k = 1.4. w = 2n/ 1256.6

f(t) = 0.5e-1.4t cos(125660

(b)

f(0.0007)0.318410 cm

f (0.0025)--0.498253 cm

f(0.3)0.328503 cnm

f(6.286) = 3.9 × 10-5 cm

f(6.29) 7.3 × 10-5 cm

f(6.58) = 4.8 × 10-5 cm

(c)First time f becomes 0 is obtained by

0.5e 14t cos(1256.6t)0

whose solution is

t = 1.25 × 10-3 sec

Yes it lies between the two values 0.0007 sec and 0.0025