Question

mperial Valley College PROJECT #3 You may work in groups of up to 4 students. Each group turns in one homework, wnitten on separate paper e,aat in tiny writing on this sheetl with llwri and all stens shoan doack All students in each group recee the same grade This assignment is due ot the begrring 덱 dass on"huidey July 27 (day of Find Exon This project is worth o total f 40points Homework will be graded not only on correctness, but also on how dlearly and coherently it is written up, so that any reader may easily follow it. Unstapled homework wll lose an automatic 5 points. Homework written in tiny writing on this sheet will not be graded Recommendation: Start as soon as pos sible. Do not procrastinate. 1. (6 points) If a person bends at the waist with a straight back, making an angle of 8 degrees with the horizontal, then the force F exerted on the back muscles can be modeled by the equation 0.6W sin(θ + 90") sin 12 where W is the weight of the person in pounds. a Calculate the force for 8 25° for a 145-lb. person (round to the nearest tenth The unit for force here is pounds) b) Use an identity to show that F is approximately equal to 2.9W cose c) For what value of θ is F maximized? Explain. Mour explanation should involve the Unit Circle and/or the range of trig functions) 2. (8 pts.) An airplane flying faster than the speed of sound sends out sound waves that form a cone [see p. 237 in 10h ed. or p. 247 in 9h ed. for a picture. The cone intersects the ground to form a hyperbola. As this hyperbola passes over a particular point on the ground, a sonic boom is heard at that point. If θ is the angle at the vertex of the oone, then 2 m where m is the Mach number for the speed of the plane (we assume that m1. The Mach number is the ratio of the speed of the plane to the speed of sound. [So, a speed of Mach 1.4 means that the plane is flying at 1.4times the speed of sound]. In a-b below, find the Mach number with the given angle; inc - d, find the angle 8 to the nearest tenth of a degree with the given Mach number 2a θ=43° a) (10 pts) An equation of the form f(t). Ae-kt sin út or f(t) . Ae-kt cos ωt describes the position 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, , A common unit for frequency is = 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 it's 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, attime . 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 ii, t 0.0025 seconds Lt03 seconds v.t 6.286 seconds v. t 6.29 seconds vi. t 6.58 seconds c. Note that the answers for () and () 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 4. (6 pts.) The formula for the up and down motion of a weight on a spring is given by (3 s(t) a sin where the constant k is unique to each spring and can be determined experimentally, s(t) is in feet, and m is the mass of the weight. [Note that this is the fomula s(t)a sin ot from Problem 3 above, except without the damping component e, and withE Suppose the spring's constant is k-6, and a mass of m-4 units is attached to the end of it. The spring is stretched and allowed to oscillate until it comes to rest a if the spring is stretched of a foot and then released fie, afind the amplitude, period, and frequency of the resulting oscillatory motion (be sure to rationalize any denominators). b. What is the equation of the motion? c. With the same spring (therefore same constant k 6), what mass must be used to produe a period of 1 second? [Leave your answer in exact form-no decimal approximations] 5. 5 pts.) Three atoms are arranged as shown in the figure on the right Atom A has radius 2.8 nm, atom B has radius 3.3 nm, and atom C has radius 3.7 nm (nanometers). The angle formed at the center A of atom Cis 37. Find the distance between the centers of atoms A and C [Round to the nearest tenth of a nanometer.] [For similar examples, see Section 7.1 Problems 31, 32 in 9 ed.] NOTE: The triangle formed is not a right triangle.] 37 6 S pts.) The bond angle for the F-0-F bond in OF is 1032 If the average length of an O-F bond is 1.418 x 10-* cm, Oxygen 1032 find the distance D between the centers of the fluorine atoms. Express your answer in scientific notation, rounded to the nearest hundredth.] Fluorine Fluorine

Answer 1

Sin 12° は6=2.5。 87xI1s) (12) o. 2&

, F when is 16-0 2. (a) 2 2- (21 . S。) m 2.728

2. (b) 3 M2 Sin(2。*) a2.124 → e) = 2 (56.44. ) 12.8 e-을 ) 4G.ST.