Monday, Trigonometry, Period 3 Lesson Overview: Warm Up Go over homework Writing Sinusoidal Functions to Model Simple Harmonic Motion 1
Recap: The past few classes, we ve been talking about simple harmonic motion meaning the motion of a point on an object that vibrates, oscillates, rotates or is moved by a wave motion. A traditional example is a mass attached to the end of a spring, which bobs up and down. The mass on the spring moves with harmonic motion. Other examples are such things as the vibrations of a guitar string, the pistons of an engine or a pendulum swinging back and forth. As it turns out, trigonometric functions are useful for describing this type of motion. And thus, we use sine and cosine functions to represent harmonic motion. A point that moves on a coordinate line is in simple harmonic motion if its distance d from the origin at time t is given by Friendly Reminders: The amplitude of an object is its maximum displacement from the zero point The period is the time it takes to complete one cycle The frequency is the number of cycles per unit of time. The frequency is equal to the reciprocal of the period. 2
Warm Up: 3
Questions on the homework? 4
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2. The Sun Problem The angle of elevation of an object above you is the angle between a horizontal line and the line of sight between you and the object. After the sun rises, its angle of elevation increases rapidly at first, then more slowly, reaching a maximum near noontime. Then the angle decreases until sunset. The next day the phenomenon repeats itself. Assume that when the Sun is up, its angle of elevation E varies sinusoidally with the time of day. Let t be the number of hours that has elapsed since midnight last night. Assume that the amplitude of this sinusoid is 60, and the maximum angle of elevation occurs at 12:45 p.m. Assume that at this time of year the sinusoidal axis is at E = 5. The period is, of course, 24 hours. a. Sketch a graph of this function. b. What is the real world significance of the t intercepts? c. What is the real world significance of the portion of the sinusoid which is below the t axis? d. Predict the angle of elevation at i. 9:27 a.m., ii. 2:30 p.m. e. Predict the time of the sunrise. 6
3. Waterwheel Problem Suppose that the waterwheel (shown in the figure) rotates at 5 revolutions per minute (rpm). You start your stopwatch. Three seconds later, point P on the rim of the wheel is at its greatest height. You are to model the distance d of point P from the surface of the water in terms of the number of seconds t, the stopwatch reads. Assuming that d varies sinusoidally with t, you can sketch a graph. Take note of the observations below. Period: One period of this graph is equivalent to point P coming back to it s original starting position. The period will be 12 seconds, since the waterwheel makes 5 complete revolutions in 60 seconds. Vertical Shift: The sinusoidal axis is 6 units above the t axis, because the center of the waterwheel is 6 feet above the surface of the water. Amplitude: The amplitude is 7 units since the point P goes 7 feet above and 7 feet below the center of the wheel. Phase Shift: The point P was at its highest when the stopwatch read 3 seconds, therefore the phase shift (for a cosine graph) will be 3 units. a. Once the graph has been sketched, you can write the equation of the function. Try to write the equation as a cosine and sine function. b. Use your calculator to verify that your equation is correct. (Adjust your windows accordingly). c. Use your calculator to find out how far point P is from the water after 5.5 seconds. 7
4. Tidal Wave Problem A tsunami (a.k.a. tidal wave) is a fast moving ocean wave caused by an underwater earthquake. The water first goes down from its normal level, then rises an equal distance above its normal level, and finally returns to its normal level. The period is about 15 minutes. Suppose that a tsunami with an amplitude of 10 meters approaches the pier at Honolulu, where the normal depth of the water is 9 meters. First sketch the graph and write the equation, then answer the following questions. a. Assuming that the depth of the water varies sinusoidally with time as the tsunami passes, predict the depth of the water at the following times after the tsunami first reaches the pier: i. 2 minutes ii. 4 minutes iii. 12 minutes b. According to your model, what will the minimum depth of the water be? How do you interpret this answer in terms of what will happen in the real world? c. The crest refers to the highest point of a wave. The wavelength is the distance a crest of the wave travels in one period. It is also equal to the distance between two adjacent crests. If a tsunami travels at 745 miles per hour, what is its wavelength? 8
5. You are at Risser s Beach, N.S. to search for interesting shells. At 2:00 p.m. on June 19, the tide is in (i.e., the water is at its deepest). At that time you find that the depth of the water at the end of the breakwater is 15 meters. At 8:00 p.m. the same day when the tide is out, you find that the depth of the water is 11 meters. Assume that the depth of the water varies sinusoidally with time. a. Derive an equation expressing depth in terms of the number of hours that have elapsed since 12:00 noon on June 19. b. Use your mathematical model to predict the depth of the water at (i) 4:00 p.m. on June 19, (ii) 7:00 a.m. on June 20, (iii) 5:00 p.m. on June 20. c. At what time will the first low tide occur on June 20? d. What is the earliest time on June 20 that the water will be at 12.7 meters deep? 9
Homework 26: Sinusoidal Functions to Model SHM Day4 Finish classwork packet Unit 4 Celebration of Knowledge on Wednesday, October 26th 10