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# Hprec8 4

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• G(t)=3cos2t+1 ; f(t)=3sin(2t-3pi/2)+1
• ### Hprec8 4

1. 1. 8.4: Simple Harmonic Motion & Modeling © 2008 Roy L. Gover(www.mrgover.com) Learning Goals: •Write a sinusoidal function to represent simple harmonic motion. •Find a sinusoidal model and use it to make predictions.
2. 2. Definition The standard forms for sine and cosine functions are: ( ) sin( )f t a bt c d= + + ( ) cos( )g t a bt c d= + + where a,b,c and d are constants.
3. 3. Important Idea In the standard form: ( ) sin( )f t a bt c d= + + ( ) cos( )g t a bt c d= + + •a controls amplitude •b controls period •c controls phase shift •d controls vertical shift Sketchpad
4. 4. Definitions Amplitude= a Period = 2 b π Phase Shift= c b − Vertical Shift = d From Lesson 7.4
5. 5. Write a sine function and a cosine function for the sinusoidal graph. Example 2π− 2π−
6. 6. Write a sine function and a cosine function for the sinusoidal graph. Try This 2π− 2π -7 1
7. 7. Solution ( ) 4sin(2 ) 3f t t= − ( ) 4cos 2 3 2 g t t π  = − − ÷  
8. 8. Definition 2π− 2π− The wave shape of these graphs are called sinusoids and their functions are called sinusoidal functions. 3 ( ) 3sin 2 1 2 f t t π  = − + ÷   Sinusoidal Function Sinusoid
9. 9. Definition ( ) sin( )f t a bt c d= + + Motion that can be described by a function of the form: ( ) cos( )g t a bt c d= + + is called simple harmonic motion. Simple harmonic motion is motion that repeats. or
10. 10. Try This What are examples of harmonic motion?
11. 11. Example A buoy in a lake bobs up and down as waves move past. The buoy moves 6 feet from its high point to its low point every 10 seconds. At t=0 the buoy is at its high point. Write an equation using cosine to describe its motion.
12. 12. Procedure 1.Find the mid-point of the motion. This value is d. 2.A is the distance from the midpoint to the highpoint. 3.Find b and c using period. 4.Substitute using a standard equation, usually cosine. 5.Make predictions using the equation.
13. 13. Example When a pianist plays middle C, the piano string vibrates at a frequency of 264 cycles per second. Write an equation of simple harmonic motion of the string when A is 1mm.
14. 14. Example The population of foxes in a certain forest vary with time. Records started being kept at t=0 when a minimum number of 200 appeared. The next maximum, 800 foxes, occurred at t=5.1 years. Predict the population when t=7,8,9 &10 years.
15. 15. Try This The Coast Guard observes a raft at the bottom of a wave bobbing up and down a total distance of 10 feet.The raft completes a full cycle every 12 seconds. Write an equation describing the motion. Where is the raft after 18 seconds?
16. 16. Solution 5cos 6 t h π  = −  ÷   At 18 sec, h=5 ft.
17. 17. Example a. Find a sinusoidal function to represent the motion of the moving weight. b. Sketch a graph of the function you found in part a. c. What is the height of the weight after 3 sec. d. When will the height of the weight be 6 cm. below the equilibium.
18. 18. Example Suppose that a weight hanging from a spring is set in motion by an upward push. It takes 5 sec. for it to complete 1 cycle of moving from its equilibrium position to 8 cm. above, then dropping to 8 cm. below, and finally returning to its equilibrium position.
19. 19. Lesson Close Harmonic motion problems occur in medicine, economics, science and engineering.