Unit 5 Harmonic Motion 4.8 JMerrill, 2007 Learning Goal To describe simple harmonic motion with a trigonometric function. Simple Harmonic Motion SIMPLE HARMONIC MOTION: motions that are periodic in nature such as oscillations of a pendulum, the orbit of a satellite, the vibrations of a tuning fork, etc., which can be described by sine or cosine curves Definition • A point that moves on a coordinate line is said to be in simple harmonic motion if its distance from the origin at time t is given by – d = a sin ωt or d = a cos ωt • where a and ω are real numbers such that ω > 0, with amplitude a, period 2π/ω, and frequency ω/(2π) Example • A ball is bobbing up and down on a spring. 10cm is the maximum distance the ball moves vertically upward or downward from its equilibrium point. It takes the ball t = 4 seconds to move from its maximum displacement above zero to its maximum displacement below zero and back again (assume no friction or resistance) Example • When the spring is at equilibrium, d = 0, when t = 0, so we will use the equation d = a sin ωt. • The maximum displacement from zero is 10 and the period is 4. • Amplitude = 10 2 • Period = 4, so 2 • So, d 10 sin t 2 Frequency Frequency 2 1 2 cycle / sec 2 4 Port Aransas Pier The graph indicates the depth of the water off a certain pier in Port Aransas, Texas, beginning at noon, March 19th. Note: t = hours after noon on March 19th y = depth of water in meters Pier Problem At what time and date is the first high tide? Noon, March 19th The second high tide? 2 am, March 20th The first low tide? 7 pm, March 19th Pier Problem How long does it take the tide to complete one cycle? 14 hours Pier Problem Amplitude = 3 A=3 14 = 2π/B B = π/7 y 3cos t 7 Pier Problem What is the depth of the water at midnight March 19? Since midnight is 12 hours after noon, let t = 12. Plug 12 into the equation for the function. y = 1.870 meters Pier Problem At what time is the depth of the tide 1-meter? Graph y1 = equation t = 2.743 and 11.257 Graph y2 = 1 2:45 pm and 11:15 pm Find intersection(s). Weight on Spring video A weight is at rest hanging from a spring. It is then pulled down 6 cm and released. The weight oscillates up and down, completing one cycle every 3 seconds. Sketch Distance above/below resting point, in cm 6 Time, in seconds 3 -6 Equation Amplitude = 6 6 A= 6 3 = 2π/B B = 2π/3 3 -6 2 y 6sin ( x 1.5) 3 Positions Determine the position of the weight at 1.5 seconds. Let x = 1.5; plug into equation for function. y = 0 cm (back at original position) Use the graph to find the time when y = 3.5 for the first time. Graph y1 = equation you wrote; graph y2 = 3.5. Find intersection. x = 1.797 seconds 3.5 is the 3.5 cm distance above the original position of the weight.
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