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Chapter 11 Elasticity And Periodic Motion

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Goals for Chapter 11 To follow periodic motion to a study of simple harmonic motion. To solve equations of simple harmonic motion. To use the pendulum as a prototypical system undergoing simple harmonic motion.

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PERIODIC MOTION

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Periodic Motion

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F x = -kx Using Newton’s second law, ma x = -kx OR a x = -(k/m)x Note the linear relationship between a x and x

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a x = -(k/m)x

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Amplitude, A Cycle Period, T Frequency, f f = 1/T SI unit : Hertz (Hz) = cycle/s = 1/s Angular frequency, ω ω = 2πf = 2π/T

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Conservation of Mechanical Energy E = (1/2) mv x 2 + (1/2)kx 2 = constant Energy in Simple Harmonic Motion

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When x = ± A, v x = 0. At this point, the energy is entirely potential energy and E = (1/2)kA 2. E = (1/2)kA 2 = (1/2) mv x 2 + (1/2)kx 2 v x = ± k/m A 2 – x 2 We can use this equation to find the magnitude of the velocity for any given position x.

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a)Force cst; b) v max and v min ; c) a max and a min ; d) v and a half-way to center; e) K, U and E in halfway

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Equations of Simple Harmonic Motion

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The relationship between uniform circular motion and simple harmonic motion.

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x = A cos θ x = A cos(ωt) x = A cos [(2π/T )t] SI unit: m ω is the angular frequency Position of the Shadow as a Function of Time

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Velocity in Simple Harmonic Motion v = -Aω sin(ωt) SI unit: m/s Maximum speed of the mass is v max = Aω

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Acceleration in Simple Harmonic Motion a = -Aω 2 cos(ωt) SI unit: m/s 2 Maximum acceleration has a magnitude a max = Aω 2

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T = 2π m / k SI unit: s Period of a Mass on a Spring From equation a x = -Aω 2 cos(ωt) a x = -Aω 2 (maximum acceleration) (1) Also, we know that a x = -kx/m a x = -kA/m (maximum acceleration) (2) From equations (1) and (2) -Aω 2 = -kx/m ω 2 = k/m ω = k/m = 2πf = 2π/T

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The Simple Pendulum

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The restoring force F at each point is the component of force tangent to the circular path at that point: F = -mgsinθ If the angle is small, sinθ is very nearly equal to θ (in radians). F = -mgθ = -mgx/L F = -(mg/L)x The restoring force F is then proportional to the coordinate x for small displacements, and the constant mg/L represents the force constant k.

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CHAPTER 12 MECHANICAL WAVES AND SOUND

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Edvard Munch The scream

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A disturbance that propagates from one place to another is referred to as a wave. Waves propagate with well-defined speeds determined by the properties of the material through which they travel. Waves carry energy.

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In a transverse wave individual particles move at right angles to the direction of wave propagation. In a longitudinal wave individual particles move in the same direction as the wave propagation.

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A wave on a string

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As a wave on a string moves horizontally, all points on the string vibrate in the vertical direction.

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Water waves from a disturbance.

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Wavelength, Frequency, and Speed

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v wave = λ /T λ f = v wave Speed of a wave

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REFLECTIONS AND SUPERPOSITION

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A reflected wave pulse: fixed end

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A reflected wave pulse: free end

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The Principle of supperposition: Whenever two waves overlap, the actual displacement of any point on the string, at any time, is obtained by vector addition of the following two displacements: 1)The displacement the point would have if ONLY the first wave were present 2) The displacement the point would have if ONLY the second wave were present

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Constructive Interference

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Destructive Interference

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Figure 14-22 Interference with Two Sources

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In phase/opposite phase: Two sources are in phase if they both emit crests at the same time. Sources have opposite phase if one emits a crest at the same time other emits a trough. Constructive interference occurs when the path length from the two sources differs by 0, λ, 2λ, 3λ, ……. Destructive interference occurs when the path length from the two sources differs by λ/2, 3λ/2, 5λ/2, …….

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Sound Waves Speed of Sound in Air v = 343 m/s

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The frequency of sound determines its pitch. High-pitched sounds have high frequencies; low-pitched sounds have low frequencies. Human hearing extends from 20 Hz to 20, 000 Hz. Sounds with frequencies above this range are referred to as ultrasonic, while those with frequencies lower than 20 Hz are classified as infrasonic.

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Waves become coherent Depending on the shape and size of the medium transmitting the wave, different standing wave patterns are established as a function of energy.

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Normal modes for a linear resonator The resonator is fixed at both ends. Wave energy increases as you go down the y axis below.

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Fundamental frequencies The fundamental frequency depends on the properties of the resonant medium. The fundamental frequency depends on the properties of the resonant medium. If the resonator is a string, cord, or wire, the standing wave pattern is a function of tension, linear mass density, and length. If the resonator is a string, cord, or wire, the standing wave pattern is a function of tension, linear mass density, and length.

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