Simple Harmonic Motion

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Emma Chambers
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spring with spring constant k. Its amplitude of oscillation is A. Therefore, its maximum speed is A) A k/m. B) A 2 k/m. C) A m/k. D) A 2 m/k. E) A m/k.

1 Please get your personal iclicker from its pigeonhole on North wall. Simple Harmonic Motion 0 t Position: x = A cos(ω t + φ) Velocity: x t = (ω A) sin(ω t + φ) { max Acceleration: t = (ω2 A) cos(ω t + φ) { a max A block of mass M oscillates on the end of a massless spring with spring constant k. Its amplitude of oscillation is A. Therefore, its maximum speed is A) A k/m. B) A 2 k/m. C) A m/k. D) A 2 m/k. E) A m/k. A block is oscillating in simple harmonic motion. Throughout a complete cycle, A) it has constant speed. B) it has arying amplitude. C) it has arying period. D) it has arying acceleration. E) it has arying mass. Simple Harmonic Motion: Energy Simple Harmonic Motion: Energy 0 x = A (maximum stretch) Potential energy of spring: U = 1 2 kx2 Kinetic energy of cart: K = 1 2 m2 K + U = constant The cart oscillates with constant amplitude since no energy is lost! At the instant of maximum stretch, the cart is at rest: U = 1 2 kx2 and K =0 A K + U = constant = 1 2 ka2

06 meters. B) 0.17 meters. C) 0.24 meters.

2 A 0.25-kilogram block oscillates on the end of a massless spring with spring constant 200 Newtons/meter. If this system has an energy of 6.0 Joules, then the amplitude of its oscillation is Is this a wae? A) 0.06 meters. B) 0.17 meters. C) 0.24 meters. D) 4.9 meters. E) 6.9 meters. Is this a wae? Is this a wae? Is this a wae? Is this a wae?

3 Is this a wae? Soliton wae in water Shallow surface wae in water Shallow surface wae in water Wae = Coupled Oscillators Stadium wae Each object (sphere) is connected to its neighbours (by the springs). Oscillation of one object leads to oscillation of its neighbours. The disturbance propagates along the connected objects a wae! a longitudinal wae

Wae = Coupled Oscillators Each object (sphere) is connected to its neighbours (by the springs). Oscillation of one object leads to oscillation of its neighbours.

4 Wae = Coupled Oscillators Each object (sphere) is connected to its neighbours (by the springs). Oscillation of one object leads to oscillation of its neighbours. The disturbance propagates along the connected objects a wae! Sound Wae = Coupled Molecules Each molecule is connected to its neighbours (by interatomic forces). Oscillation of one molecule leads to oscillation of its neighbours. The disturbance propagates along the pipe a sound wae! a transerse wae Sound waes do not trael through A) solids. Wae Speed Follow one crest as the disturbance propagates. The crest moes rightward at the wae speed. B) liquids. C) gases. D) plasmas. E) a acuum. a longitudinal wae Wae Properties Amplitude A (same as for indiidual oscillators, each in simple harmonic motion) Angular frequency ω and ordinary frequency f = ω/(2π) (same as for indiidual oscillators, each in simple harmonic motion) Wae speed (not the same as the time-dependent elocity of indiidual oscillators) Waelength λ (no such property for indiidual oscillators) λ Wae Speed Wae speed relates the frequency and waelength = f λ and is the same speed for any frequency (or any waelength). Wae speed on a stretched string is determined by the tension force F and the mass per unit length M/L F = M/L snapshot of traelling wae snapshot of traelling wae

A sound wae in air has a frequency ( pitch ) of 600 Hertz. A second sound wae in air has a frequency ( pitch ) of 300 Hertz.

D) double that of the first wae s waelength. E) four times that of the first wae s waelength. A transerse wae is propagating along a -e-r-y long stretched string at speed.

Wae Interference Two separate waes can interfere (add or subtract) to create a net wae. E.g., one pulse moes rightward; the other moes leftward.

Notice that the yellow wae doesn t seem to trael leftward nor rightward. It is a standing wae.

5 A sound wae in air has a frequency ( pitch ) of 600 Hertz. A second sound wae in air has a frequency ( pitch ) of 300 Hertz. The second wae s waelength is A) one-quarter that of the first wae s waelength. B) one-half that of the first wae s waelength. C) the same as the first wae s waelength. D) double that of the first wae s waelength. E) four times that of the first wae s waelength. A transerse wae is propagating along a -e-r-y long stretched string at speed. The period of oscillation of any single point on the string is T. The wae s frequency of oscillation is A) T. B) /T. C) T/. D) 1/T. E) 1/. Wae Interference Two separate waes can interfere (add or subtract) to create a net wae. E.g., one pulse moes rightward; the other moes leftward. Wae Interference Two separate waes can interfere (add or subtract) to create a net wae. The cyan wae traels leftward. The green wae traels rightward. The yellow wae is the superposition. Notice that the yellow wae doesn t seem to trael leftward nor rightward. It is a standing wae. Creating Standing Waes The hand on the left disturbs the stretched spring at four different frequencies in succession. These frequencies were chosen ery carefully to guarantee that there would be an integer multiple of half-waelengths between the two hands: L =(n)( 1 2 λ) λ = 2L ( ) f =(n) n 2L for n =1, 2, 3, 4,... Standing Waes on Stretched String n=1 ( first harmonic or fundamental ) n=2 ( second harmonic or first oertone ) n=3 ( third harmonic or second oertone ) n=4 ( fourth harmonic or third oertone )

6 Standing Waes on Stretched String Stringed Instruments

Physics 207 Lecture 28

Physics 207 Lecture 28 Goals: Lecture 28 Chapter 20 Employ the wae model Visualize wae motion Analyze functions of two ariables Know the properties of sinusoidal waes, including waelength, wae number, phase, and frequency. Work