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# Midterm 2 Study Guide PHYS 132

Cal Poly

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This 9 page Study Guide was uploaded by Caroline Hodes on Tuesday February 16, 2016. The Study Guide belongs to PHYS 132 at California Polytechnic State University San Luis Obispo taught by Fariha Nasir in Winter 2016. Since its upload, it has received 17 views. For similar materials see General Physics II in Physics 2 at California Polytechnic State University San Luis Obispo.

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Date Created: 02/16/16

Useful Equations for Physics (Part 2) Chapters 14, 20, 21, and 22 C HAPTER 14 Frequency and Period: 1 ???? = ???? F is the frequency, or the number of cycles per second. 1 Hz = 1 cycle per second = 1 s ???? = 1 ???? T is the period, or the time to complete one full second. Kinematic of Simple Harmonic Motion 2???????? ???? ???? = ????cos( ) ???? ???? ???? = ????cos 2???????????? ) ???? ???? = ????cos ???????? ) x(t) indicated that the position x is a function of time t. A is the amplitude, or the object’s maximum displacement from equilibrium. Cosine function is always in radians. In this context, ω is the angular frequency. 2???????? ???????????????? = = 2???????????? = ???????? ???? Vmaxis the maximum speed and thus is a positive number. The Phase Constant ????(????) = ????cos ???????? + ???? 0) ???????????? = −????????sin ???????? + ???? 0) = −???? ????????????sin ???????? + ???? 0) Uniform circular motion projected onto one dimension is simple harmonic motion. Cycles and oscillations are not true units. Different values of the phase constant correspond to different starting points on the circle and thus to different initial conditions. The Dynamics of Simple Harmonic Motion 2 ????????= −???? ???? The acceleration is proportional to the negative of the displacement. (???? ) = −???????????? ???????????? ???? The negative sign indicates that the spring force is a restoring force, a force that always points back toward the equilibrium position. ???? ???? ???? ????????= = − ???? ????????2 ???? This is the equation of motion for a mass on a string. This is a second-order differential equation. ???? ???? = 2???????? = √ ???? The theory of oscillation, based on Hooke’s law for a spring and Newton’s second law, is in good agreement with the experimental observations. Vertical Oscillations ???????? ???????? = ???? This is the distance the spring stretches when the block is attached to it. The vertical oscillations of a mass on a spring are the same simple harmonic motion as those of a block on a horizontal string. The Pendulum ???? ???? = 2???????? = √ ???? The angular frequency is determined by the length of the string. The frequency, and hence the period, is independent of the mass. A restoring force that is directly proportional to the displacement from equilibrium is called a linear restoring force. Any system with a linear restoring force will undergo simple harmonic motion around the equilibrium position. Damped Oscillations An oscillation that runs down and stops is called a damped oscillation. ???? = −???????? The damping constant b depends in a complicated way on the shape of the object and on the viscosity of the air or other medium in which the particle moves. ???? ???? = ???????? −???????? ????????cos ???????? + 0) ???? ????2 ????2 ???? = √ − = √ ????0− ???? 4???? 2 4???? 2 ???? ???? = ???? The oscillator’s mechanical energy decays exponentially with time constant τ. C HAPTER 20 The Wave Model ???????? ???????????????????????????? √ ???? ???? ???? = ???? μ is the string’s linear density. The wave speed is a property of the medium. One-Dimensional Waves A graph that shows the wave’s displacement as a function of position at a single instant of time is called a snapshot graph. A graph that shows the wave’s displacement as a function of time at a single position in space is called a history graph. ???? ????,???? = the displacement at time t of a particle at position x Sinusoidal Waves A wave source that oscillates with a simple harmonic motion (SHM) generates a sinusoidal wave. The frequency f of the wave is the frequency of the oscillating source. The amplitude A of the wave is the maximum value of the displacement. The distance spanned by one cycle of the motion is called the wavelength (symbolized by λ) of the wave. During a time interval of one period T, each crest of a sinusoidal wave travels forward of a distance of exactly one wavelength λ. ???? = ???????????????????????????????? = ???? ???????????????? ???? ???? = ???????? A wave moves forward a distance of one wavelength during a time interval of one period. Waves in Two and Three Dimensions The phase difference between two points on a wave depends on only the ratio of their separation Δx to the wavelength λ. The phase difference between two adjacent wave fronts is ???????? = 2???? rad. Sound and Light ???????????????????????? = 343 ????/???? (sound speed in air at 20˚C) ???? = ???? = 299,792,458 ????/???? ????????????ℎ???? (electromagnetic wave speed in vacuum) ???? = ???????????????????? ???????? ????????????ℎ???? ???????? ???? ???????????????????????? = ???? ???????????????????? ???????? ????????????ℎ???? ???????? ????ℎ???? ???????????????????????????????? ???? The speed of light in a material is characterized by the material’s index of refraction n. The frequency of a wave is the frequency of the source. It does not change as the wave moves from one medium to another. Power, Intensity, and Decibels The power of a wave is the rate, in joules per second, at which the wave transfers energy. ???? ???????????????????????????? ???? = = 2 ???? 4???????? The intensity of a wave is proportional to the square of its amplitude. The Doppler Effect After a wave crest leaves the source, its motion is governed by the properties of the medium. ???? = ????0 + 1 − ???????????? (approaching source) ???? ????−= 0 1 − ???????????? (receding source) The situation is different when the source is stationary and the observer is moving. ????+= 1 + ???? ????0???? ) 0 (observer approaching a source) ????−= 1 − ???? ????0???? ) 0 (observer receding from a source) C HAPTER 21 The Principle of Superposition When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacement due to each individual wave. Standing Waves Standing waves happen when crests and troughs “stand in place” as the wave oscillates. Nodes are points on the standing-wave pattern that never move. They are spaced λ/2 apart. The points of maximum amplitude are called antinodes. The intensity is maximum at points of constructive interference and zero at points of destructive interference. ???? ????,???? = ???? ???? cos???????? ???? ???? = 2???? sin???????? Standing Waves on a String The amplitude of a wave from a boundary is unchanged. The wavelength and frequency of a reflected sinusoidal wave are unchanged. A boundary condition is a mathematical statement of any constraining that must be obeyed at the boundary or edge of a medium. 2???? ????????= ???? A standing wave can exist only if its wavelength is one of the values. ???? ???? ???? ????????= = = ???? ???????? 2???? ???? 2???? Standing Sound Waves and Musical Acoustics Open-Open or Closed-Closed Tube Open-Closed Tube 2???? 4???? ????????= ???? ???? { ???? { ???? ???? = ???? ???? = ???????? ???? = ???? ???? = ???????? ???? 2???? 1 ???? 4???? 1 m = 1, 2, 3, 4, … m =1, 3, 5, 7, … Interference in One Dimension Maximum constructive interference: ???????? = 2???? ???????? + ???????? = ???? ∙ 2???? rad ???? 0 m = 0, 1, 2, 3, … Two identical sources produce maximum constructive interference when the path- length difference is an integer number of wavelengths. Perfect destructive interference: ???????? 1 ???????? = 2???? + ???????? 0 (???? + ) ∙ 2???? rad ???? 2 m = 0, 1, 2, 3, … The Mathematics of Interference A light wave that reflects from a boundary at which the index of refraction increases has a phase shift of π rad. Interference in Two and Three Dimensions Maximum constructive interference: Perfect destructive interference: ???????? = 2???? ???????? + ???????? = ???? ∙ 2???? ???????? = 2???? ????????+ ???????? = (???? + ) ∙ 2???? ???? 0 ???? 0 2 C HAPTER 22 The Interference of Light Young’s Double Slit Experiment There are two slits of light and two waves overlap and interfere, resulting in a pattern of light and dark bands on the screen. Light spreads out behind each slit. Alternating bright and dark bands of light due to constructive and destructive interference are called interference fringes. The brightest fringe, at the midpoint of the viewing screen, with m = 0, is called the central maximum. th The m bright fringe occurs where the wave from one slit travels m wavelengths farther than the wave from the other slit and thus ???????? = ????????. The m = 1 fringes occur at points on the screen where the light from one slit travels exactly one wavelength farther than the light from the other slit. The interference pattern is a series of equally spaced bright lines on the screen. The dark fringes are located exactly halfway between the bright fringes. Angles of Bright Fringes Positions of Bright Fringes ???? ???????????? ???????? = ???? ???? ????????= ???? m = 0, 1, 2, 3, … m = 0, 1, 2, 3, ... Interference Pattern at Position Y 2 ???????? ????????????????????????????= 4????1cos ( ????) ???????? The Diffraction of Grating N light waves, from N different slits, will are in phase with each other when they arrive at a point in the screen at a????gle ???? such that ???????????????????? = ???????? (m = 0, 1, 2, 3, …). The position???????? (position of bright fringes) of the m maxim????m is ???? = ???????????????????? . Single-Slit Diffraction Huygens’s Principle 1. Each point on a wave front is the source of a spherical wavelet that spreads out at wave speed. 2. At a later time, the shape of the wave front is the line tangent to all the wavelets. Every point on the wave front can be paired with another point distance a/2 away. Position of Dark Fringes Single Slit Angles of Dark Fringes ???? ???? = ???????????? ???? = 2???????? ????????= ???? ???? ???? ???? ???? p = 1, 2, 3, … p = 1, 2, 3, … The width of the central maximum is twice the spacing λL/a between the dark fringes on either side. The smaller the opening you squeeze a wave through, the more it spreads out on the other side. Circular-Aperture Diffraction 2.44???????? ???? = 2????1= 2????tan???? ≈1 ???? The factor that determines how much a wave spreads out behind an opening is the ratio λ/a, the size of the wavelength compared to the size of the opening. Interferometers The number of bright-dark-bright fringe shifts as mir2or M moves distan2e ∆???? is ∆???? = ∆???? 2 ????/2

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