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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Ch 15: Oscillations: Simple harmonic motion sections 1-4 Snapshots at equal time intervals. Vectors show speed which is max at x=0, minimum at x = +-xm If we define t=0 when x = xm then it returns at t=T to x=xm, i.e. the period or time to complete one cycle is T. The frequency, f, is the number of complete oscillations per second. T = 1/f SI unit of f is hertz, Hz = s-1 Ch15-1/18 periodic/harmonic motion Any motion which repeats itself at regular intervals. We are interested in a particular form: x(t): the displacement at time t (m) xm: the amplitude (m) (positive, maximum displacement) ω: the angular frequency (s-1) t: time (second) φ : phase constant or phase angle (radian) ωt + φ: phase This is simple harmonic motion. Ch15-2/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 1
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Angular frequency We know x(t) must return to the same value after T (and simplifying by taking φ = 0): The cosine function repeats when 2π is added to argument: SI unit of angular frequency is is the radian per second. φ is in radians Ch15-3/18 Examples xm’ (red) > xm T’ (red) = T/2 ω’ = 2 ω φ = -π/4 for red This is effect of a phase difference. There is a phase difference of 2π for 1 period T Ch15-4/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 2
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Velocity of SHM vm is velocity amplitude Ch15-5/18 Acceleration of SHM In SHM, acceleration is proportional to displacement but opposite in sign and related by ω2 am is acceleration amplitude Ch15-6/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 3
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons The force law for SHM From Newton’s second law, knowing acceleration => force known: This is a restoring force proportional to the displacement, i.e., Hooke’s law for a spring SHM is executed by a particle subject to a restoring force proportional to displacement of particle. Ch15-7/18 Linear simple harmonic oscillator Block-spring system: block mass m, frictionless surface. No force when x = 0. Once pushed or pulled away from x=0, it moves in SHM. It is a linear simple harmonic oscillator (linear oscillator). Linear since F proportional to x, not x2 etc. Using gives the angular frequency of motion as: and period as: Ch15-8/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 4
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Question Which of the following Force relationships implies simple harmonic oscillation? (a) F = -5 x (b) F = -400x2 (c) F = 10x (d) F = 3x2 Ch15-9/18 Question Which of the following Force relationships implies simple harmonic oscillation? (a) F = -5 x (b) F = -400x2 (c) F = 10x (d) F = 3x2 Ans: (a) since F is proportional to the displacement AND is restorative. To identify SHM, either a = -(some +ve constant) x or F= -(some +ve constant) x Ch15-10/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 5
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Question Which of a, b or c is equivalent to ? Ch15-11/18 Question Which of a, b or c is equivalent to ? Ans: (a) just because it is, by definition Ch15-12/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 6
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Sample problem 15-2 For a block-spring system, given x(0) = -8.50 cm, v(0) = - 0.920 m/s and a(0) = +47.0 m/s2. Determine angular frequency, phase constant, amplitude. Key ideas: Write down: from which we get: Also Ch15-13/18 Sample problem 15-2 There are two solutions: which will lead to two values of xm: Which is correct? Ch15-14/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 7
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Sample problem 15-2 There are two solutions: which will lead to two values of xm: Which is correct? Ans: The 155o solution since the amplitude must be positive. Ch15-15/18 Energy in simple harmonic motion The potential energy of a linear oscillator is associated with the spring – how much it is compressed or extended. From work with Hooke’s law we know so that here we have The kinetic energy is associated entirely with the block (we assume a massless spring) so that: substitute using and hence: Ch15-16/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 8
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.Rogers: Lectures based on Halliday, Resnick and Walker’sFundamentals of Physics, Copyright 2005 by Wiley and Sons Energy in simple harmonic motion (cont) So total mechanical energy in system is: Oscillating systems store potential energy in their springiness and kinetic energy in their inertia. Ch15-17/18 Energy in simple harmonic motion (cont) E as a function of time: Note both U and K peak twice per period T (sin2, cos2) E as a function of displacement: Ch15-18/18Material, including many figures, is used with permission of John Wiley and Sons, Inc.Material is not to be further distributed in any format and is subject to CopyrightProtection. 9
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