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Chapter 13 Periodic Motion
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Special Case: Simple Harmonic Motion (SHM)
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Only valid for small oscillation amplitude But SHM approximates a wide class of periodic motion, from vibrating atoms to vibrating tuning forks...
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Starting Model for SHM: mass m attached to a spring Demonstration
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Simple Harmonic Motion (SHM) x = displacement of mass m from equilibrium Choose coordinate x so that x = 0 is the equilibrium position If we displace the mass m, a restoring force F acts on m to return it to equilibrium (x=0)
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Simple Harmonic Motion (SHM) By ‘SHM’ we mean Hooke’s Law holds: for small displacement x (from equilibrium), F = – k x ma = – k x negative sign: F is a ‘restoring’ force (a and x have opposite directions) Demonstration: spring with force meter
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What is x(t) for SHM? We’ll explore this using two methods The ‘reference circle’: x(t) = projection of certain circular motion A little math: Solve Hooke’s Law
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The ‘Reference Circle’ P = mass on spring: x(t) Q = point on reference circle P = projection of Q onto the screen
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The ‘Reference Circle’ P = mass on spring: x(t) Q = point on reference circle A = amplitude of x(t) (motion of P) A = radius of reference circle (motion of Q)
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The ‘Reference Circle’ P = mass on spring: x(t) Q = point on reference circle f = oscillation frequency of P = 1/T (cycles/sec) = angular speed of Q = 2 /T (radians/sec) = 2 f
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What is x(t) for SHM? P = projection of Q onto screen. We conclude the motion of P is: See additional notes or Fig. 13-4 for
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Alternative: A Little Math Solve Hooke’s Law: Find a basic solution: Solve for x(t)
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v = dx/dt v = 0 at x = A |v| = max at x = 0 a = dv/dt |a| = max at x = A a = 0 at x = 0 See notes on x(t), v(t), a(t)
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Show expression for
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going from 1 to 3, increase one of A, m, k (a) change A : same T (b) larger m : larger T (c) larger k : shorter T Do demonstrations illustrating (a), (b), (c)
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Summary of SHM for an oscillator of mass m A = amplitude of motion, = ‘phase angle’ A, can be found from the values of x and dx/dt at (say) t = 0
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Energy in SHM As the body oscillates, E is continuously transformed from K to U and back again See notes on v max
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E = K + U = constant Do Exercise 13-17
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Summary of SHM x = displacement from equilibrium (x = 0) T = period of oscillation definitions of x and depend on the SHM
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Different Types of SHM horizontal (have been discussing so far) vertical (will see: acts like horizontal) swinging (pendulum) twisting (torsion pendulum) radial (example: atomic vibrations)
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Horizontal SHM
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Now show: a vertical spring acts the same, if we define x properly.
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Vertical SHM Show SHM occurs with x defined as shown Do Exercise 13-25
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‘Swinging’ SHM: Simple Pendulum Derive for small x Do Pendulum Demonstrations
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‘Swinging’ SHM: Physical Pendulum Derive for small Do Exercises 13-39, 13-38
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Angular SHM: Torsion Pendulum (fiber-disk)
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Application: Cavendish experiment (measures gravitational constant G). The fiber twists when blue masses gravitate toward red masses
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Angular SHM: Torsion Pendulum (coil-wheel) Derive for small
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Radial SHM: Atomic Vibrations Show SHM results for small x (where r = R 0 +x)
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Announcements Homework Sets 1 and 2 (Ch. 10 and 11): returned at front Homework Set 5 (Ch. 14): available at front, or on course webpages Recent changes to classweb access: see HW 5 sheet at front, or course webpages
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Damped Simple Harmonic Motion See transparency on damped block-spring
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SHM: Ideal vs. Damped Ideal SHM: We have only treated the restoring force: F restoring = – kx More realistic SHM: We should add some ‘damping’ force: F damping = – bv Demonstration of damped block-spring
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Damping Force this is the simplest model: damping force proportional to velocity b = ‘damping constant’ (characterizes strength of damping)
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SHM: Ideal vs. Damped In ideal SHM, oscillator energy is constant: E = K + U, dE/dt = 0 In damped SHM, the oscillator’s energy decreases with time: E(t) = K + U, dE/dt < 0
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Energy Dissipation in Damped SHM Rate of energy loss due to damping:
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What is x(t) for damped SHM? We get a new equation of motion for x(t): We won’t solve it, just present the solutions.
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Three Classes of Damping, b small (‘underdamping’) intermediate (‘critical’ damping) large (‘overdamping’)
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‘underdamped’ SHM
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‘underdamped’ SHM: damped oscillation, frequency ´
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‘underdamping’ vs. no damping underdamping: no damping (b=0):
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‘critical damping’: decay to x = 0, no oscillation can also view this ‘critical’ value of b as resulting from oscillation ‘disappearing’: See sketch of x(t) for critical damping
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‘overdamping’: slower decay to x = 0, no oscillation See sketch of x(t) for overdamping
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Application Shock absorbers: want critically damped (no oscillations) not overdamped (would have a slow response time)
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Forced Oscillations (Forced SHM)
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Forced SHM We have considered the presence of a ‘damping’ force acting on an oscillator: F damping = – bv Now consider applying an external force: F driving = F max cos d t
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Forced SHM Every simple harmonic oscillator has a natural oscillation frequency ( if undamped, ´ if underdamped) By appling F driving = F max cos d t we force the oscillator to oscillate at the frequency d (can be anything, not necessarily or ´)
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What is x(t) for forced SHM? We get a new equation of motion for x(t): We won’t solve it, just present the solution.
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x(t) for Forced SHM If you solve the differential equation, you find the solution (at late times, t >> 2m/b)
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Amplitude A( d ) Shown (for = 0): A( d ) for different b larger b: smaller A max Resonance: A max occurs at R, near the natural frequency, = (k/m) 1/2 Do Resonance Demonstrations
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Resonance Frequency ( R ) A max occurs at d = R (where dA/d d =0):
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natural, underdamped, forced: > ´ > R natural frequency: underdamped frequency: resonance frequency:
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Introduction to LRC Circuits (Electromagnetic Oscillations) See transparency on LRC circuit
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Electric Quantity Counterpart charge Q(t) x(t) current I = dQ/dt v = dx/dt (moving charge) (generates a magnetic field, B)
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Electrical Concepts electric charge: Q current (moving charge): I = dQ/dt resistance (Q collides with atoms): R voltage (pushes Q through wire): V = RI
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Voltage (moves charges) resistance R causes charge Q to lose energy: V = RI (voltage = potential energy per unit charge) C and L also cause energy (voltage) changes
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Circuit Element (Voltage) R = resistanceV R = RI (Q collides with atoms) C = capacitanceV C = Q/C (capacity to store Q on plate) L = inductanceV L = L(dI/dt) (inertia towards changes in I)
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Change in Voltage = Change in Energy voltage = potential energy per unit charge recall, around a closed loop:
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Which looks like:
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Circuit Element Counterpart 1/C = 1/capacitance k L = inductance m R = resistance b (Extra Credit: Exercise 31-35) Use this table to write our damped SHM as damped electromagnetic oscillations
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In the LRC circuit, Q(t) acts just like x(t)! underdamped, critically damped, overdamped
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Driven (and resonance): V driving = V max cos d t
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