Chapter 15 Oscillations

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1 Chapter 5 Oscillations Any motion or event that repeats itself at reular intervals is said to be periodic. Oscillation: n eneral, an oscillation is a periodic fluctuation in the value of a physical quantity above and below central or equilibrium value. Examples: Mechanical and non-mechanical oscillations. Galileo probably made the first qualitative observations of oscillations, which led to the property of isochronism. We confine our attention on simple harmonic oscillation. Damped and forced oscillations are treated as supplementary material. 5. Simple Harmonic Oscillation The displacement from equilibrium is iven by x( Asinωt where A is the amplitude and ω, measured in rad/s, is called the anular frequency, rather than anular velocity. What is the difference between frequency and anular frequency?

2 Simple Harmonic Oscillation A complete form of simple harmonic oscillation: x( Asin( ωt + Φ) The arument ωt+φ is called the phase, while φ is called the phase constant (or phase anle), measured in radians. π ω πf T 3 Simple Harmonic Oscillation () x( Asin( ωt + Φ) A simple harmonic oscillator has the followin characteristics:. Simple: the amplitude is constant.. sochronism: the period is independent of amplitude. 3. Harmonic: The time dependence of the fluctuatin quantity can be expressed in terms of a sinusoidal function of a sinle frequency. d x + ω x This differential equation characterizes all types of simple harmonic oscillation. 4

3 Example 5.: The position of a particle movin alon the x-axis is iven by x.8 sin(t+.3) m, where t is in seconds. (a) What are the amplitude and period of the motion? (b) Determine the position, velocity, and acceleration at t.6 s. Solution: (a) The amplitude A is.8 m and the period T is π/.5 s. (b) x( Asin( ωt v( Aω cos( ωt a( Aω sin( ωt 5 5. The Block-Sprin System F sp kx Fsp k a x m m d x k a x m d x k + x m This differential equation is merely another way of writin Newton s second law. k π ω T π m, ω m k 6

4 Example 5.: A -k block is attached to a sprin for which k N/m. t is held at an extension of 5 cm and then release at t. Find: (a) the displacement as a function of time; (b) the velocity when x+a/; (c) the acceleration when x+a/. Solution: (a) k N/m, m k, ω rad/s, Tπ/5 s, A.5 m and φπ/ (obtained from initial condition). x(.5sin(t + π / ) (b) and (c) x( A/ sin( ωt /, cos( ωt ± v(.5 ( ± a(.5 / ) two velocities, why? 7 Example 5.4: Show that a block hanin from a vertical sprin, as shwon in Fi. 5.7, executes simple harmonic motion. Solution: mkx o, Fm-kx-k(x-x o )-kx, where x xxo is the displacement from the equilibrium position. Since the restorin force is linearly proportional to the displacement from equilibrium, the motion will be simple harmonic, Gravitation force plays what role in this case? 8

U K E kx mv U + K ka ka ka sin mω A cos ( ωt cos ( ωt ( ωt The total enery of any simple harmonic oscillator is

5 5.3 Enery in Simple Harmonic Motion Since the force exerted by an ideal sprin is conservative, the enery of the block-sprin system is constant. U K E kx mv U + K ka ka ka sin mω A cos ( ωt cos ( ωt ( ωt The total enery of any simple harmonic oscillator is constant and proportional to the square of amplitude. 9 Enery in Simple Harmonic Motion All SHM is characterized by a parabolic potential well. The variation of the kinetic, potential, and total enery as a function of time.

6 Example 5.4: vertical block-sprin system Show that a block hanin from a vertical sprin, as shwon in Fi. 5.7, executes simple harmonic motion. Solution: U U K sp since E U mv x kx kx + U k( x mx m( x sp + mx& m / k + K mx& + x ) + x ) 5.4 Pendulums A simple pendulium is an idealized system in which a point mass is suspended at the end of a massless strin. Newton s second law applied alon this directions is: d s m sinθ m ts physical meanin is that the component of the weiht acts as a restorin force. For small anle, sin. Substitutin this toether with sl into above equation to find d θ + θ L

7 Pendulums () d θ + ω T θ L L π / ω π θ θ sin( ωt o Note here that the anular frequency ω should not be confused with the instantaneous anular velocity d /. L 3 The rotational form of d θ + T md md The Physical Pendulum d θ md sinθ Small- anle approximation, sinθ θ ω π / ω θ π md Newton's second law, 4

8 The Torsional Pendulum The restorin torque obeys Hooke's law, d θ κθ d θ κ + θ κ ω T π / ω π κ 5 Exercises and Problems Ch.5: Ex. 4, Prob. 3, 4, 5, 6,, 3, 4 6

Chapter 14. PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman. Lectures by Wayne Anderson

Chapter 14. PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman. Lectures by Wayne Anderson Chapter 14 Periodic Motion PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 14 To describe oscillations in