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**Fundamentals of Physics**

Chapter 13 Oscillations Oscillations Simple Harmonic Motion Velocity of SHM Acceleration of SHM The Force Law for SHM Energy in SHM An Angular Simple Harmonic Oscillator Pendulums The Simple Pendulum The Physical Pendulum Measuring āgā SHM & Uniform Circular Motion Damped SHM Forced Oscillations & Resonance Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

Oscillations Oscillations - motions that repeat themselves. Oscillation occurs when a system is disturbed from a position of stable equilibrium. Clock pendulums swing Boats bob up and down Guitar strings vibrate Diaphragms in speakers Quartz crystals in watches Air molecules Electrons Etc. Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

Oscillations Oscillations - motions that repeat themselves. Physics 2111, 2008 Fundamentals of Physics

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**Simple Harmonic Motion**

Harmonic Motion - repeats itself at regular intervals (periodic). Frequency - # of oscillations per second 1 oscillation / s = 1 hertz (Hz) Period - time for one complete oscillation (one cycle) T T Physics 2111, 2008 Fundamentals of Physics

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**Simple Harmonic Motion**

Position Time Physics 2111, 2008 Fundamentals of Physics

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**Simple Harmonic Motion**

Angles are in radians. Physics 2111, 2008 Fundamentals of Physics

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**Amplitude, Frequency & Phase**

The frequency of SHM is independent of the amplitude. Physics 2111, 2008 Fundamentals of Physics

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**Velocity & Acceleration of SHM**

The phase of v(t) is shifted Ā¼ period relative to x(t), In SHM, a(t) is proportional to x(t) but opposite in sign. Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

The Force Law for SHM Simple Harmonic Motion is the motion executed by a particle of mass m subject to a force proportional to the displacement of the particle but opposite in sign. Hookeās Law: āLinear Oscillatorā: F ~ - x SimpleHarmonicMotion/HorizSpring.html Physics 2111, 2008 Fundamentals of Physics

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**The Differential Equation that Describes SHM**

Simple Harmonic Motion is the motion executed by a particle of mass m subject to a force proportional to the displacement of the particle but opposite in sign Hookeās Law! Newtonās 2nd Law: The general solution of this differential equation is: Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

What is the frequency? k = 7580 N/m m = kg f = ? Physics 2111, 2008 Fundamentals of Physics

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**xm without m falling off?**

m = 1.0 kg M = 10 kg k = 200 N/m ms = 0.40 Maximum xm without slipping Physics 2111, 2008 Fundamentals of Physics

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**Simple Harmonic Motion**

SimpleHarmonicMotion/HorizSpring.html Physics 2111, 2008 Fundamentals of Physics

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**Vertical Spring Oscillations**

Physics 2111, 2008 Fundamentals of Physics

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**Energy in Simple Harmonic Motion**

Physics 2111, 2008 Fundamentals of Physics

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**Energy in Simple Harmonic Motion**

range of motion turning point turning point Physics 2111, 2008 Fundamentals of Physics

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**Gravitational Pendulum**

Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L. SimpleHarmonicMotion/pendulum Physics 2111, 2008 Fundamentals of Physics

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**Simple Pendulum SHM for small q**

Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L. acceleration ~ - displacement SHM SHM for small q Physics 2111, 2008 Fundamentals of Physics

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**A pendulum leaving a trail of ink:**

Physics 2111, 2008 Fundamentals of Physics

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**acceleration ~ - displacement SHM**

Physical Pendulum A rigid body pivoted about a point other than its center of mass (com). SHM for small q Pivot Point acceleration ~ - displacement SHM Center of Mass quick method to measure g Physics 2111, 2008 Fundamentals of Physics

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**Angular Simple Harmonic Oscillator**

Torsion Pendulum: t ~ q Hookeās Law Spring: Physics 2111, 2008 Fundamentals of Physics

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**Simple Harmonic Motion**

Any Oscillating System: āinertiaā versus āspringinessā Physics 2111, 2008 Fundamentals of Physics

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**SHM & Uniform Circular Motion**

The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM. The execution of uniform circular motion describes SHM. Physics 2111, 2008 Fundamentals of Physics

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**SHM & Uniform Circular Motion**

The reference point Pā moves on a circle of radius xm. The projection of xm on a diameter of the circle executes SHM. radius = xm x(t) UC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations Physics 2111, 2008 Fundamentals of Physics

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**SHM & Uniform Circular Motion**

The reference point Pā moves on a circle of radius xm. The projection of xm on a diameter of the circle executes SHM. x(t) v(t) a(t) radius = xm Physics 2111, 2008 Fundamentals of Physics

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**SHM & Uniform Circular Motion**

The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM. Measurements of the angle between Callisto and Jupiter: Galileo (1610) earth planet Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

Damped SHM SHM in which each oscillation is reduced by an external force. Restoring Force SHM Damping Force In opposite direction to velocity Does negative work Reduces the mechanical energy Physics 2111, 2008 Fundamentals of Physics

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**differential equation**

Damped SHM differential equation Physics 2111, 2008 Fundamentals of Physics

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Damped Oscillations 2nd Order Homogeneous Linear Differential Equation: Eq Solution of Differential Equation: where: b = 0 ļ SHM Physics 2111, 2008 Fundamentals of Physics

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**Damped Oscillations āthe natural frequencyā**

Exponential solution to the DE Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

Auto Shock Absorbers Typical automobile shock absorbers are designed to produce slightly under-damped motion Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

Forced Oscillations Each oscillation is driven by an external force to maintain motion in the presence of damping: wd = driving frequency Physics 2111, 2008 Fundamentals of Physics

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**āthe natural frequencyā**

Forced Oscillations Each oscillation is driven by an external force to maintain motion in the presence of damping. 2nd Order Inhomogeneous Linear Differential Equation: āthe natural frequencyā Physics 2111, 2008 Fundamentals of Physics

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**Forced Oscillations & Resonance**

2nd Order Homogeneous Linear Differential Equation: Steady-State Solution of Differential Equation: where: w = natural frequency wd = driving frequency Physics 2111, 2008 Fundamentals of Physics

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**Forced Oscillations & Resonance**

The natural frequency, w, is the frequency of oscillation when there is no external driving force or damping. less damping more damping w = natural frequency wd = driving frequency When w = wd resonance occurs! Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

Oscillations Physics 2111, 2008 Fundamentals of Physics

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**Fundamentals of Physics**

Resonance Physics 2111, 2008 Fundamentals of Physics

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**Stop the SHM caused by winds on a high-rise building**

400 ton weight mounted on a spring on a high floor of the Citicorp building in New York. The weight is forced to oscillate at the same frequency as the building but 180 degrees out of phase. Physics 2111, 2008 Fundamentals of Physics

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**Forced Oscillations & Resonance**

Mechanical Systems e.g. the forced motion of a mass on a spring Electrical Systems e.g. the charge on a capacitor in an LRC circuit Physics 2111, 2008 Fundamentals of Physics

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