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Chapter 13 Vibrations and Waves

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**Periodic motion Periodic (harmonic) motion – self-repeating motion**

Oscillation – periodic motion in certain direction Period (T) – a time duration of one oscillation Frequency (f) – the number of oscillations per unit time, SI unit of frequency 1/s = Hz (Hertz) Heinrich Hertz ( )

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**Motion of the spring-mass system**

Hooke’s law: The force always acts toward the equilibrium position: restoring force The mass is initially pulled to a distance A and released from rest As the object moves toward the equilibrium position, F and a decrease, but v increases

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**Motion of the spring-mass system**

At x = 0, F and a are zero, but v is a maximum The object’s momentum causes it to overshoot the equilibrium position The force and acceleration start to increase in the opposite direction and velocity decreases The motion momentarily comes to a stop at x = - A

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**Motion of the spring-mass system**

It then accelerates back toward the equilibrium position The motion continues indefinitely The motion of a spring mass system is an example of simple harmonic motion

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**Simple harmonic motion**

Simple harmonic motion – motion that repeats itself and the displacement is a sinusoidal function of time

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Amplitude Amplitude – the magnitude of the maximum displacement (in either direction)

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Phase

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Phase constant

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Angular frequency

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Period

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**Velocity of simple harmonic motion**

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**Acceleration of simple harmonic motion**

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**The force law for simple harmonic motion**

From the Newton’s Second Law: For simple harmonic motion, the force is proportional to the displacement Hooke’s law:

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**Energy in simple harmonic motion**

Potential energy of a spring: Kinetic energy of a mass:

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**Energy in simple harmonic motion**

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**Energy in simple harmonic motion**

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Chapter 13 Problem 11 A simple harmonic oscillator has a total energy E. (a) Determine the kinetic and potential energies when the displacement is one-half the amplitude. (b) For what value of the displacement does the kinetic energy equal the potential energy?

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**Pendulums Simple pendulum: Restoring torque:**

From the Newton’s Second Law: For small angles

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Pendulums Simple pendulum: On the other hand

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Pendulums Simple pendulum:

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Pendulums Physical pendulum:

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Chapter 13 Problem 32 An aluminum clock pendulum having a period of 1.00 s keeps perfect time at 20.0°C. (a) When placed in a room at a temperature of –5.0°C, will it gain time or lose time? (b) How much time will it gain or lose every hour?

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**Simple harmonic motion and uniform circular motion**

Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

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**Simple harmonic motion and uniform circular motion**

Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

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**Simple harmonic motion and uniform circular motion**

Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

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**Simple harmonic motion and uniform circular motion**

Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

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**Damped simple harmonic motion**

Damping force Damping constant

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**Forced oscillations and resonance**

Swinging without outside help – free oscillations Swinging with outside help – forced oscillations If ωd is a frequency of a driving force, then forced oscillations can be described by: Resonance:

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**Forced oscillations and resonance**

Tacoma Narrows Bridge disaster (1940)

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**Wave motion A wave is the motion of a disturbance**

All waves carry energy and momentum

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Types of waves Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.) Electromagnetic – governed by electricity and magnetism equations, may exist without any medium Matter – governed by quantum mechanical equations

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Types of waves Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: Transverse – if the direction of displacement is perpendicular to the direction of propagation Longitudinal – if the direction of displacement is parallel to the direction of propagation

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Types of waves Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: Transverse – if the direction of displacement is perpendicular to the direction of propagation Longitudinal – if the direction of displacement is parallel to the direction of propagation

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**Superposition of waves**

Superposition principle – overlapping waves algebraically add to produce a resultant (net) wave Overlapping solutions of the linear wave equation do not in any way alter the travel of each other

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Sinusoidal waves One of the most characteristic solutions of the linear wave equation is a sinusoidal wave: A – amplitude, φ – phase constant

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Wavelength “Freezing” the solution at t = 0 we obtain a sinusoidal function of x: Wavelength λ – smallest distance (parallel to the direction of wave’s travel) between repetitions of the wave shape

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Wave number On the other hand: Angular wave number: k = 2π / λ

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**Angular frequency Considering motion of the point at x = 0**

we observe a simple harmonic motion (oscillation) : For simple harmonic motion: Angular frequency ω

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Frequency, period Definitions of frequency and period are the same as for the case of rotational motion or simple harmonic motion: Therefore, for the wave velocity

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Wave velocity v is a constant and is determined by the properties of the medium E.g., for a stretched string with linear density μ = m/l under tension T

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Chapter 13 Problem 41 A harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum travels 425 cm along the rope in 10.0 s. What is the wavelength?

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Interference of waves Interference – a phenomenon of combining waves, which follows from the superposition principle Considering two sinusoidal waves of the same amplitude, wavelength, and direction of propagation The resultant wave:

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**Interference of waves If φ = 0 (Fully constructive)**

If φ = π (Fully destructive) If φ = 2π/3 (Intermediate)

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**Reflection of waves at boundaries**

Within media with boundaries, solutions to the wave equation should satisfy boundary conditions. As a results, waves may be reflected from boundaries Hard reflection – a fixed zero value of deformation at the boundary – a reflected wave is inverted Soft reflection – a free value of deformation at the boundary – a reflected wave is not inverted

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Questions?

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**Answers to the even-numbered problems**

Chapter 13 Problem 2 1.1 × 102 N The graph is a straight line passing through the origin with slope equal to k = 1.0 × 103 N/m.

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**Answers to the even-numbered problems**

Chapter 13 Problem 8 575 N/m 46.0 J

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**Answers to the even-numbered problems**

Chapter 13 Problem 12 2.61 m/s 2.38 m/s

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**Answers to the even-numbered problems**

Chapter 13 Problem 16 0.15 J 0.78 m/s 18 m/s2

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