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Chapter 15 Oscillations
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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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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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Chapter 15 Problem 16
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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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Chapter 15 Problem 37
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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 15 Problem 51
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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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Answers to the even-numbered problems
Chapter 15: Problem 2 10 N; (b) 1.2 × 102 N/m
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Answers to the even-numbered problems
Chapter 15: Problem 28 200 N/m; (b) 1.39 kg; (c) 1.91 Hz
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Answers to the even-numbered problems
Chapter 15: Problem 38 12 s
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Answers to the even-numbered problems
Chapter 15: Problem 42 0.499 m; (b) mJ
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Answers to the even-numbered problems
Chapter 15: Problem 58 6.0%
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