Chapter 15 Oscillations

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 (1857-1894)

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

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

Phase

Phase constant

Angular frequency

Period

Velocity of simple harmonic motion

Acceleration of simple harmonic motion

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:

Chapter 15 Problem 16

Energy in simple harmonic motion Potential energy of a spring: Kinetic energy of a mass:

Energy in simple harmonic motion

Chapter 15 Problem 37

Pendulums Simple pendulum: Restoring torque: From the Newton’s Second Law: For small angles

Pendulums Simple pendulum: On the other hand

Pendulums Simple pendulum:

Pendulums Physical pendulum:

Chapter 15 Problem 51

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

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

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

Damped simple harmonic motion Damping force Damping constant

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:

Answers to the even-numbered problems Chapter 15: Problem 2 10 N; (b) 1.2 × 102 N/m

Answers to the even-numbered problems Chapter 15: Problem 28 200 N/m; (b) 1.39 kg; (c) 1.91 Hz

Answers to the even-numbered problems Chapter 15: Problem 38 12 s

Answers to the even-numbered problems Chapter 15: Problem 42 0.499 m; (b) 0.940 mJ

Answers to the even-numbered problems Chapter 15: Problem 58 6.0%