Chapter 15 Oscillations Periodic motion Periodic harmonic motion

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

Chapter 15 Problem 5 A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2. 00 cm, and the frequency is 1. 50 Hz. (a) Show that the position of the particle is given by x = (2. 00 cm) sin (3. 00 π t). Determine (b) the maximum speed and the earliest time (t > 0) at which the particle has this speed, (c) the maximum acceleration and the earliest time (t > 0) at which the particle has this acceleration, and (d) the total distance traveled between t = 0 and t = 1. 00 s.

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:

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

Energy in simple harmonic motion

Chapter 15 Problem 17 A 50. 0 -g object connected to a spring with a force constant of 35. 0 N/m oscillates on a horizontal, frictionless surface with an amplitude of 4. 00 cm. Find (a) the total energy of the system and (b) the speed of the object when the position is 1. 00 cm. Find (c) the kinetic energy and (d) the potential energy when the position is 3. 00 cm.

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 27 A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length R. That is,

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

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:

Questions?

Answers to the even-numbered problems Chapter 15 Problem 2 (a) 4. 33 cm (b) − 5. 00 cm/s (c) − 17. 3 cm/s 2 (d) 3. 14 s; 5. 00 cm

Answers to the even-numbered problems Chapter 15 Problem 16 (a)0. 153 J (b) 0. 784 m/s (c) 17. 5 m/s 2

Answers to the even-numbered problems Chapter 15 Problem 26 1. 42 s; 0. 499 m