Department

Physics and AstrophysicsCourse Code

PHYA11H3Professor

allChapter

14Ch 14 Oscillations

Oscillatory motion – a repetitive motion back and forth about an equilibrium position

oSwinging motions and vibrations of all kinds

oAll oscillatory motion is periodic

Oscillators – objects or systems that undergo oscillatory motion

o2 characteristics:

Oscillation takes place about an equilibrium position

Motion is periodic

Oscillations – An object oscillates if the position as a function of time is a periodic function, i.e. it repeats

Ex. Of d-t graphs for oscillating systems [see diagram below]

oOscillations takes place around an equilibrium position – the positive x-axis

oThe motion is periodic. One cycle takes time T

oThe 3rd oscillation is sinusoidal

Period (T) – the time to complete one complete cycle

oThe time in seconds required to complete one full cycle or oscillation

Frequency (f) – number of cycles per second

oNumber of cycles or oscillations completed in one second

oUnits: Hertz (Hz)

o1Hz = 1 cycle per second = 1s-1

oMeasured in s-1 or Hz (Hertz) and read as “cycles per second”.

Frequency Period

103 Hz = 1 kHz 1ms

106 Hz = 1 MHz 1μs

109 Hz = 1 GHz 1ns

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Simple harmonic motion SHM

Simple harmonic motion – sinusoidal oscillation that is the most basic type of all oscillatory motions

Harmonic motion is described by a trigonometric function ... cos or sin

Amplitude (A) – maximum displacement from the equilibrium position

oThe objects position oscillates from x= – A and x=A

oThe distance from the axis to the maximum, not the distance from the minimum to the maximum

The instantaneous velocity is zero at the points where x= ± A

oThese are the turning points in the motion

The maximum speed vmax is reached as the object passes through the equilibrium position at x=0 m

oThe velocity is positive as the object moves to the right but negative as the object moves toward the left

oA mass oscillating on a spring is the prototype of simple harmonic motion

Can we “solve” for the motion of the mass?

What is the position x of the mass at any given time t? What is x(t)?

Kinematics of simple harmonic motion

***The equations mentioned in this section are for , implying that ***

***The particle starts from rest at the point of maximum displacement***

The positions versus time graph is clearly a cosine function with period T

An object’s position can be written as

Where x(t) indicates that the position x is a function of time t

The cosine function must be used in radians mode

Angular frequency

oIf the oscillation is harmonic, we can “convert” the time to angles (radians)

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oAnd then instead of cycles or oscillations per second we talk about radians per second

oThis defines the angular frequency ω

x=R cos ωt y=R sin ωt

The velocity versus time graph is clearly an “upside-down” sine function with the same period T

An object’s velocity can be written as

is the maximum speed and thus a positive number

The minus sign is needed to turn the sine function upside down

Velocity is the time derivative of position

oThe object moves faster if you stretch the spring farther and give the oscillation a larger amplitude

14.2 simple harmonic motion and circular motion

The above equations are for an oscillation in which the object just happened to be at xo=A at t=0

For different initial conditions:

oThe very close connection between simple harmonic motion and circular motion can be used

oUniform circular motion projected onto one dimension is simple harmonic motion

oThe x-component of a particle in uniform circular motion is simple harmonic motion

As φ increases, the particle’s x-component is

When used to describe oscillatory motion, ω is called the angular frequency rather than the angular velocity

The angular frequency of an oscillator has the same numerical value, in rad/s, as the angular velocity of the

corresponding particle in circular motion

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###### Document Summary

Oscillatory motion a repetitive motion back and forth about an equilibrium position: swinging motions and vibrations of all kinds, all oscillatory motion is periodic. Oscillators objects or systems that undergo oscillatory motion: 2 characteristics: Oscillation takes place about an equilibrium position. Oscillations an object oscillates if the position as a function of time is a periodic function, i. e. it repeats. Of d-t graphs for oscillating systems [see diagram below: oscillations takes place around an equilibrium position the positive x-axis, the motion is periodic. One cycle takes time t: the 3rd oscillation is sinusoidal. Period (t) the time to complete one complete cycle: the time in seconds required to complete one full cycle or oscillation. Simple harmonic motion sinusoidal oscillation that is the most basic type of all oscillatory motions. Harmonic motion is described by a trigonometric function cos or sin.

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