1

Oscillations

Time variations that repeat themselves at regular intervals - periodic or cyclic behaviour

Examples: Pendulum (simple); heart (more complicated)

Terminology:

Period: time for one cycle of motion [s]

Frequency: number of cycles per second [s-1 = hertz (Hz)]

LECTURE 2

CP Ch 14

How can you determine the mass of a single E-coli bacterium or a DNA molecule ? CP458

2

Signal from ECG

period T

Period: time for one cycle of motion [s]Frequency: number of cycles per second [s-1 = Hz hertz]

CP445

1 kHz = 103 Hz 106 Hz = 1 MHz 1GHz = 109 Hz

3

Example: oscillating starsB

righ

tnes

s

Time CP445

4

Simple harmonic motion SHM

• object displaced, then released

• objects oscillates about equilibrium position

• motion is periodic

• displacement is a sinusoidal function of time (harmonic)

T = period = duration of one cycle of motion

f = frequency = # cycles per second

• restoring force always acts towards equilibrium position

x

eFspringrestoring force

eF k x

CP447

x = 0

+X

5

+ xmax- xmax

origin 0equilibrium position

displacementx [m]

velocityv [m.s-1]

accelerationa [m.s-2]

ForceFe [N]

CP447

Vertical hung spring: gravity determines the equilibrium position – does not affect restoring force for displacements from equilibrium position – mass oscillates vertically with SHM

Fe = - k y

Motion problems – need a frame of reference

6

= d/dt = 2 / T

One cycle: period T [s]Cycles in 1 s: frequency f [Hz] angular frequency -

T = 1 / ff = 1 / T = 2 f=2 / T

amplitude A

A

CP453

Connection SHM – uniform circular motion

[rad.s-1]

7

SHM & circular motion

max max maxcos 2 cos 2 cost

x x x f t x tT

t

uniform circular motionradius A, angular frequency

Displacement is sinusoidal function of time

x component is SHM:

max

cosx A t

x A

22 f

T

CP453

v

xA xmax

8

Simple harmonic motion

time

disp

lace

men

t

Displacement is a sinusoidal function of time

T

x xmax cos 2 t

T

xmax cos 2ft xmax cos t

amplitude

xmax

By how much does phase change over one period? CP451

TT

9

Simple harmonic motion

equation of motion m a k x

substitute1

22

k k mf T

m m k

(restoring force)

oscillation frequency and period

max

max

2max

2

cos

sin

cos

x x t

v x t

a x t

a x

CP457

xmax A

10

a 2x

2 2 2 2max

2 2 2max

2 2 2max

2 2max

sin

(1 cos )

( )

( )

v x t

x t

x x

v x x

acceleration is rad (180) out of phase with displacement

At extremes of oscillations, v = 0

When passing through equilibrium, v is a maximum

CP457

Simple harmonic motion

11

0 10 20 30 40 50 60 70 80 90 100-10

0

10SHM

posi

tion

x

0 10 20 30 40 50 60 70 80 90 100-5

0

5

velo

city

v

0 10 20 30 40 50 60 70 80 90 100-1

0

1

acce

lera

tion

a

time t

How do you describe the phase relationships between displacement, velocity and acceleration? CP459

12

Problem 2.1

If a body oscillates in SHM according to the equation

where each term is in SI units. What are(a) the amplitude(b) the frequency and period(c) the initial phase at t = 0?(d) the displacement at t = 2.0 s?

5.0cos(0.40 0.10) mx t

13

Problem 2.2

An object is hung from a light vertical helical spring that subsequently stretches 20 mm. The body is then displaced and set into SHM. Determine the frequency at which it oscillates.

14

Answers to problems

2.1 5.0 m 0.064 Hz 16 s 0.10 rad 3.1 m

2.2 = 22 rad.s-1 f = 3.5 Hz