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MassachusettsInstituteofTechnologyOpenCourseWare

8.03SC Fall2012

ProblemSet#2

Problem2.1Drivenoscillatorwithdamping

An

object

of

mass

0.2

kg

is

hung

from

a

spring

whose

spring

constant

is

80

N/m.

The

body

is

subjecttoaresistiveforcegivenbybv,wherev isitsvelocity(m/sec)andb=4Nm1 sec.

a) Setupthedifferentialequationofmotionforfreeoscillationsofthesystem,andfindtheperiod

ofsuchoscillations.

b) Theobject issubjectedtoasinusoidal forcegivenbyF(t) =F0sin(t),whereF0 =2Nand

=30sec1. Inthesteadystate,whatistheamplitudeoftheforcedoscillation?

Insteadofadrivingforce(see(b)),wenowoscillatetheendofthespringatthetopverticallywith

a

harmonic

displacementX=X0sin(t).

c)

Set

up

the

differential

equation

of

motion

for

this

driven

oscillator.

d) What istheamplitudeofthemass insteadystatefor=0,30and300sec1? X0 = 0.5cm

inallcases.

Problem2.2(French4-6)1 Seismograph

ImagineasimpleseismographconsistingofamassM hung

from a spring on a rigid frame attached to the earth (see

figure). Thespringforceandthedampingforcedependon

thedisplacementandvelocityrelativetotheearthssurface,

but the relevant acceleration (Newtons 2nd law) of M is

relative

to

the

fixed

stars.

a) UsingytodenotethedisplacementofM relativetotheearthandtodenotethedisplacement

oftheearthssurfacerelativetothefixedstars,showthattheequationofmotionisd2y dy d2

+ +0

2y=dt2 dt dt2

b) Solvefory(steady-state)if=Ccos(t).

c) SketchagraphoftheamplitudeAofthedisplacementy asa functionof (supposingC the

sameforall).

d) A typical long-period seismometer has a period of about 30 sec and a Q of about 2. As the

result

of

a

violent

earthquake

the

earths

surface

may

oscillate

with

a

period

of

about

20

min

andwithanamplitudesuchthatthemaximumacceleration isabout109 m/sec2. Howsmall

avalueofAmustbeobservableifthisistobedetected?

1ThenotationFrench indicateswherethisproblem is located inoneof the textbooksused for8.03 in2004:

French,

A.

P.

Vibrations

and

Waves.

The

M.I.T.

Introductory

Physics

Series.

Cambridge,

MA:

Massachusetts

Institute

of

Technology,

1971.

ISBN-10:

0393099369;

ISBN-13:

9780393099362.

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e) Makesureyouappreciatethebigdifferencebetweenthisproblemandproblem2.1. Compare

the

amplitudes

of

the

two

problems

at

very

low

and

very

high

frequencies.

Problem

2.3

(French

4-10)

Power

dissipation

Thepowerinputtomaintainforcedvibrationscanbecalculatedbyrecognizingthatthispoweris

themeanrateofdoingworkagainsttheresistiveforcebv.

a)

Satisfy

yourself

that

the

instantaneous

rate

of

doing

work

against

this

force

is

equal

to

bv2.

b) Usingx=Acos(t),showthatthemeanrateofdoingworkisb2A2/2.

c) Substitute the value of A at any arbitrary frequency and hence obtain the expression for P.

TheresultwillbethesameasgiveninEq.(4-23)onpage98inFrench.

Problem

2.4

Transient

behavior

Considerthesimpledampedspring-masssystemshown in

the first figure. The mass is driven by an external force

givenbyF(t) =F0cos(t+).

The

mass

is

at

rest

at

its

equilibrium

position,

x

=

0,

when

the force is turned on instantaneously at t = 0. The re

sponse of the mass to this driving force in shown in the

second figure. Assuming that the mass is m = 1 kg, use

thetimeseriesforx(t)togetestimates (within20%)for:

a) The natural frequency of the

undamped oscillator, 0/(2)

inHz. Hint: Youmayassume

that

is

small,

so

that

22/40.1 0

b) The damping coefficient, b in

Ns/m.

c) The frequency of the driving

force,/(2)inHz.

d) The amplitude of the driving

force,F0 inN.

e) Whatis?! " #! #" $!

#$

#!

%

&

'

$

!

$

'

&

%

()*+ -./

0).1234+*+56-*/

MITOCW8.03SC 2 ProblemSet#2

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Problem2.5DrivenRLCcircuit

AgeneratorofemfV(t) =V0cos(t)isconnected inserieswitharesistanceR,an inductanceL,

andacapacitanceC.

a) UsingFaradays law,writedownthedifferentialequationsforthecurrentI inthecircuitand

forthecharge,q,onthecapacitor.

b)

Solve

for

q(,

t).

c) SolveforI(,t).

Inwhatfollows,V0 = 3V,R=50,L=100mH,andC= 0.01F.

d)

Plot

the

amplitude

of

the

current

(Io)asafunctionof.

e) Atwhatvalueof is(Io)amaximum?

f) Plotqo asafunctionof.

g) Atwhatfrequency,,isthevoltageacrossthecapacitoramaximum?

MITOCW8.03SC 3 ProblemSet#2

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MIT OpenCourseWarehttp://ocw.mit.edu

8.03SC Physics III: Vibrations and Waves

Fall 2012

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