OsCILLATIONS JEE-MAIN ADDITIONAL SYLLABUS

(FOR ADV STUDENTS) PHYSICS-VOL- VI

DAMPEDAND FORCED OSCILLATION

Damped oscillations Relaxation time for velocity
* The oscillations ofa body whose amplitude The time intervalduring which the velocityof
goes on decreasing with time are defined as harmonic oscillator reduces to l/e ofits initial
damped oscillations. velocity is defined as relaxation time of
Ifv be the velocty of the oscillator, then
damping forcd F, =-b)vhere b' is d¡mping velocity(z,).
constant
The resulting force acting on damped
harmonic oscillator is
V=Ve";when =T,,
2m
then a
F=F.restoring =-Kx-by Relaxation time for amplitude
+Fjanping,d'x b dx + K-X when= Ty
2m
then A=
ma =-Kx-by +
e
d m dt m
Difierential equation of damped h¡rmonic Relaxation time for energy
d'x d m
oscillator is +2+ax=|
dt
where when= Tg = then E7 =
b
2y =, a Ty=I, = 2T, (or) TE
m m
Solution to above differential equation is
ie. the relaxation time for energyis halfofthat
Ax=Acos(o't+ ) =xe cos(o'tt ) for amplitude.
In these oscillations, the ampiitude decreases Quality factor
exponentially due to damping force like average energy stored
frictional force, viscous force, hysteresis etc. Q=27x
A=xewhere Y=:
energy loss in one cycle = @T
2m Forced oscillations
displacement The oscillations in which a body oscillates under
the influence of an external periodic force
(driver) are knownas forced oscillations.
-A time The amplitude ofoscillator decreases due to
damping forces but on account of the energy
gained from the external source it remains
constant.
The resultant force acting on the oscillator
In these oscillations the frequency decreases F=F + F + F external
damping restoring
ie. p=o, - ’F=-by- Kx +F, cos 0,t
2t where o, is frequency ofexternal periodic force
Time period ofthe oscillator Diferential equationof the oscillator
d'x
this is greater than the_tËme period of the +2y +a,x= F, cos 0,1
2n
dt dt
harmonic oscillatoro b K
where 2y =-and
m
o, = m
Due to decrease in amplitude, the energy of Solution to above differential equation
the oscillator also goes on decreasing
exponentially. x= Asin(o,t +¢)
=-K'e-brim F,/m
E7 = Exe m where amplitud+ 4
2 (ba
m

NARAYANAGROUP
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 forced
amplitude
The
Amplitude
resonance Amplitude
(a-a frequency.
resonant
Smaller
orthe is ofrate The T= Theknown is amplitude
In from InThis thenResonance is where or DAMPED
frequence larger externmal magnitude
Whenofexternal
frequency remain In
force.constant
resonance, resonance, equal these
oscillator offallinamplitude term 27_27 time phenomenon
the the b A=
is the oscillations, =2y AND
of the periodicsharpness period asofmotiondriver amplitude theto
frequency
extermal darnping,
danping the of FORCED Vo-a;
+4y'a
4y'w; + A=
becomes 21,-=resonant the maximum natural amplitude
of and
resonance)
(sharp frequency to increases
is with
isminimum. dampingLowdampingNo
force of oscillator becomes known phase F/m
source flatter m the ofexternal
frequency OSCILLATION
sharper resonance with respect the
maximum resonance)
damping
(flat High frequency. energy
driven. amplitude
F/m change as to depends =
is is on in
ofthe
maximum
the the either resonance.
maximum ofthe to tan
resonance. n resonance driver(,)
Hence will
periodic time.
resonance frequencymeans
when side transfer oscillator, upon and bo,Im
value. force But
energy
of the the the the
is
Sol:
period
damping block
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with what WE-02:
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where
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avalue
toWE-01: minimum.i.., this SYLLABJQ
JEE-MAIN
force f o'reduces a=el5b2m Hence,becomesaA. After10 do,L
Thu_ADDITIONAL
ofmass
20kg is constant = a becomes maximumVelocitVelocity
y ADV
of
constant factor with in
constant
oscillation (2m 50 A aA, more
(ie.,sec A.e times 5s.
damped n decrease
Hz, =5 mass decreases mo;-o)4ajy' +velocity o;-o)-20,)
(or STUDENTS)
b force
'=(e=(0.9)
=0.729. = In The =0
maximum =V=
=Vs-3
by4Hz = =does its
2" another F energY)d
90N/m 12 of
Ae-|15)/2m
is 1Hz afteroriginalamplitude
oscillatory when
is
constant 2kg to +4y? V=
(b40g/s.
) suspended or introduced the
t=15 5sec
when YSICSPH-VOL-VI
P in 20% 10s 0.9 +4y(20,
at
timne a frequency a
oscillates timnes Ao,
Find
mediunand damping =A,ls6/e2m sec)its 0.9 it of denominator Ao,
resonance
50 magnitude, will
taken to originalits a cos ,
(a) motion Nm. amplitude =p decreasedamped )
sprtg a ? (o,+*) =0
Jor time force on sb/2m
By 0r
a of ie

ROUP Sol:-
e
OSCILLATIONS
retardation
proportional average significant
spherical of
sec 0WE-04: y2 -biy2. Initial value c)
=0.693Typ x=2.302 Take dropped
b)amplitude period Massvalue. its
initial amplitude
energy=kfe m Let -b(Ti2)
2m T=2K timea)JKdamping
m mechanical
original)
to =0.693 -bT,2
==n(2)x=0.693x after the natural m initial
t= energy to =v90x0.2 =200g=0.2
energy x0.3010× halfofits m
bob life amplitude
t If x a constant
to sec, only 0.04 0.-2 E, time 2m logarithm 2 A
=2r-0.3s energyvalue of
its (due of of a = oscillation
in =3.46s
=KA;At 2: is initial V90
velocity,pendulum. the then simple dropped =0.693 x 2m A,e |0.2 =18 bkg, (c)
the = to
pendulumn to t (up /b on 2m
e value 40g/s Force time
period 2m
bl both -bTi2 drop
viscous may
pendulum has
to b time to0.042x0.2 n(2) m kg to
with a halfofits sides
after Let Is, =0.04constant taken drop
befactor to
between t2 -=6.93s the
amplitude half
'b' suffersWhencalled time b<<Km. kg'sK-90 to
as
drag) of m
for
initial I,,. N/m of half
the the the t=l/e
is its its of
a
1. Sol: (FOR
E represents But = restoring is JEE-MAN
l)i iv) Among :.differential For but the the Retarding
Iadisplacement constant
time
and Amplitude amplitude
Angular moment
point small) ADV
2 =0,e*n =0,e small =-mgl=
ii -=1 smg
in 9 of
2)ii VVV Vtime the dt d'o m> oftorque thependulum STUDENTS)
damping,
equation inADDITIONAL
SYLLABUS
of
damped
EXERCISE or sin suspension, sin of force seconds
and following, si=-
n+ 6
inertial of proportionality,
t= (ot ml'a
iv mg
+ =-mgl the
+¢) wil the mbvl =
3)i, time harmonic of mby cosmg0
tìme solution
be: =-mgl th en pendulum is
PHYSICS-VOL
i, time the bv where the sin (assuming
iü mbù
and figure sin
pendulum+mbvl. When the
iv motion of a average
the =ang. is
4)i mbvl + angular damping
and which above about If ,
is acc. I'net life VI
iv is