Transcript File

Presented by CHITRA JOSHI
PGT (PHYSICS)
KV OFD RAIPUR DEHRADUN
SIMPLE HARMONIC MOTION
Simple harmonic motion is a special type
of periodic motion, in which a particle
moves to and fro repeatedly about a
mean(equilibrium position under a
restoring force, which is always directed
towards the mean position and whose
magnitude at any instant is directly
propotional to the displacement of the
particle from the mean position at that
instant.
GENERAL CHARACTERS OF
SIMPLE HARMONIC MOTION
Displacement-The displacement of a particle
executing s.h.m. at any instant is defined as the
distance of the particle from the mean position at
that 1 instant.
y=asinθ=asinωt
AMPLITUDE
The maximum displacement on either side
of mean position is called amplitude of
motion.as the maximum value of sinθ or
cosθ=1
VELOCITY
The velocity of the particle executing
s.h.m. at any instant, is defined as the time
rate of change of its displacement at that
instant.
Thus, the velocity in s.h.m. at the instant
t is,
v=wacos(ωt+ø0)
=wa √(1-sin2(ωt+ø))
V0 =wa =velocity amplitude
ACCELERATION
The acceleration of the particle executing
s.h.m. at any instant is defined as the time
rate of change of its velocity at that instant.
OSCILLATIONS
PERIODIC MOTION
Periodic motion of a body is that motion
which is repeated identically after a fixed
interval of time.The fixed interval of time
after which the motion is repeated is called
periodic motion.
E.g.-The revolution of earth around the
sun is a periodic motion.Its period of
revolution is one year.
OSCILLATORY MOTION
Oscillatory or vibratory motion is that
motion in which a body moves to and fro
or back and forth repeatedly about a fixed
point (called mean position or equilibrium
position) in a definite interval of time.
E.g-The motion of the pendulum of a wall
clock is oscillatory motion.
HARMONIC OSCILLATION
Harmonic oscillation is that oscillation
which can be expressed in terms of single
harmonic function(i.e.sine function or
cosine function).
“A harmonic oscillation of constant
amplitude and of single frequency is called
simple harmonic oscillation.
Mathematically,
y=asineωt=asin2πt/T
y=acosωt=acos2πt/T
Here y=displacement of body from mean
of position at any instant t.
a=maximum displacement
ω=angular frequency=2πv=2π/T
v,Tare frequency and time period of
harmonic oscillation.
NON HARMONIC OSCILLATION
Non-harmonic oscillation is that oscillation
which cannot be expressed in terms of
single harmonic function.
Mathematically,
y=asinωt+bsinωt
Or
y=asin2πt + bsin4πt
T
T
PERIODIC FUNCTIONS
Periodic functions are those functions
which are used to represent periodic
motion. A function f(t) is said to be
periodic, if f(t)=f(t+T)=f(t+2t)
PHASE
Phase of a vibrating particle, at any instant
is a physical quantity which completely
expresses the position and direction of
motion of the particle at that instant with
respect to its mean position.
it is denoted by ø.
ø=2πt/T+ø0
SIMPLE HARMONIC MOTION
Simple harmonic motion is a special type
of periodic motion, in which a particle
moves to and fro repeatedly about a
mean(i.e. equilibrium) position under a
restoring force, which is always directed
towards the mean position and whose
magnitude at any instant is directly
propotional to the displacement of the
particle from the mean position at that
instant.
GENERAL CHARACTERS OF
SIMPLE HARMONIC MOTION
Displacement- The displacement of a
particle executing s.h.m. at any instant is
defined as the distance of the particle from
the mean position at that 1 instant.
y=asinθ=asinωt
AMPLITUDE
The maximum displacement on either side of mean
position is called amplitude of motion. As the maximum
value of sinθ or cosθ=1
Velocity-The velocity of the particle executing s.h.m. at
any instant, is defined as the time rate of change of its
displacement at that instant.Thus, the velocity in s.h.m.
at the instant t is,
v=wacos(ωt+ø0)
=wa√1-sin2(ωt+ø0)
=wa√1-(y2/a2)
v=w√a2-y2
ACCELERATION
The acceleration of the particle executing S.H.M
at any instant is defined as the time rate of
change of its velocity at that instant.
Time period- it is defined as the time taken by
the particle executing S.H.M.to complete one
vibration.
A=ω2/y
Or ω=√A/y
Time period,T=2π/ω=2π√y/A
T=2π√displacement/acceleration
TOTAL ENERGY IN S.H.M.
A particle executing S.H.M. possesses two
types of energy.
Potential energy-This is an account of the
displacement of the particle from its mean
position.
Kinetic energy-This is an account of the
velocity of the particle.
POTENTIAL ENERGY
Consider a particle of mass m, executing
linear S.H.M.with amplitude a and constant
angular frequency ω.suppose t second
after starting from the mean position, the
displacement of the particle is y.
y=asinωt
1
Therefore velocity of the particle at instant
t,v=dy/dt=awcosωt
2
Acceleration of the particle at this instant,
A=dv/dt=-aw2sinωt=-ω2y
3
Restoring force, f=mass x acceleration
f=-mw2y=-ky
f=-ky
mw2=k force constant
The work done
dw=-fdy =-(-ky)dy
dw=kydy
Total work done for displacing the particle
from the mean position to a position of
displacement y will be
W = 0∫y kydy =1/2 ky2
This work done appears as potential energy U of
the given instant.
U=1/2 ky2
4
U=1/2 mw2y2=1/2mw2a2sin2wt
U=1/2mw2a2sin2wt
5
Kinetic energy – K.E. of the particle at the instant t,
is given by
K=1/2mv2
K=1/2m(awcoswt)2
K=1/2mw2a2cos2wt
6
K=1/2ka2cos2wt
K=1/2ka2(1-sin2wt)
K=1/2ka2(1-y2/a2)
=1/2k(a2-y2)
K=1/2mw2(a2-y2)
7
Total energy of the particle at the instant it will be
E=U+K=1/2ky2+1/2k(a2-y2)
E=1/2ka2=1/2mw2a2
E=1/2m(2 ‫ ת‬a)2
W=2 ‫ ת‬a
E=2m ‫ ת‬2v2a2
Waves
Wave motion : A wave motion is a means
of transferring energy and momentum frm
one point to another without any actual
transportation of matter between these
points.
Types of waves
We usually come across three types of
waves-They are
Mechanical waves
Electromagnetic waves
Matter waves
Mechanical waves-These are not
familiar.for e.g. waves on water surface,
waves on string, sound waves e.t.c. These
waves are governed by Newton’s laws of
motion.
ELECTROMAGNETIC WAVES
Electromagnetic waves,which require no
material medium for their production and
propagation i.e. they can pass through
vaccum and other material medium.
MATTER WAVES(DE-BROGLIE
WAVE )
These waves are associated with moving particles of
matter, like electrons, protons, neutrons, atoms,
molecules e.t.c.
Types of mechanical wave motion-The mechanical
waves are of two types1)Transverse wave motion
2)Longitudinal wave motion
Transverse wave motion-A transverse motion is that
wave motion in which individual particles of the medium
execute simple harmonic motion about their mean
position in a direction perpendicular to the direction of
propagation of wave motion.
A
E
C
λ
Normal level
partical
λ
B
D
LONGITUDNAL WAVE MOTION
A longitudnal wave motion is that motion in
which individual particles of the madium execute
simple harmonic motion about their mean
position along the same direction along which
the wave is propagated.
For e.g.1)Sound waves travel through air in the
form of longitudnal waves.
2)Vibrations of air column in organ pipes are
longitudnel.
RELATION BETWEEN PARTICLE VELOCITY AND
WAVE VELOCITY-The equation of a plane progressive
wave travelling with a velocity v along the direction of xaxis.
Y(x,t)=r sin[2π/λ(vt-x)+ø0 ]
1
If initial phase,ø0=0 then
2
Y(x,t)=r sin[2π/λ(vt-x)]
At any position x, velocity of particle is the rate of change
of displacement of the particle with time; it is represented
by u(x,t),
u(x,t)=d/dt[y(x,t)
=d/dt[r sin{2π/λ(vt-x)}]
3
u(x,t)=r cos{2π/λ(vt-x)}*2πv/λ
Also,d/dt[y(x,t)=d/dt[r sin{2π/λ(vt-x)}]
=r cos{2π/λ(vt-x)}(-2π/λ)
4
Dividing eqn 3 and 4
U(x,t)
=-v
d/dx{y(x,t)}
Or u(x,t)=-v d/dx{y(x,t)}
Some important terrms connected with wave
motionAMPLITUDE-The amplitude of a wave is the
magnitude of maximum displacement of the
element from their equilibrium position,as the
wave passes through them.itis represented by r.
WAVE LENGTH-It is equal to the distance
travelled by the wave during the time, any one
particle of the medium completes one vibration
about its mean position.
ANGULAR WAVE NUMBER
Angular wave number of a wave is 2π times of the no. of
waves that can be commodated per unit length.
k=2π/λ
S.i. unit of k=radian m-1
FREQUENCY-It may be define as frequency of a wave
as the number of complete wave lengths traversed by
the wave in one second.it is represented by v and is
measured in heartz.
Angular frequency of the wave is 2π times the frequency
of the wave.it is represented by ω and is measured in
rad s-1
ω=2πv
TIME PERIOD
Time period of a wave is equal to the time
taken by the wave to travel a distance
equal to one wave length. It is represented
as T.
T=I/v
DOPPLER EFFECT IN SOUND
According to Doppler’s effect, whenever there
is a relative motion between a source of
sound and listener, the apparent frequency of
sound heard by the listener is different from
the actual frequency of sound emitted by the
source.
EXPRESSION FOR APPARENT
FREQUENCY
Suppose medium, source and listener move
in same direction then apparent wavelength
becomes
λ’={(v+vm)-vs)
ᶹ
These waves of wave length λ’ travel towards
the listener.
Relative velocity of sound waves with respect
to listener=(v+vm)-Vl .This is the distance
available in one second to waves of wave length
λ’.
Apparent frequency of sound waves heard by
listener is
ᶹ’ =(v+vm)-Vl
λ’
putting value of λ’
In case the medium is stationary vm=0
ᶹ ’=(v-vL)ᶹ
(v-vs)
Specials cases1)if the source is moving
towards the listener but the listener is at
rest,then vs is positive and vL=0,
ᶹ’= ( v) ᶹ
v-vs
i.e ᶹ’> ᶹ
2)if the source is moving away from the listener
is at rest, then vs is negative and vL=0
ᶹ’= ( v) ᶹ
= ( v) ᶹ
v-(-vs)
v+vs
i.e. ᶹ’<ᶹ
3)if the source is at rest and listener is moving
away from the source, then vs=0 and vL=positive
ᶹ’=(v-vL) ᶹ
v
i.e. ᶹ’<ᶹ
If the source is at rest and listener is moving
towards the source, then vs=0 vL=negetive.
ᶹ’={v-(-vL)} ᶹ
v
i.e. ᶹ’>ᶹ
4)if the source and listener are approaching
each othert, then vs is positive and Vl is neg.
ᶹ’={v-(-vL)} ᶹ
=(v+vL) ᶹ
v-vs
(v-vs)
i.e. ᶹ’>ᶹ
6)if the source and listener are moving away
from each other, then vs is nag and vL is posi
ᶹ’=(v-vL) ᶹ.
(v+vs)
i.e. ᶹ’<ᶹ
7)if the source and listener are both in motion in
the same direction and with same velocity, then
vs=vL = v’ i.e. ᶹ’=ᶹ
8)if the source and listener move at right angles
to the direction of wave propagation.vs cos 90
°=0 and vL cos 90°=0 and ᶹ’=ᶹ
THANK YOU