Simple harmonic motion is a special type of oscillation in which the particle oscillates on a straight line, the acceleration of the particle is always directed towards a fixed point on the line and its magnitude is proportional to the displacement of the particle from this point.
The fixed point is called centre of oscillation.
If we take centre of oscillation as the origin and the line of motion as the X axis, SHM can be defined by the equation
a = -ω²x ... (1)
Where ω² is a positive constant.
If x is positive, a is negative and if x is negative, a is positive. It means that the acceleration if always directed towards to the centre of oscillation.
As acceleration is; a = F/m
We can write SHM equation as
F/m = -ω²x
F = -mω²x
F = -kx ...(2)
Force constant or spring constant
The constant k = mω² is called the force constant or spring constant.
The resultant force on the particle is zero when it is at the centre of oscillation
Equation of motion of SHM
Terms Associated with SHM
a. Amplitude
b. Time period
c. Frequency and angular frequency
d. Phase
e. Phase constant
Thursday, April 24, 2008
Concept review Ch. 13 Fluid Mechanics
Fluid
Pressure in fluids
Pascal's law
Atmospheric pressure
Barometer
Archimedes' principle
Buoyancy
Floatation
Flow of fluids: Steady flow, turbulent flow
Incompressible fluid
nonviscous fluid
Equation of continuity
Bernoulli's equation
Ventury tube
Aspirator pump
Pressure in fluids
Pascal's law
Atmospheric pressure
Barometer
Archimedes' principle
Buoyancy
Floatation
Flow of fluids: Steady flow, turbulent flow
Incompressible fluid
nonviscous fluid
Equation of continuity
Bernoulli's equation
Ventury tube
Aspirator pump
Concept review Ch. 14 Mechanical Properties of Matter
Elasticity
macroscopic reason of elasticity
Stress
Volume stress
Strain
Shearing strain
Hooke's law
Modulii of elasticity
Relation between longitudinal stress and strain
Elastic potential energy of a strained body
Surface tension
Surface energy
Contact angle
Viscosity
Poiseuille's equation
Stokes' law
Critical velocity
Reynold's number
macroscopic reason of elasticity
Stress
Volume stress
Strain
Shearing strain
Hooke's law
Modulii of elasticity
Relation between longitudinal stress and strain
Elastic potential energy of a strained body
Surface tension
Surface energy
Contact angle
Viscosity
Poiseuille's equation
Stokes' law
Critical velocity
Reynold's number
Concept review Ch.15 Wave Motion and Waves on a String
Particles carry kinetic energy with themselves and transfer energy to other particles in collisions. In this instance, the particle travels for some distance in space and then collides with another particle (Borrowing the terminology from heat and mass transfer we may say there is both mass transfer and energy transfer).
Wave motion is another way of transporting energy. When we say hello to our friend, no material particle is ejected by us from our lips that falls on our friend's ear. We create some disturbance in the part of air close to our lips. Energy is transferred to these air particles by sometimes pushing them ahead or sometimes pulling them back. This affects the density of ther air near our mouth. This disturbance is transferred to the next layer of air and so on till the ear of the listener gets disturbed. Only the disturbance produced in the air travels and the air itself does not move This tope of motion of energy is called a wave motion.
Wave motion on a string.
Consider a long string with one end fixed to a wall and the other held by a person. If the string is held tight, and a small bump is made near the end held by the person, the disturbance travels down the string with a constant speed.
For an elastic and homogeneous string, the bump moves with constant speed to cover equal distances in equal time periods. The shape of the bump is not altered as it moves provided the bump is small.
Equation of a travelling wave
In the string the wave travels from the hand end to the wall end. Let us assue that hand held end is at the left side and the end fixed in the wall is in the right. Hence the X-axis is along the string towards right.
Let function f(t) represent the displacement y of the particle at x = 0 as a function of time
y(x=0,t) = f(t)
As the disturbance is travelling on the string towards right with a constant speed v, the displacement in the y direction produced at the left end at time t, reaches the point x at time t+(x/v)(it takes time equal to x/v). We can express the statement in a different way. The displacement of the particle at point x at time time was originated at the left hand at the time t-(x/v). If the left end is taken as x=0, the displacement at time t-(x/v) will be f(t-x/v) which is y(x=0,t-x/v).
Hence y(x,t) = y(x=0,t-x/v) = f(t-x/v)
y(x,t) represents the displacementof the particle at x at time t. It is generally abbreviated as y and the wave equation is written as
y = f(t-x/v) ...(1)
The equation represents a wave travelling in the positive x direction with a constant speed v.
Function f depends on how the source is providing disturbance.
If the wave is travelling in negative x-direction, with speed v it may be written as
y = f(t+x/v)
Alternative forms of wave equation
y = A sinω(t-x/v)
y = A sin(ωt-kx) …(i)
Where k = ω/v
Remember λ = 2 π v/ ω = 2 π/k or
π/k = λ/2
y = A sin k(vt-x) as kv = ω …(ii)
Interference of waves going in the same direction
Let the amplitudes of the two waves be A1 and A2 and the two waves differ in phase by an angle δ.
Their equations may be written as
y1 = A1 sin(kx-ωt)
y2 = A2 sin(kx-ωt+δ)
By trigonometric properties, the combination of waves (suuperposition of waves) is represented by
= A sin(kx-ωt+ε)
A² = A² cos² ε + A² sin² ε
=
A1² +A2² +2A1A2cosδ
where A1 +A1cosδ = A cosε
A2sinδ = A sinε
Resultant amplitude is maximum, when cosδ = +1 or δ = 2nπ. The maximum value is A1+A2.
Resultant amplitude is minimum, when cosδ = -1 or cosδ = (2n+1)π . The minimum value is A1-A2.
(These results are used in interference of light.)
Standing waves
Standing waves are produced when two sine waves of equal amplitude and frequency propagate on a long string in opposite directions.
The equations of two waves can be:
y1 = A sin(ωt-kx)
y2 = A sin(ωt+kx)
The equations of the resultant wave will be:
y = y1+y2
= A[sin(ωt-kx) + sin(ωt+kx)]
= 2Asin ωt cos kx
= (2A cos kx) sin ωt
Interpretation of the equation: The amplitude of the wave is |2A cos kx|, but the amplitude is not equal for all the particles.
There are some points where the amplitude |2A cos kx| = 0 all the time.
This will at points where
cos kx = 0
=> kx = (n + ½) π
=> x = (n + ½) λ/2
where n is an integer.
Although these points are not physically clamped, they remain fixed as the two waves pass them simultaneously. These points whose amplitude is zero always are called nodes.
At the points where |cos kx| = 1, the maximum amplitude will be present. These points at which the maximum amplitude is obtained are called antinodes.
When sin ωt = 0, the amplitude of all points is zero which means that all points are at their normal positions or mean positions.
When sin ωt = 1, all points for which cos kx is positive reach their positive maximum displacement. At the same time, all points for which cos kx is negative reach their negative maximum displacement.
From the equation is also clear that the separation between consecutive nodes or consecutive antinodes is λ/2.
As the particles at the nodes do not move at all, energy cannot be transmitted across them.
Wave motion is another way of transporting energy. When we say hello to our friend, no material particle is ejected by us from our lips that falls on our friend's ear. We create some disturbance in the part of air close to our lips. Energy is transferred to these air particles by sometimes pushing them ahead or sometimes pulling them back. This affects the density of ther air near our mouth. This disturbance is transferred to the next layer of air and so on till the ear of the listener gets disturbed. Only the disturbance produced in the air travels and the air itself does not move This tope of motion of energy is called a wave motion.
Wave motion on a string.
Consider a long string with one end fixed to a wall and the other held by a person. If the string is held tight, and a small bump is made near the end held by the person, the disturbance travels down the string with a constant speed.
For an elastic and homogeneous string, the bump moves with constant speed to cover equal distances in equal time periods. The shape of the bump is not altered as it moves provided the bump is small.
Equation of a travelling wave
In the string the wave travels from the hand end to the wall end. Let us assue that hand held end is at the left side and the end fixed in the wall is in the right. Hence the X-axis is along the string towards right.
Let function f(t) represent the displacement y of the particle at x = 0 as a function of time
y(x=0,t) = f(t)
As the disturbance is travelling on the string towards right with a constant speed v, the displacement in the y direction produced at the left end at time t, reaches the point x at time t+(x/v)(it takes time equal to x/v). We can express the statement in a different way. The displacement of the particle at point x at time time was originated at the left hand at the time t-(x/v). If the left end is taken as x=0, the displacement at time t-(x/v) will be f(t-x/v) which is y(x=0,t-x/v).
Hence y(x,t) = y(x=0,t-x/v) = f(t-x/v)
y(x,t) represents the displacementof the particle at x at time t. It is generally abbreviated as y and the wave equation is written as
y = f(t-x/v) ...(1)
The equation represents a wave travelling in the positive x direction with a constant speed v.
Function f depends on how the source is providing disturbance.
If the wave is travelling in negative x-direction, with speed v it may be written as
y = f(t+x/v)
Alternative forms of wave equation
y = A sinω(t-x/v)
y = A sin(ωt-kx) …(i)
Where k = ω/v
Remember λ = 2 π v/ ω = 2 π/k or
π/k = λ/2
y = A sin k(vt-x) as kv = ω …(ii)
Interference of waves going in the same direction
Let the amplitudes of the two waves be A1 and A2 and the two waves differ in phase by an angle δ.
Their equations may be written as
y1 = A1 sin(kx-ωt)
y2 = A2 sin(kx-ωt+δ)
By trigonometric properties, the combination of waves (suuperposition of waves) is represented by
= A sin(kx-ωt+ε)
A² = A² cos² ε + A² sin² ε
=
A1² +A2² +2A1A2cosδ
where A1 +A1cosδ = A cosε
A2sinδ = A sinε
Resultant amplitude is maximum, when cosδ = +1 or δ = 2nπ. The maximum value is A1+A2.
Resultant amplitude is minimum, when cosδ = -1 or cosδ = (2n+1)π . The minimum value is A1-A2.
(These results are used in interference of light.)
Standing waves
Standing waves are produced when two sine waves of equal amplitude and frequency propagate on a long string in opposite directions.
The equations of two waves can be:
y1 = A sin(ωt-kx)
y2 = A sin(ωt+kx)
The equations of the resultant wave will be:
y = y1+y2
= A[sin(ωt-kx) + sin(ωt+kx)]
= 2Asin ωt cos kx
= (2A cos kx) sin ωt
Interpretation of the equation: The amplitude of the wave is |2A cos kx|, but the amplitude is not equal for all the particles.
There are some points where the amplitude |2A cos kx| = 0 all the time.
This will at points where
cos kx = 0
=> kx = (n + ½) π
=> x = (n + ½) λ/2
where n is an integer.
Although these points are not physically clamped, they remain fixed as the two waves pass them simultaneously. These points whose amplitude is zero always are called nodes.
At the points where |cos kx| = 1, the maximum amplitude will be present. These points at which the maximum amplitude is obtained are called antinodes.
When sin ωt = 0, the amplitude of all points is zero which means that all points are at their normal positions or mean positions.
When sin ωt = 1, all points for which cos kx is positive reach their positive maximum displacement. At the same time, all points for which cos kx is negative reach their negative maximum displacement.
From the equation is also clear that the separation between consecutive nodes or consecutive antinodes is λ/2.
As the particles at the nodes do not move at all, energy cannot be transmitted across them.
Concept review Ch. 16 Sound waves
Nature and propagation
Sound is produced in a material medium by a vibrating source.
Sound waves constitute alternate compression and rarefaction pulses traveling in the medium.
Sound is audible only if the frequency of alternation of pressure is between 20 Hz to 20,000 Hz.
Displacement wave and Pressure Wave
A longitudinal wave in a fluid can be described either in terms of the longitudinal displacement suffered by the particles of the medium or in terms of the excess pressure generated due to the compression or rarefaction.
Speed of a sound wave
v = √(B/ ρ)
where
v = velocity
B = Bulk modulus of the material.
ρ = normal density of the fluid
Hence the velocity of a longitudinal wave in a medium depends on its elastic properties and inertial properties of the medium.
Newton’s formula for speed of sound in a gas
v = √(P/ρ)
The density of air at temperature 0°C and pressure 76 cm of mercury column is ρ = 1.293 kg/m³
So P = .76m*(13.6*10^3 kg/ m³)*(9.8 m/s²) = 101292.8
Hence P/ ρ = 78339.37
√(P/ρ) = 279.8917 m/s
The velocity of sound in air comes as 280 m/s.
But the measured value of speed of sound in air is 332 m/s
Laplace's correction
Laplace suggested a correction. With Laplace’s correction the formula is
v = √( γ P/ρ)
where γ = Cp/Cv (Cp and Cv are molar heat capacities at constant pressure and constant volume respectively)
With this new formula the value comes out to be 331.1723 closer to 332 m/s.
Effect of pressure, temperature and humidity on speed of a sound wave
The speed of sound is not affected by the change in pressure provided the temperature is kept constant. If pressure is changed but the temperature is kept constant, the density varies proportionately and P/ρ remains constant.
Speed of sound increases with increasing humidity. The density of water vapour is less than dry air at the same pressure. Thus, the density of moist air is less than that of dry air.
Intensity of sound waves
The intensity of a sound wave is defined as the average energy crossing a unit cross sectional area perpendicular to the direction of propagation of the wave in unit time.
The loudness of sound that we feel is mainly related to the intensity of sound. It also depends on the frequency to some extent.
Appearance of sound to human ear
The appearance of sound to human ear is characterised by three parameters.
1. pitch
2. loudness
3. quality
1. Pitch: Higher the frequency, higher will be the pitch.
2. Loudness: Loudness that we sense is related to the intensity of sound though it is not proportional to it.
3. A sound generated by a source contains a number of frequency components in it. Certain sounds have well defined frequencies which have considerable amplitude. Such sounds are particularly pleasant to the ear.
Interference of sound waves
Resultant change in pressure due to superposition of two sound waves.
p1 = p01 sin(kx- ωt)
p2 = p02 sin[k(x + ∆x)- ωt]
= p02 sin{(kx- ωt)+ δ]
Where δ = k∆x = 2 π ∆x/ λ
P = p0sin[(kx- ωt)+ ε]
where
p0² = p01² + p02² + 2 p01 p02 cos δ
tan ε = p02 sin δ /( p01 + p02cos δ)
the resultant amplitude is maximum when δ = 2n π and is minimum when δ = (2n+1) π.
Hence when δ = 2n π there is constructive interference
When δ = (2n+1) π there is destructive interference.
Beats
The phenomenon of periodic variation of intensity of sound when two sound waves of slightly different frequencies interfere, is called beats.
Bending of waves from an obstacle or an opening is called diffraction.
Diffraction effects are appreciable when the dimensions of openings or the obstacles are comparable or smaller than the wave length of the wave.
Doppler effect
If the source of sound or the observer or both, move with respect to the medium, the frequency observed may be different from the frequency of the source. This apparent change in frequency of the wave due to motion of the source or the observer is called Doppler effect.
Mach number
Mach Number = µs/v
µs = speed of source creating the sound wave
v = velocity of sound wave
Sound is produced in a material medium by a vibrating source.
Sound waves constitute alternate compression and rarefaction pulses traveling in the medium.
Sound is audible only if the frequency of alternation of pressure is between 20 Hz to 20,000 Hz.
Displacement wave and Pressure Wave
A longitudinal wave in a fluid can be described either in terms of the longitudinal displacement suffered by the particles of the medium or in terms of the excess pressure generated due to the compression or rarefaction.
Speed of a sound wave
v = √(B/ ρ)
where
v = velocity
B = Bulk modulus of the material.
ρ = normal density of the fluid
Hence the velocity of a longitudinal wave in a medium depends on its elastic properties and inertial properties of the medium.
Newton’s formula for speed of sound in a gas
v = √(P/ρ)
The density of air at temperature 0°C and pressure 76 cm of mercury column is ρ = 1.293 kg/m³
So P = .76m*(13.6*10^3 kg/ m³)*(9.8 m/s²) = 101292.8
Hence P/ ρ = 78339.37
√(P/ρ) = 279.8917 m/s
The velocity of sound in air comes as 280 m/s.
But the measured value of speed of sound in air is 332 m/s
Laplace's correction
Laplace suggested a correction. With Laplace’s correction the formula is
v = √( γ P/ρ)
where γ = Cp/Cv (Cp and Cv are molar heat capacities at constant pressure and constant volume respectively)
With this new formula the value comes out to be 331.1723 closer to 332 m/s.
Effect of pressure, temperature and humidity on speed of a sound wave
The speed of sound is not affected by the change in pressure provided the temperature is kept constant. If pressure is changed but the temperature is kept constant, the density varies proportionately and P/ρ remains constant.
Speed of sound increases with increasing humidity. The density of water vapour is less than dry air at the same pressure. Thus, the density of moist air is less than that of dry air.
Intensity of sound waves
The intensity of a sound wave is defined as the average energy crossing a unit cross sectional area perpendicular to the direction of propagation of the wave in unit time.
The loudness of sound that we feel is mainly related to the intensity of sound. It also depends on the frequency to some extent.
Appearance of sound to human ear
The appearance of sound to human ear is characterised by three parameters.
1. pitch
2. loudness
3. quality
1. Pitch: Higher the frequency, higher will be the pitch.
2. Loudness: Loudness that we sense is related to the intensity of sound though it is not proportional to it.
3. A sound generated by a source contains a number of frequency components in it. Certain sounds have well defined frequencies which have considerable amplitude. Such sounds are particularly pleasant to the ear.
Interference of sound waves
Resultant change in pressure due to superposition of two sound waves.
p1 = p01 sin(kx- ωt)
p2 = p02 sin[k(x + ∆x)- ωt]
= p02 sin{(kx- ωt)+ δ]
Where δ = k∆x = 2 π ∆x/ λ
P = p0sin[(kx- ωt)+ ε]
where
p0² = p01² + p02² + 2 p01 p02 cos δ
tan ε = p02 sin δ /( p01 + p02cos δ)
the resultant amplitude is maximum when δ = 2n π and is minimum when δ = (2n+1) π.
Hence when δ = 2n π there is constructive interference
When δ = (2n+1) π there is destructive interference.
Beats
The phenomenon of periodic variation of intensity of sound when two sound waves of slightly different frequencies interfere, is called beats.
Bending of waves from an obstacle or an opening is called diffraction.
Diffraction effects are appreciable when the dimensions of openings or the obstacles are comparable or smaller than the wave length of the wave.
Doppler effect
If the source of sound or the observer or both, move with respect to the medium, the frequency observed may be different from the frequency of the source. This apparent change in frequency of the wave due to motion of the source or the observer is called Doppler effect.
Mach number
Mach Number = µs/v
µs = speed of source creating the sound wave
v = velocity of sound wave
Concept review Ch. 17 Light waves
Nature of light waves
Huygen's principle
Young's double hole experiment
Young's double slit experiment
Alternate bright and dark rings observed.
Fringe width = w = Dλ/d
D = distance betwee slits and the screen
λ = wave length of light wave
d = distance between slits.
visit for past JEE objective questions on Young's double slit experiment.
(This is an important topic on which many questions were asked.)
http://iit-jee-physics-ps.blogspot.com/2008/05/past-jee-oq-wave-motion-light-waves-1.html
Optical path
Interference
Fresnel's biprism
Coherent light source
There will be a constant phase difference between the two light waves in young's double experiment
Incoherent light source
the phase differene will be random - No interference will be there
Diffraction of light
Fraunhoffer diffraction by a single slit
Fresnel diffraction
Limit of resolution
Scattering of light
Polarization of light
Polaroids
Huygen's principle
Young's double hole experiment
Young's double slit experiment
Alternate bright and dark rings observed.
Fringe width = w = Dλ/d
D = distance betwee slits and the screen
λ = wave length of light wave
d = distance between slits.
visit for past JEE objective questions on Young's double slit experiment.
(This is an important topic on which many questions were asked.)
http://iit-jee-physics-ps.blogspot.com/2008/05/past-jee-oq-wave-motion-light-waves-1.html
Optical path
Interference
Fresnel's biprism
Coherent light source
There will be a constant phase difference between the two light waves in young's double experiment
Incoherent light source
the phase differene will be random - No interference will be there
Diffraction of light
Fraunhoffer diffraction by a single slit
Fresnel diffraction
Limit of resolution
Scattering of light
Polarization of light
Polaroids
Concept review Ch. 18 Geometrical Optics
Reflection at smooth surfaces
Spherical mirrors
Refraction at plane surfaces
Critical angle
Optical fibre
Prism
Angle of minimum deviation
Refraction at spherical surfaces
Refractoin through thin lenses
Len maker's formula
Lens formula
Power of a lens
Defects of images
a. spherical aberration
b. Coma
c. Astigmatism
d. Curvature
e. Distortion
Chromatic aberrations
Spherical mirrors
Refraction at plane surfaces
Critical angle
Optical fibre
Prism
Angle of minimum deviation
Refraction at spherical surfaces
Refractoin through thin lenses
Len maker's formula
Lens formula
Power of a lens
Defects of images
a. spherical aberration
b. Coma
c. Astigmatism
d. Curvature
e. Distortion
Chromatic aberrations
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