Kinematics and Laws of Motion
Motion in straight linev=u+ats=ut+12at2v2=u2+2asSn=u+12a(2n-1)
Relative VelocityVelocity of A with respect to B=VA-VB
Linear Momentum=p=mv
Force = Rate of change of momentum (here mass is considered to be constant)F=dpdt=d(mv)dt=mdvdt=ma (if mass is not constant)F=dpdt=d(mv)dt=mdvdt+vdmdt
Elastic CollisionIn elastic collision total kinetic energy of the two bodies remains the same. m1u1+m2u2=m1v1+m2v212m1u12+12m2u22=12m1v12+12m2v22 v1=m1-m2m1+m2u1+2m2m1+m2u2 v2=m2-m1m1+m2u2+2m1m1+m2u1
Coefficient of restitution (e)e=v2-v1u1-u2 Newton’s law of restitution applies to the component of the velocity of the sphere perpendicular to the surface.
F=-dUdx U=Potential energyF=Force
Condition for stable equilibriumi) dUdx=0 ii) d2Udx2>0
Condition for unstable equilibriumi) dUdx=0 ii) d2Udx2<0
Condition for neutral equilibriumi) dUdx=0 ii) d2Udx2=0
Tangential velocity vv=𝜔r
Moment of inertia : Ring or thin wall hollow cylinderAxis: passing thru center along cylinder axisI=mr2
Moment of inertia : Disc or solid cylinderAxis: passing thru center along cylinder axis I=12mr2
Moment of inertia : Spherical ShellAxis: passing thru centerI=23mr2
Moment of inertia : solid sphereAxis: passing thru centerI=25mr2
Moment of inertia : rodAxis: passing thru center (and perpendicular to axis)I=mL212
Gravitational ForceF=Gm1m2r2
Gravitational field intensity (E)E(r)=GMr2
Acceleration due to gravity on surface of earth=g=GMeRe2
Variation of Acceleration due to gravity with depth=gd=g(1-dRe)
Variation of Acceleration due to gravity with height (h)gh=GMe(Re+h)2=gRe2(Re+h)2=g(1+hRe)-2gh=g(1+hRe)-2g(1-2hRe)
Gravitational potential (V)V(r)=-GMr
Gravitational potential Energy (U)U=-GMmr
Gravitational Self energy of uniform SphereU=35GM2R
Escape Velocity: The minimum velocity with which a body must be projected up so as to enable it to just overcome the gravitational pull, is known as escape velocity. vescape=2gRe=(2)vorbital
Binding energy of satellite: The minimum enery required for satellite to escape from earth gravitational influence is called binding energy of satellite.
Binding energy of satellite revolving around circular orbit round the earth = 12GMmr
Kepler’s first law ( law of elliptical orbit) :A planet moves round the sun in an elliptical orbit with sun situated at one of its foci. Kepler’s second law ( law of areal velocities) :A planet moves round the sun in such a way that its areal velocity is constant. Kepler’s third law ( law of time period) :A planet moves round the sun in such a way that the square of its period is proportional to the cube of semi major axis of its elliptical orbit.
Simple Harmonic Motion
Damped Periodic Motion
Speed of waves in a string v=T𝜇where v=speed of waveT=Tension in string𝜇=mass per unit length
Energy associated with 1 wavelength of string waveE𝜆=12𝜇𝜔2A2𝜆 where 𝜆=wavelength of waveA = Wave amplitude𝜇=mass per unit length𝜔=angular wave frequency
Power transported by a string waveP=12𝜇𝜔2A2v where v=speed of waveA = Wave amplitude𝜇=mass per unit length𝜔=angular wave frequency
Due to interference of two waves redistribution of energy takes place in medium y1=A1sin(kx-𝜔t)y2=A2sin(kx-𝜔t+𝛿) y=y1+y2y=Asin(kx-𝜔t+𝜖)whereA=A12+A22+2A1A2cos𝛿tan𝜖=A2sin𝛿A1+A2cos𝛿
Simple Harmonic Progressive Wavey=Asin(𝜔t-2𝜋x𝜆)
Speed of longitudinal waveSound wave travel in compressions and rarefactionsv=B𝜌where v = speed of waveB = Bulk Modulus𝜌= mass density For linear medium (like solid bar), speed of soundv=Y𝜌 where v = speed of waveY = Young's Modulus𝜌= mass density
Speed of sound in gasv=𝛾P𝜌
Waves in String : String fixed at both endsSpeed of waves : v=T𝜇where𝜇=massperunitlengthT=TensioninString
Kinetic Theory
The average distance a molecule can travel without colliding is called mean free path.Meanfreepath=𝜆=12n𝜋d2 where d = diameter of each moleculen=number of molecule per unit volumem-mass of one molecule𝜌=densityofgas
Boltzmannconstant=KB=PVNTwhere N=number of molecules (count of molecules)P=PressureV=VolumeT=Temperature PV=KBNTPV=NAKBNNAT𝜇=numberofmoles=NNANAKB=RPV=𝜇RT
Avogadro’s hypothesis : The number of molecules per unit volume is samefor all gases.
KB=Bolzmannconstant=PVNT where P=PressureV=VolumeN- Total Number of molecule (count of molecule)T=TemnperaturePV=KBNTPV=NAKBNNAT 𝜇=numberofmoles=NNA NAKB=R PV=𝜇RT
Van der Waals equation(p+an2V2)(V-nb)=nRTwhere an2V2=Pressurecorrection nb=Volumecorrection
Specific Heat Capacity : Monoatomic Gas1. Monoatomic Gas has three degree of freedom.2. Total internal energy of 1 mole of gas= UU=32KBNAT=32RT3. Cv(monoatomicgas)=dUdT=32R4. Cp(monoatomicgas)=Cv+R=52R5. Ratioofspecificheat=𝛾=CpCv=53
Specific Heat Capacity : Diatomic Gas (Rigid Diatomic gas) 1. Rigid Diatomic Gas has five degree of freedom.2. Total internal energy of 1 mole of gas= UU=52KBTNA=52RT3. Cv(RigidDiatomicgas)=dUdT=52R4. Cp(RigidDiatomicgas)=Cv+R=72R5. Ratioofspecificheat=𝛾=CpCv=75
Specific Heat Capacity : Diatomic Gas which is not rigid1. Non rigid Diatomic Gas has five degree of freedom and additional vibrational mode2. Total internal energy of 1 mole of gas= UU=(52KBT+KBT)NA=72RT3. Cv(NonrigidDiatomicgas)=dUdT=72R4. Cp(NonrigidDiatomicgas)=Cv+R=92R 5. Ratioofspecificheat=𝛾=CpCv=97
Polyatomic Gas
Non rigid Diatomic Gas has six degrees of freedom and certain (f) vibrational mode
Total internal energy of 1 mole of gas U=(62KBT+fKBT)NA
Cv(Polyatomic gas)=dUdT=(3+f)R
Ratio of specific heat 𝛾=CpCv=4+f3+f
Cp-Cv=R ( this is true for any ideal gas, monoatomic, diatomic or polyatomic)
Speed of sound=c=𝛾P𝜌
Pressure=P=13nmvrms2 where n=number of molecules per unit volumem=mass of moleculevrms =Root mean square speed
PV=23×(12nVmvrms2) wheren=number of molecules per unit volume
Internalenergy=E=12nVmvrms2( This is total Internal energy in volume V gas)wheren=number of molecules per unit volume
E=32PV( This is total Internal energy in volume V gas)
E=32KBNT( N=total number of molecules)
Average Kinetic Energy ( of one molecule)=EN=32KBT
Average Kinetic Energy=12mvrms2where,m=mass of one molecule
vrms2=3KBTm=3RTMwhere,M=Molar mass of molecule
Root mean square velocity vrms=3RTM where,M=Molar mass of molecule
Average velocityvaverage=8RT𝜋M
Most probable velocity vp=2RTM
Mean Free Path: Average distance covered by molecule between 2 successive collisionl¯=12n𝜋d2wheren = number density of gasd = diameter of molecule
Root Mean Square Velocity (Vrms)Vrms2=v12+v22+v32+...+vn2Nwhere N is number of collision
Average Velocity (V¯)V=v1+v2+v3+...+vnNwhere N is number of collision
Youngs Modulus Y=LongitudinalStresslogitudinalstrain=𝜎𝜖
Shear modulus or Modulus of Rigidity 𝜂=ShearStressShearstrain
Bulk modulusK=B=VolumetricStressVolumetricstrain
Force Constant: FvsLcurve(dFdx) is the effective value of the force constant at any point.
Poisson ratio is the ratio of transverse contraction ( or expansion) strain to longitudinal extension strain in the direction of stretching force.Poissonratio=𝜎=-𝜖transverse𝜖longitudinal
Relation between Young Modulus, Modulus of rigidity, Bulk modulus and Poisson Raio𝜂=Y2(1+𝜎)B=Y3(1-2𝜎)9Y=1B+1𝜂𝜂=3B-2𝜂2𝜂+6B
Strain Energy: The work done by the external applied force during stretching is stored as potential energy ( U) in the wire and is called as strain energy in the wire. U=12LoadElongation=12Y(Strain)2Volume
BrittleA material that has a tendency to break easily or suddenly without any extension first. It has small range of plastic extension.
Cohesive Force: Force of attarction between two molecules of same substance.
Adhesive Force: Force of attarction between two molecules of different substance
Surface Tension: Surface tension is the attractive force in liquids that pulls surface moleculesinto the rest of the liquid, minimizing the surface area.The surface tension T = force along a line of unit length. = surfece energy per unit area.
Surface Tension and Bubbles :Pressure is more on concave side For bubble with two surfaces,Pressuredifference=𝛥p=4TR
Pressure Difference for droplet which has one surface, 𝛥p=2TR
Total Surface energy in bubble with two surfaces=8T𝜋R2
Total Surface energy in drop ( one surfaces )=4T𝜋R2
Bernoulli's equationp+12𝜌v2+𝜌gh=constant
Capillary ActionHeight to which capillary action lift liquid h=2Tcos𝜃𝜌rgif𝜃iscloseto0°,thencos𝜃=1,then,h=2T𝜌rg
Viscosity : Viscosity is the resistance of a fluid to flow due to internal friction. When liquid flows over flat surface, a backward viscous force acts tangentially to every layer.This force depends upon the area of the layer, velocity of the layer, and the distance of the layer from the surface.FAdvdxF=𝜂Advdx𝜂=StressStrainRate where 𝜂 = coefficient of viscosity
Laminar Flow Confined to Tubes - Poiseuille’s Law:It is relationship between horizontal flow and pressure.Q=p2-p1R where Q=Flow Ratep1 and p2 are pressure difference between two points.R is the resistance to flow ( everything except pressure)
The resistance R ( or X) to laminar flow of an incompressible fluid having viscosity 𝜂 through a horizontal tube of uniform radius r and length l , is given byR=8𝜂l𝜋r4 where r = radius of the tubel = length of the tube
Flow of a liquid thru cylindrical pipe: v=(p2-p1)4𝜂L(R2-r2)where L = length of pipeR = radius of tuber = distance r from center where velocity needs to be determined
Poiseuille found that if a pressure difference ( P) is maintained across the two ends of a capillary tube of length 'l' and radius r, then the volume of liquid coming out of the tube per second is( i) Directly proportional to the pressure difference ( P) .( ii) Directly proportional to the fourth power of radius ( r) of the capillary tube( iii) Inversely proportional to the coefficient of viscosity ( 𝜂 ) of the liquid.( iv) Inversely proportional to the length ( l) of the capillary tube.
According to Stokes’ law, the backward viscous force acting on a small spherical body of radius r moving with uniform velocity v through fluid of viscosity η is given byF=6𝜋𝜂rv where r=radius of sphere𝜂 = cooeficient of viscosityF =Retarding forcev=velocity of sphere
When viscous force plus buoyant force becomes equal to force due to gravity, the net force becomes zero. The sphere then descends with a constant terminal velocity (vt) .6𝜋𝜂rvt=43𝜋r3(𝜌-𝜎)g where𝜌=densityofsphere𝜎=densityofliquid
Laminar flow occurs when the fluid flows in infinitesimal parallel layers with no disruption between them.
Steady Flow: A flow that is not a function of time is called steady flow. Steady state flow refers to the condition where the fluid properties at a point in the system do not change over time.
Critical velocity: Velocity of liquiid flow below which its flow is streamline and above which its flow becomes turbulent is called Critical Velocity.
The Venturimeter is a device to measure theflow speed of incompressible fluid.
REYNOLDS NUMBERRe=𝜌vd𝜂where d=internaldiameterofpipev=speedoffluid𝜌=densityoffluid
Heat Transfer
Heat transfe rate (heat current) thru cross sectiondQdt=-kAdTdxwhere K=thermal conductivity of material
Amount of heat flowing per second Heat Current =i=𝛥Q𝛥t=KAT1-T2x
Thermal Resistance=RR=xKA
Heat Current =i=𝛥Q𝛥t=T1-T2R where R is thermal resistance
Thermal Resistance in seriesR=R1+R2+R3+...
Thermal Resistance in parallel1R=1R1+1R2+1R3+...
Emmisive Power: It is energy emmited per unit area, per unit time, per unit soldit angle along normal to area.E=𝛥U(𝛥A)(𝛥𝜔)(𝛥t)
Absorptive Power: It is defined as fraction of the incident radiation that is absorbed by the body.AbsorptivePower=𝛼=energyabsorbedenergyincident
Kirchhoff's Law: The ratio of emmisive power to absorptive power is the same for all bodies at a given temperature and is equal to the emmisive power of a black body at that temperature.E(body)𝛼(body)=E(blackbody)
Thermal RadiationAs the temperature is increased, the wavelength corresponding to the highest intensity decreases.
Wien's displacement law𝜆maxT=bfor black body, Wien constant= b=0.288cmK𝜆max= wavelength at which Intensity is maximum.
Stepan Bolzmann LawThe energy of the thermal radiation emitted per unit time by a black body
Thermal Expansion Lt=L0(1+𝛼𝛥T)At=A0(1+𝛽𝛥T)Vt=V0(1+𝛾𝛥T) where𝛼 =linear coefficient of thermal expansion𝛽 =superficial coefficient of thermal expansion𝛾=volumetric coefficient of thermal expansion
First Law of thermodynamics𝛥Q=𝛥U+𝛥Wwhere 𝛥Q = heat supplied to the system.𝛥W = work done by the system.𝛥U = change in internal energy in the system.
Work done by the system against constant P is𝛥W=P𝛥V
Specific Heat Capacity (s) s=Sm=1𝛥Qm𝛥T
Work done by system in isothermal process 𝛥W=𝜇RTlnV2V1
Work done by system in adiabetic process𝛥W=𝜇R(T1-T2)𝛾-1 𝛥Wadiabetic=(P2V2-P1V1)1-𝛾=𝜇R(T1-T2)𝛾-1
Work done in isochoric process𝛥W=0( as volume is constant)
Critical TemperatureCritical temperature ( Tc ) of a substance is the temperature at and above which vapor of the substance cannot be liquified, no matter how much pressure is applied.
Critical Pressure:The pressure required to liquify a gas at its critical temperature is called Critical Pressure (Pc ) .
Critical Volume:The volume of he gas at critical temperature and critical pressure is called Critical Volume ( Vc )
RTcPcVc=83This is called critical coefficient. It is same for all gases.
Polarization : dipole moment per unit volume P=𝜒eEwhere𝜒e=Electric susceptibility of dielectric material. Polarisation is also defined as amount of induced surface charge per unit area So,P=QinducedA
Energy stored in capacitor=E=12CV2=12QV=Q22C
Electric field intensity at a point near and outside the surface of a charged conductor of any shape:E=𝜎𝜖0kwhere𝜎=surfacechargedensity
Electric field due to uniformly charged infinite plane sheet:E=𝜎2𝜖0where𝜎=surfacechargedensity
Mechanical force acting on unit area of a charged conductor =f=𝜎22𝜖0
The electrostatic energy per unit volume is called energy density=12𝜖0E2
Electric field due to electric dipole making angle 𝜃 with dipole E=p4𝜋𝜖0r33cos2𝜃+1
Current Electricity
V = I R
ResistanceR=𝜌lAwhere 𝜌=resistivity
Current Densityj=IA
I=neAVdI = currentn = number of electrons per unit volumee = charge of electronA = Cross sectional AreaVd = drift velocity
Mobility is drift velocity per unit volume𝜇=|vd|E
Temperature dependance of resistivity𝜌T=𝜌0[1+𝛼(T-T0)]
Magnetic force on current carrying conductor (F)F=(nAl)qVdB where n = number of charge per unit volumeA=cross-sectional area of conductorl = length of conductor\Vd = drift\ velocityB = magnetic field F=IlBwhereI=currentflowingthruconductorl=vectormagnitudeoflengthofconductorB=magneticfield
Magnetic force on a moving charge (F)F=qvB
Magnetic field due to current element (Biot-Savarat Law)𝛥B=𝜇04𝜋I𝛥lsin𝜃r2
Magnetic field surrounding a thin straight current carrying conductor at point PB=𝜇04𝜋Id(sin𝛼+sin𝛽) where d=perpendicular distance d from wire𝛼,𝛽=angle subtended by end of wire ar point P against perpendicular distance
Magnetic field surrounding a infinetely long thin straight current carrying conductor at point PB=𝜇02𝜋Id
Magnetic field on the axis of a circular current loopB=𝜇0IR22(x2+R2)3/2
Magnetic field at the center of the loop B=𝜇0I2R
Magnetic field at the center due to an arc B=𝜇04𝜋IR𝜃 where 𝜃=anglesubtendedbyarcinradianR=radius of arc
Magnetic field due to long solenoid (infinite length)B=𝜇onIwhere n=number of turns per unit lengthI=current thru solenoid
Magnetic field at one end of solenoid (infinite length)B=12𝜇onI where n=number of turns per unit lengthI=current thru solenoid
Magnetic field at at point P of solenoid (finite length)B=𝜇onI2(sin𝛼+sin𝛽) where n=number of turns per unit lengthI=current thru solenoid𝛼,𝛽=angle subtended by end of solenoid ar point P against perpendicular.
Magnetic moment of current loopm=NIA where,I = current in loopA = area of the loop (vector)N = Number of turns
Magnetic field along the axis of the current carrying loop B=𝜇0R22x3=𝜇04𝜋2mx3 where m=magnetic moment
Magnetic field in the plane of the current carrying loop B=𝜇0R22x3=𝜇04𝜋mx3 where m=magnetic moment
Force between two long parallel conductors per unit lengthF=𝜇0IaIb2𝜋d
Torque on Current Loop𝜏=IAB𝜏=m×B where, I=current in loopA=area of the loop ( vector)B=Magnetic fieldm= Magnetic moment
Magnetic Pole Strength: It strength of the magnetic pole to attract magnetic materials towards itself.M=magnetic dipole momentm=pole strength2l=length of bar magnetthenM=(m)2l)Unit of pole strength=Am
If magnet is cut into two equal pieces, such that length of each piece becomes half, then pole strength does not change.
If magnet is cut into two equal pieces, such that width of each piece becomes half, then pole strength becomes half.
Coulomb's law for magnetismF=𝜇04𝜋m1×m2r2
Magnetic field strength at any point is defined as force experienced by hypothetical north pole of unit pole strengthB=Fm
LR Circuit: growth of currenti=i0(1-e-RLt)i0=ER at steady stateTime constant=𝜏L=LR Time constant refers to time constant when current rises about 63% of final value.
LR Circuit: decay of currenti=i0e-RLti0=ER at steady state
Cyclotron frequency of revolution of particle=𝜈c𝜈c=qB2𝜋m
Coefficient of coupling = K : It measures the manner in which 2 coils are coupledK=ML1L2M=mutual inductanceK=0, if there is no couplingK=1, for maximum coupling0<K<1ifthereisairgap
Coefficient of coupling K=magneticfluxlinkedinsecondarycoilmagneticfluxlinkedinprimarycoil
Combination of mutual inductanceIn seriesLs=L1+L2±2M
Combination of inductanceIn parallel <