Motion in a straight line with constant a:

V=u t at, s= ut + at, 12- =2as

Relative Velocity: TAJB = ÜA ÜB

0.1: Physical Constants
Speed of light 3 x 10® m/s
Planck constant h 6.63 x 10-34 Js
hc 1242 eV-nm H
Gravitation constant G 6.67x 10-" m³ kgs-2 Projectile Motion:
Boltzmann constant 1.38 x 10-2 J/K u cos

Molar gas constant 8.314 J/(mol K) R

Avogadro's number NA 6.023 x 1023 mol-l

Charge of electron 1.602 x 10-19 C 2= ut cOs , y=ut sin 0-gt?
Permeability of vac 47 × 10-7 N/A? y= tan -
uum 2u cos?
Permitivity of vacuum 8.85 x 10-12 F/m 2u sin u sin 20 u sin?
T= R H=
Coulomb constant 4TeD 9x 10 Nm²/C2 2g
Faraday constant F 96485 C/mol
Mass of electron e 9.1 x 10-31 kg
1.3: Newton's Laws and Friction
Mass of proton 1.6726 × 10-27 kg
Mass of neutron 1.6749 × 10-2" kg Linear momentum: p= má
Atomic mass unit 1.66 x 10-2 kg
Atomic mass unit 931.49 MeV/c2 Newton's first law: inertial frame.
Stefan-Boltznann 5.67x 10-s W/ (n² K*) Newton's second law: F= , F= mã
constant
Rydberg constant Roo 1.097 x 10 m-!
Newton's third law: FAB = -FBA
Bohr magneton PB 9.27 x 10- J/T
0.529 x 10-10,
Bohr radius m Frictional force: ftatic, max = LsN, fkinetie =*N
Standard atnosphere atm 1.01325 x 10° Pa
Wien 2.9 x 10-3 m K Banking angle: = tan , u+tan
displacement rg 1-u tan b
constant
Centripetal force: F = "mu²,

1 MECHANICS Pseudo force: Fpseudo =-7m¯o, Foentrifugal =U

1.1: Vectors Minimum speed to complete vertical circle:

Notation: ä = agi t ay,j + ay k

Umin, bottom = V5gl, U'min, top = Vgl

Magnitude: a = la = /a+ a + a?
Dot product: ab= agby + ayby t azb, = ab cos Conical pendulum: T=2n/Lcos
mg

Cross product:
1.4: Work, Power and Energy
Work: W= F.5= FS cos 8, WN= [Y.da
| x Qj = ab sin Kinetic energy: K = mu =
Potential energy: F=-U/Ôz for conservative forces.
1.2: Kinematics
Ugravitational = mgh, Uspring = ka?
Average and Instantaneous Vel. and Accel.:

Day = AY/At, Dnst dr/dt Work done by conservative forces is path indepen
ainst = di/dt dent and depends only on initial and final points:
aav = At/At Fconservative d= 0.

Work-energy theorem: W = AK
Mechanical energy: E=U+K. Conserved if forces are Rotation about an axis with constant a:
conservative in nature.
w= Wo t at, e= wt +at, w?- w?=2ae
Power Pav AW
A
Pnst = F.

1.5: Centre of Mass and Collision Moment of Inertia: I = L, mr:2, I=frPdm

Centre of mass: Tem Tcm =Tdn mr² +mr? mr² mr² m? mr mr

CM of few useful configurations: ring disk shell sphere rod hollow solid rectangle

1. m1, m2 separated by r:
mË +m2
Theorem of Parallel Axes: I, = len + md?
CII

2. Triangle (CM = Centroid) ye =4

Theorem of Perp. Axes: I, = I,+ 1,
3. Semicircular ring: ye =
Radius of Gyration: k = I/m
4. Semicircular disc: ye =
Angular Moment um: L=Yx, L= l3
5. Hemispherical shell: ye = Ci Torque: Y=YxF, = T= la

6. Solid Hemisphere: ye =
Conservation of L: Text =0 ’ L= const.
7. Cone: the height of CM from the base is h/4 for Equilibrium condition: SF =6, Si=0
the solid cone and h/3 for the holow cone.
Kinetic Energy: Krot =lu?
Motion of the CM: M =)m; Dynamics:

Fost Tem = Iema, Fext = macm Pem = mem

Vem m; Pem = Micms lcm
M M K=mvem + Lemw' L= lemo +Yem Xmïem
Impulse: J= fYdt Aj 1.7: Gravitation
Before collision, After collision
Collision: m m F Fm
Gravitational force: F = G "
Momentum conservation: mju1 tm2U2 = m1UË tm2v
GMm
Elastic Colision: mqv4mzUy?=m1u+mzv Potential energy: U=
Coefficient of restitution:
Gravitational acceleration: g=
1, completely elastic
e =
0, completely in-elastic Variation of g with depth: 1nstdo g(1-)
Variation of g with height: goutside g(1 )
f vz = 0 and mË < m2 then v =U1.
If v2 =0 and m1 > m2 then u, =2U1. Effect of non-spherical earth shape on g:
Elastic collision with m1 = m2 : u = U2 and v = U1. Jat pole > gat equator(. Re - R, N 21 km)

Effect of earth rotation on apparent weight:

1.6: Rigid Body Dynamics
Angular velocity: wv = , w=, =öxF
Angular Accel.: aay = , a = , i= x
 A
Superposition of two SHM's:
mw?Rcos
mg =mng mw?Rcos?
R
T1 = A, sin wt. T2 = A2 sin(wt + 6)
= 01 t æ = Asin(wt + e)
Orbital velocity of satellite: vo =
A= yA,?+A,° + 2A1Ag cos ß
Az sin &
Escape velocity: Ve = /2GM tane
A1 + Ag cos ð

Kepler's laws: 1.9: Properties of Matter

First: Elliptical orbit with sun at one of the focus. Modulus of rigidity: Y = B= -VA, =8
Second: Areal velocity is constant. (. dLjdt = 0).
An
Third: T² x a'. In circular orbit T?= GM Compressibility: K
==-+
lateral strain AD/D
Poisson's ratio: o = pngitudinal strainAIJI
1.8: Simple Harmonic Motion Elastic energy: U=; stress x strain X volume
Hooke's law: F=-kz (for small elongation z.)

Acceleration: a = = m
2=-w Surface tension: S= F/l
Time period: T = 2 = 2r Surface energy: U = SA
Displacement: z = Asin{wt + ) Excess pressure in bubble:

Velocity: v= Aw cos(wt + ¢) = twvA-2 Apair =2S/R, Apsoap =4S/R

Capillary rise: h= 2Scos
rpg

Potential energy: U=~kr"

-A
Hydrostatic pressure: p= pgh
Kinetic energy K= m1? Buoyant force: Fg = pVg = Weight of displaced liquid
-A 0 A
Equation of continuity: A1UË = Agv2
Total energy: E =U +K=muA?
Bernoulli's equation: p+ spu' + pgh = constant
Torricelli's theorem: vefflux = V2gh
Simple pendulum: T= 2n/ Viscous force: F= -nA

Physical Pendulum: T= 2r/a Stoke's law: F= 6rnrv

Poiseuilli's eguation: Volume ffow

Torsional Pendulum T= 2r/ time 8n

Terminal velocity: v = 2r"(p-a)g

Springs in series:

Springs in parallel: keg = kË + k2

2 Waves 4. 1st overtone/2nd harmonics: Ë =
2.1: Waves Motion 5. 2nd overtone/3d harmonics: v2 =
General equation of wave: = . 6. All harmonics are present.

Notation: Amplitude A, Frequency v, Wavelength À, Pe

riod T, Angular Frequency w, Wave Number k,
27T 27
T== v=vÀ, k = String fixed at one end:
A

A/2
Progressive wave travelling with speed v:
1. Boundary conditions: y =0 at æ
y=f(t -a/v), ’ t2; y=f(t + æ/v), -*
2. Allowed Freq.: L =(2n +1), v=n /
V , n=
0.1,2,. ...

Progressive sine wave:

3. Fundamental /1tharnonics: = v
4. 1 overtone/3rd harmonics: vË =
y= Asin(ka wt) = Asin(2r(z/)- t/T)
5. 2nd overtone/5th harmonics: v2 =
2.2: Waves on a String 6. Only odd harmonics are present.
Speed of waves on a string with mass per unit length u
and tension T: v= VT/u
Transmitted power: Pay = 2r²4wA2
Sonometer: vx t,va VT, vxa. v=/
Interference: 2.3: Sound Waves

y1 = Aj sin(kz wt), y2 = Az sin(kr wt + o) Displacement wave: s = %sin w(t - æ/v)

y=y1 + y2 = Asin(kz -wt +e)| Pressure wave: p= po cos w(t - a/v), po = (Buw/v)so
A= /A+ A? + 2A1 Azcos& Speed of sound waves:
Az sin ð
tan e = B
A1 + A2 cos 8 Vliquid = Vsolid =
2n7, constructive;
1(2rn + 1)7, destructive.
Intensity: I= 2 s = 2pv

Standing Waves:
Standing longitudinal waves:
A/4
P1 = Po sin w(t - */v), p2 = Po sin w(t + æ/v)
V1 = A1 sin(kæ wt), y2 = Ag sin(kæ + wt) p=P1 t p2 = 2po cos kæ sin wt
y=y1 t y2 = (2A cos kæ) sin wt
m= (7+)$ nodes; n=0, 1,2,...
n-, antinodes. n = 0, 1, 2,...
Closed organ pipe:

String Axed at both ends:

A/2 1. Boundary condition: y =0at a =0
1. Boundary conditions: y =0 at a =0 and at =L 2. Allowed freq.: L = (2rn +1) , v= (2n + 1), n =
0,1,2,...
2. Allowed Freq.: L=n¡, v= / , n=1,2,3, ... 3. Fundamental/1st harmonics: =
3. Fundamental/1st harmonics: vo = 4. 1st overtone/3rd harmonics: =3v0 =
 5. 2nd overtone/5'h harmonics: v2= 5 = SO
Path difference: Az =
6. Only odd harnonics are present. S D

Phase difference: 8 = Az
Interference Conditions: for integer n,
Open organ pipe:
2rT, constructive;
O (2n +1)m, destructive,
1. Boundary condition: y =0at a =0 constructive;
Allowed freq.: L = n, v= n n=l, 2,... Aæ =
(n+) A, destructive
2. Fundamental/1*t harmonics: V =
3. 1t overtone/2nd harnonics: V= 2uo = E Intensity:
4. 2nd overtone/3rd harmonics: y = 3u0 = I=h+h+ 2/Ihlh cos 8,
5. All harmonics are present. Imta (Vh- VE)
h=h:l= 4lo cos",Imax =4lo, Imin = 0
Fringe width: w=
Resonance column: Optical path: Ar'= uAr

Interference of waves transmitted through thin ilm:

h td=, atd= U= 2(l2-l)
nd, constructive;
Beats: two waves of almost equal frequencies wË w2 Ar = 2yd = 1(n+) A, destructive.
P1= Po s0n w1(t a/), P2 =PO sin wz(t - a/v)
p=p1+P2 = 2po cos Aw(t - /v) sin w(t-*/v) Diffraction from a single slit:
w = (wË t wz)/2, Aw = w- wz (beats freg.)
For Minima: nÀ = bsin® b(y/D)
Doppler Effect:
Resolution: sin =
Law of Malus: I = I cos?
where, v is the speed of sound in the medium, uo is
the speed of the observer w.r.t. the mediumn, consid
ered positive when it moves towards the source and
negative when it moves away from the source, and us
is the speed of the source w.r.t. the medium, consid
ered positive when it moves towards the observer and
negative when it moves away from the observer.

2.4: Light Waves

Plane Wave: E = Eosin w(t -#), I= %

Spherical Wave: E= Eo sinwt-), I =

Young's double slit experiment

3 Optics Lens maker's formula: =(u-1)
3.1: Reflection of Light
normal Lens formula: - = m=
Laws of reflection:
incident i!r reflected ()
Incident ray, reflected ray, and normal lie in the same
plane (ii) Zi= Zr Power of the lens: P =, P in diopter if f in metre.
Two thin lenses separated by distance d:
Plane mirror:

d
(i) the image and the object are equidistant from mir
ror (ii) virtual image of real object
f2

Spherical Mirror:
3.3: Optical Instruments
Simple microscope: m= D/f in normal adjustment.
1. Focal length f = R/2 Objective Eyepiece
2. Mirror equation: +=
3. Magnification: m= Compound microscope:

3.2: Refraction of Light D

Speed of light in vacuum

Refractive index: #= speed of light in medium 1. Magnification normal adjustment: m=
incident reflected 2. Resolving power: R= = sn
sin i
Snell's Law: sin r
fo
refracted

real depth Astronomical telescope:

Apparent depth: = apparent depth

Critical angle: . =sin

1. In normal adjustment: m = -4, L= fotie
2. Resolving power: R=

3.4: Dispersion
Deviation by a prism:
Cauchy's equation: =0 + A> 0
Dispersion by prism with small A and i:
8=i+i- A, general result 1. Mean deviation: y, = (y -1)A
=
sin i=i for minimum deviation 2. Angular dispersion: = (y -r)A
sin
Dispersive power: w = B (if Aand i small)
On =(u- 1)A, for small A
Dispersion without deviation:

2 (Hy- 1)A +(, - 1)A' = 0

Refraction at spherical surface: Deviation without dispersion:
(hy -r)A= (W, - ,) A'

m=
4 Heat and Thermodynamics 4.4: Theromodynamic Processes

4.1: Heat and Temperature First law of thermodynamics: AQ= AU + AW

Temp. scales: F =32+ , K=C+ 273.16 Work done by the gas:

V
Ideal gas equation: pV = nRT, n: number of moles pdy
AW =pAV, W=
van der Waals equation: (p+ )(V-b) = nRT
Thermal expansion: L= Lo(1+ aAT),
A= Ao(1+ BAT), V=Vo(1+yAT), y=28 =3a
Wisothermal = nRT ln
()
Wisobarie =p(V2-)
Thermal stress of a material: = Y Wadiabatie =
y-1
Wisochorle = 0
4.2: Kinetic Theory of Gases
General: M= mNA, k = R/N
+Q1
Efficiency of the heat engine:
Maxwell distribution of speed:
work done by the engine
F=
heat supplied to it Q1
RMS speed: Urms 3kT Tz
= Vm lcarnot=1
T1
8k1 SRT
Average speed: = VM

Most probable speed: Vp =V 2KTm Coeff. of performance of refrigerator: + W

4Q2
2

Pressure: p= pums
COP = =
Equipartition of energy: K = }kT for each degree of
freedom. Thus, K = Tfor molecule having f de Entropy: AS =, Sf - S = f
grees of freedoms.
Const. T: AS = , Varying T : AS = ms ln
Internal energy of n moles of an ideal gas is U = nRT.
Adiabatic process: AQ =0, pV = constant
4.3: Specific Heat
4.5: Heat Transfer
Specific heat: s= máT
Conduction: 4 =-KAAT
Latent heat: L = Q/m
Thermal resistance: R= KA
Specific heat at constant volumne: Cy = AT..
Raetes = Ry +Ry =+(+)
Specific heat at constant pressure: Cp =
K2 A2
Relation between G and Cy: Cp - Cy =R Hparallel =t=(K1A1 + K2A2)

Ratio of specific heats: Y=Cp/O,

emissi ve power Ebody
Relation between U and C,: AU = nC,AT Kirchhoff's Law: absorptive pOwer Gbody
= Eblackbody

Specific heat of gas mixture:

n1Cu1 + n2Cv2 Wien's displacement law: AmT = b
Cy = y=
n1Cu1 + ngCy2

Molar internal energy of an ideal gas: U = RT, Stefan-Boltzmann law: =eAT4

f=3 for monatomic and f=5 for diatomnic gas.
Newton's law of cooling: =-bA(T -T,)
5 Electricity and Magnetism 5.3: Capacitors
5.1: Electrostatics Capacitance: C = g/V
Coulomb's law: F=g2 f
Parallel plate capacitor: C= toA/d
d
1
Electric field: E(F) =

Electrostatic energy: U= 4TTEp r Spherical capacitor: C = ATeoriTz

T2-T1

Electrostatic potential: V=g

dV =-:F, V)=-|ÉdF Cylindrical capacitor: C=2neal

In(ra/r1)

Electric dipole moment: j= qd

-qt d
Capacitors in parallel: Ceg O +Cz
B
1 pcos 8
Potential of adipole: V= Trco
Capacitors in series: =t+ AB
Field of a dipole:
E, Force between plates of a parallel plate capacitor:
E9
F=92Aco

1_2p cos 1 psin

Energy stored in capacitor: U= BCv = =v
Torque on a dipole placed in Ä: F=px Ä Energy density in electric feld E: U/V= eoE?
Capacitor with dielectric: C = oKA
Pot. energy of a dipole placed in E: U=-pE d

5.2: Gauss's Law and its Applications 5.4: Current electricity

Electric flux: =f da Current density: j =i/A =aE
Gauss's law: fE- dS = qin/¬0 Drift speed: va=T= eA

Field of a uniformly charged ring on its axis: Resistance of a wire: R= l/A, where p =1l/o
Temp. dependence of resistance: R= Ro(1 + aAT)
Ohm's law: V = iR
E and V of a uniformly charged sphere:
1_Qr for r <R Kirchhoff's Laws: () The Junction Lau: The algebraic
E= sum of all the currents directed towards a node is zero
for r > R
R i.e., Enode I; = 0. (ii) The Loop Law: The algebraic
V=spg3- e). tor r<R
for r > R
V
sum of all the potential differences along a closed loop
in a circuit is zero i.e., loopA Vi = 0.
4TEn r
R

E and V of a uniformly charged spherical shell: Resistors in parallel: =t

0, for r < R
E= E
4TEn for r > R Resistors in series: Reg = RË + R
R

for r < R R1 R2
V= 1_4 for r > R
R Wheatstone bridge: R
R
Field of aline charge: E= r
Balanced if R/Ry = Rs/Ra.
Field of an infinite sheet: E=
Electric Power: P= V/R= I'R= IV
Field in the vicinity of conducting surface: E =
 Energy of a magnetic dipole placed in B:
Galvanometer as an Ammeter: U= -iB
i-i,
iG= (i- i,)s B
Hall effect: V,. = Bi
ned
Galvanometer as a Voltmeter:

VAB =i(R+ G) 5.6: Magnetic Field due to Current

Charging of capacitors: Biot-Savart law: dÄ - 4n idixr

at) = Cv1-e te
Field due to a straight conductor:
Discharging of capacitors: g(t) = goe c g(t)

B= 4(cos 1 - cos 62)

Time constant in RC circuit: T= RC
Field due to an infinite straight wire: B =
Peltier effect: emf e= A0 Peltier heat
Garge transfer red Force between parallel wires: d = oiiz

Seeback effect: T
To T
Field on the axis of a ring:
1, Thermo-emf: e= aT +}6T?
2. Thermnoelectric power: de/dt = a + bT.
3. Neutral temp.: T, = -a/b.
4. Inversion temp.: T;=-2a/b.
Thomson effect: emf e= = cThomson
aed heat
= oAT.
Faraday's law of electrolysis: The mass deposited is
ia?
Bp = 2(a2+d²y3/2

Field at the centre of an arc: B= otb

4Ta
oi
Field at the centre of aring: B =
m= Zit =+Eit
Ampere's law: fB.dl =on
where i is current, t is time, Z is electrochemical equiv
alent, E is chemical equivalent, and F= 96485 C/g is Field inside a solenoid: B = uoi, n = 0000000000
Faraday constant.

5.5: Magnetism Field inside a toroid: B= r

Lorentz force on a moving charge: F= qi x B+ q¼

Charged particle in a uniform magnetic field: Field of a bar magnet:
N+ B
gB
B, = o 211 B2 =
4T d3

Horizontal . . B
Force on a current carrying wire: Angle of dip: B, = B cos & N
B.

Tangent galvanometer: B,tan = i=Ktan

Magnetic moment of a current loop (dipole): Moving coil galvanometer: niAB = k9, i=¤
à =iÃ
Time period of magnetometer: T= r/ B,
Torque on a magnetic dipole placed in B: 7= ixB Permeability: B
= ui
5.7: Electromagnetic Induction
RC circuit:
Magnetic flux: =fB- da eo sinwt
R

Faraday's law: e= d¢ Z= /R+(1/wC, tan =aCR

Lenz's Law: Induced current create a B-field that op R

poses the change in magnetic flux. LR circuit:

wL

eo sin wt
Z= R+wL', tan =
Motional emf: e = Blu

LCR Circuit: -wL

Self inductance: ¢= Li, e=-L4 ep sln wt

wL R

Self inductance of a solenoid: L = pg n² (rr²!) Z= /R2 +(ao-wL) tan = -wl

Growth of current in LR circuit: i =

Power factor: P=ermglrms COs
0.63
Transformer: =, eji =egi2
1

Decay of current in LR circuit: i= ige L7R Speed of the EM waves in vacuum: c=l/uo¬o
to

0.37io

Time constant of LR circuit: T= L/R

Energy stored in an inductor: ð=Li2
B
Energy density of Bfield: u==
Mutual inductance: =Mi, e==-M

EMF induced in a rotating coil: e= NABw sin wt

Alternating current:

i= io sin(wt +), T=2r/w

Average current in AC: i= + i dt = 0

RMS current: ima +so Pàt-#

Energy: E= imsRT
Capacitive reactance: X,=
Inductive reactan ce: X, = wL

Imepedance: Z = eo/io
6 Modern Physics
6.1: Photo-electric effect
Population at time t: N = NÍe-At NG

ti/2
Photon's energy: E = hu = hc/A
Photon's momentum: p = h/) = E/c Half life: t/2 = 0.693/A
Max. KE of ejected photo-electron: Kmax = hu Average life: tay =1/)
Threshold freq. in photo-electric effect: v = /h Population after n half lives: N = No/2".
V Mass defect: Am = (Zm, + (A- Z)m] - M
Stopping potential: V, =e () Binding energy: B= |Zm, + (A - Z)m, - M]e
Q-value: Q= U;-U,
de Broglie wavelength: A= h/p Energy released in nuclear reaction: AE Amc?
where Am == mreactants mproducts

6.2: The Atom

Energy in nth Bohr's orbit: 6.4: Vacuum tubes and Semiconductors

mZe 13.6Z2
eV
En = Seoh2n2 En = Half Wave Rectifier:
R$Output
Radius of the nth Bohr's orbit:
eoh²n2 n²ao Full Wave Rectifier:
ao = 0.529 A
TmZe

Quantization of the angular momentum: l = Grid

Triode Valve: Catho de

Photon energy in state transition: E, - E, = hu Filament -Plate

Ez

E E Plate resistance of a triode: Tp AV

Ai, AV.=0
Emission Absorption

Wavelength of emitted radiation: for a transition

Transconductance of a triode: gm =
from nth to mth state:

Amplification by atriode: =- AV
Relation between rps L, and gm: = Ip X Jm
Ka

X-ray spectrum: Amin =G

Current in a transistor: I, = I,+I
min

Moseley's law: D= a(Z- b)

a and 8 parameters of a transistor: a = , B =
X-ray diffraction: 2dsin0= nÀ , B= a
Heisenberg uncertainity principle: Al.
ApAr > h/ (2r), AEAt > h/ (2r) Transconductance: gm = AVbe
Logic Gates:
AND OR NAND NOR KOR
6.3: The Nucleus A AB A+B AB A+B AB +
0 1 1 0
1 1 1

Nuclear radius: R= RÍA'/3, Ro l.lx l0-15 m 1

1

1
1

1 0
1

Decay rate: dt
=-AN