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# Physics Formula Chart JEE Notes | EduRev

## JEE : Physics Formula Chart JEE Notes | EduRev

``` Page 1

0.1: Physical Constants
Speed of light c 310
8
m=s
Planck constant h 6:6310
34
J s
hc 1242 eV-nm
Gravitation constant G 6:6710
11
m
3
kg
1
s
2
Boltzmann constant k 1:3810
23
J=K
Molar gas constant R 8:314 J=(mol K)
23
mol
1
Charge of electron e 1:60210
19
C
Permeability of vac-
uum
0 410
7
N=A
2
Permitivity of vacuum 0 8:8510
12
F=m
Coulomb constant
1
4
0
910
9
N m
2
=C
2
Mass of electron me 9:110
31
kg
Mass of proton mp 1:672610
27
kg
Mass of neutron mn 1:674910
27
kg
Atomic mass unit u 1:6610
27
kg
Atomic mass unit u 931:49 MeV=c
2
Stefan-Boltzmann
constant
 5:6710
8
W=(m
2
K
4
)
Rydberg constant R1 1:09710
7
m
1
Bohr magneton B 9:2710
24
J=T
10
m
Standard atmosphere atm 1:0132510
5
Pa
Wien displacement
constant
b 2:910
3
m K
1 MECHANICS
1.1: Vectors
Notation: ~ a =a
x
^ { +a
y
^ | +a
z
^
k
Magnitude: a =j~ aj =
q
a
2
x
+a
2
y
+a
2
z
Dot product: ~ a
~
b =a
x
b
x
+a
y
b
y
+a
z
b
z
=ab cos
Cross product:
~ a
~
b
~ a
~
b

^ {
^ | ^
k
~ a
~
b = (a
y
b
z
a
z
b
y
)^ {+(a
z
b
x
a
x
b
z
)^ |+(a
x
b
y
a
y
b
x
)
^
k
j~ a
~
bj =ab sin
1.2: Kinematics
Average and Instantaneous Vel. and Accel.:
~ v
av
= ~ r=t; ~ v
inst
=d~ r=dt
~ a
av
= ~ v=t ~ a
inst
=d~ v=dt
Motion in a straight line with constant a:
v =u +at; s =ut +
1
2
at
2
; v
2
u
2
= 2as
Relative Velocity: ~ v
A=B
=~ v
A
~ v
B
Projectile Motion:
x
y
O
u sin
u cos
u

R
H
x =ut cos; y =ut sin
1
2
gt
2
y =x tan
g
2u
2
cos
2

x
2
T =
2u sin
g
; R =
u
2
sin 2
g
; H =
u
2
sin
2

2g
1.3: Newton's Laws and Friction
Linear momentum: ~ p =m~ v
Newton's rst law: inertial frame.
Newton's second law:
~
F =
d~ p
dt
;
~
F =m~ a
Newton's third law:
~
F
AB
=
~
F
BA
Frictional force: f
static, max
=
s
N; f
kinetic
=
k
N
Banking angle:
v
2
rg
= tan,
v
2
rg
=
+tan
1 tan
Centripetal force: F
c
=
mv
2
r
; a
c
=
v
2
r
Pseudo force:
~
F
pseudo
=m~ a
0
; F
centrifugal
=
mv
2
r
Minimum speed to complete vertical circle:
v
min, bottom
=
p
5gl; v
min, top
=
p
gl
Conical pendulum: T = 2
q
l cos
g
mg
T
l


1.4: Work, Power and Energy
Work: W =
~
F
~
S =FS cos; W =
R
~
F d
~
S
Kinetic energy: K =
1
2
mv
2
=
p
2
2m
Potential energy: F =@U=@x for conservative forces.
U
gravitational
=mgh; U
spring
=
1
2
kx
2
Work done by conservative forces is path indepen-
dent and depends only on initial and nal points:
H
~
F
conservative
 d~ r = 0.
Work-energy theorem: W = K
Mechanical energy: E =U +K. Conserved if forces are
conservative in nature.
Power P
av
=
W
t
; P
inst
=
~
F~ v

Page 2

0.1: Physical Constants
Speed of light c 310
8
m=s
Planck constant h 6:6310
34
J s
hc 1242 eV-nm
Gravitation constant G 6:6710
11
m
3
kg
1
s
2
Boltzmann constant k 1:3810
23
J=K
Molar gas constant R 8:314 J=(mol K)
23
mol
1
Charge of electron e 1:60210
19
C
Permeability of vac-
uum
0 410
7
N=A
2
Permitivity of vacuum 0 8:8510
12
F=m
Coulomb constant
1
4
0
910
9
N m
2
=C
2
Mass of electron me 9:110
31
kg
Mass of proton mp 1:672610
27
kg
Mass of neutron mn 1:674910
27
kg
Atomic mass unit u 1:6610
27
kg
Atomic mass unit u 931:49 MeV=c
2
Stefan-Boltzmann
constant
 5:6710
8
W=(m
2
K
4
)
Rydberg constant R1 1:09710
7
m
1
Bohr magneton B 9:2710
24
J=T
10
m
Standard atmosphere atm 1:0132510
5
Pa
Wien displacement
constant
b 2:910
3
m K
1 MECHANICS
1.1: Vectors
Notation: ~ a =a
x
^ { +a
y
^ | +a
z
^
k
Magnitude: a =j~ aj =
q
a
2
x
+a
2
y
+a
2
z
Dot product: ~ a
~
b =a
x
b
x
+a
y
b
y
+a
z
b
z
=ab cos
Cross product:
~ a
~
b
~ a
~
b

^ {
^ | ^
k
~ a
~
b = (a
y
b
z
a
z
b
y
)^ {+(a
z
b
x
a
x
b
z
)^ |+(a
x
b
y
a
y
b
x
)
^
k
j~ a
~
bj =ab sin
1.2: Kinematics
Average and Instantaneous Vel. and Accel.:
~ v
av
= ~ r=t; ~ v
inst
=d~ r=dt
~ a
av
= ~ v=t ~ a
inst
=d~ v=dt
Motion in a straight line with constant a:
v =u +at; s =ut +
1
2
at
2
; v
2
u
2
= 2as
Relative Velocity: ~ v
A=B
=~ v
A
~ v
B
Projectile Motion:
x
y
O
u sin
u cos
u

R
H
x =ut cos; y =ut sin
1
2
gt
2
y =x tan
g
2u
2
cos
2

x
2
T =
2u sin
g
; R =
u
2
sin 2
g
; H =
u
2
sin
2

2g
1.3: Newton's Laws and Friction
Linear momentum: ~ p =m~ v
Newton's rst law: inertial frame.
Newton's second law:
~
F =
d~ p
dt
;
~
F =m~ a
Newton's third law:
~
F
AB
=
~
F
BA
Frictional force: f
static, max
=
s
N; f
kinetic
=
k
N
Banking angle:
v
2
rg
= tan,
v
2
rg
=
+tan
1 tan
Centripetal force: F
c
=
mv
2
r
; a
c
=
v
2
r
Pseudo force:
~
F
pseudo
=m~ a
0
; F
centrifugal
=
mv
2
r
Minimum speed to complete vertical circle:
v
min, bottom
=
p
5gl; v
min, top
=
p
gl
Conical pendulum: T = 2
q
l cos
g
mg
T
l


1.4: Work, Power and Energy
Work: W =
~
F
~
S =FS cos; W =
R
~
F d
~
S
Kinetic energy: K =
1
2
mv
2
=
p
2
2m
Potential energy: F =@U=@x for conservative forces.
U
gravitational
=mgh; U
spring
=
1
2
kx
2
Work done by conservative forces is path indepen-
dent and depends only on initial and nal points:
H
~
F
conservative
 d~ r = 0.
Work-energy theorem: W = K
Mechanical energy: E =U +K. Conserved if forces are
conservative in nature.
Power P
av
=
W
t
; P
inst
=
~
F~ v

1.5: Centre of Mass and Collision
Centre of mass: x
cm
=
P
ximi
P
mi
; x
cm
=
R
xdm
R
dm
CM of few useful congurations:
1. m
1
, m
2
separated by r:
m
1
m
2
C
r
m
2
r
m
1
+m
2
m
1
r
m
1
+m
2
2. Triangle (CM Centroid) y
c
=
h
3
C
h
3
h
3. Semicircular ring: y
c
=
2r

C
2r

r
4. Semicircular disc: y
c
=
4r
3 C
4r
3
r
5. Hemispherical shell: y
c
=
r
2
C
r
r
2
6. Solid Hemisphere: y
c
=
3r
8
C
r
3r
8
7. Cone: the height of CM from the base is h=4 for
the solid cone and h=3 for the hollow cone.
Motion of the CM: M =
P
m
i
~ v
cm
=
P
m
i
~ v
i
M
; ~ p
cm
=M~ v
cm
; ~ a
cm
=
~
F
ext
M
Impulse:
~
J =
R
~
F dt = ~ p
Collision:
m
1
m
2
v
1
v
2
Before collision After collision
m
1
m
2
v
0
1
v
0
2
Momentum conservation: m
1
v
1
+m
2
v
2
=m
1
v
0
1
+m
2
v
0
2
Elastic Collision:
1
2
m
1
v
1
2
+
1
2
m
2
v
2
2
=
1
2
m
1
v
0
1
2
+
1
2
m
2
v
0
2
2
Coecient of restitution:
e =
(v
0
1
v
0
2
)
v
1
v
2
=

1; completely elastic
0; completely in-elastic
If v
2
= 0 and m
1
m
2
then v
0
1
=v
1
.
If v
2
= 0 and m
1
m
2
then v
0
2
= 2v
1
.
Elastic collision with m
1
=m
2
: v
0
1
=v
2
and v
0
2
=v
1
.
1.6: Rigid Body Dynamics
Angular velocity: !
av
=

t
; ! =
d
dt
; ~ v =~ !~ r
Angular Accel.:
av
=
!
t
;  =
d!
dt
; ~ a =~ ~ r
Rotation about an axis with constant :
! =!
0
+t;  =!t +
1
2
t
2
; !
2
!
0
2
= 2
Moment of Inertia: I =
P
i
m
i
r
i
2
; I =
R
r
2
dm
ring
mr
2
disk
1
2
mr
2
shell
2
3
mr
2
sphere
2
5
mr
2
rod
1
12
ml
2
hollow
mr
2
solid
1
2
mr
2
rectangle
m(a
2
+b
2
)
12
a
b
Theorem of Parallel Axes: I
k
=I
cm
+md
2
cm
I
k
d
Ic
Theorem of Perp. Axes: I
z
=I
x
+I
y
x
y
z
p
I=m
Angular Momentum:
~
L =~ r~ p;
~
L =I~ !
Torque: ~  =~ r
~
F; ~  =
d
~
L
dt
;  =I
O
x
y
P
~ r
~
F

Conservation of
~
L: ~
ext
= 0 =)
~
L = const.
Equilibrium condition:
P
~
F =
~
0;
P
~  =
~
0
Kinetic Energy: K
rot
=
1
2
I!
2
Dynamics:
~
cm
=I
cm
~ ;
~
F
ext
=m~ a
cm
; ~ p
cm
=m~ v
cm
K =
1
2
mv
cm
2
+
1
2
I
cm
!
2
;
~
L =I
cm
~ ! +~ r
cm
m~ v
cm
1.7: Gravitation
Gravitational force: F =G
m1m2
r
2
m
1
m
2 F F
r
Potential energy: U =
GMm
r
Gravitational acceleration: g =
GM
R
2
Variation of g with depth: g
inside
g

1
2h
R

Variation of g with height: g
outside
g

1
h
R

Eect of non-spherical earth shape on g:
g
at pole
>g
at equator
(*R
e
R
p
 21 km)
Eect of earth rotation on apparent weight:
mg
0

=mgm!
2
R cos
2

R
m!
2
R cos
mg
~ !

Orbital velocity of satellite: v
o
=
q
GM
R
Escape velocity: v
e
=
q
2GM
R

Page 3

0.1: Physical Constants
Speed of light c 310
8
m=s
Planck constant h 6:6310
34
J s
hc 1242 eV-nm
Gravitation constant G 6:6710
11
m
3
kg
1
s
2
Boltzmann constant k 1:3810
23
J=K
Molar gas constant R 8:314 J=(mol K)
23
mol
1
Charge of electron e 1:60210
19
C
Permeability of vac-
uum
0 410
7
N=A
2
Permitivity of vacuum 0 8:8510
12
F=m
Coulomb constant
1
4
0
910
9
N m
2
=C
2
Mass of electron me 9:110
31
kg
Mass of proton mp 1:672610
27
kg
Mass of neutron mn 1:674910
27
kg
Atomic mass unit u 1:6610
27
kg
Atomic mass unit u 931:49 MeV=c
2
Stefan-Boltzmann
constant
 5:6710
8
W=(m
2
K
4
)
Rydberg constant R1 1:09710
7
m
1
Bohr magneton B 9:2710
24
J=T
10
m
Standard atmosphere atm 1:0132510
5
Pa
Wien displacement
constant
b 2:910
3
m K
1 MECHANICS
1.1: Vectors
Notation: ~ a =a
x
^ { +a
y
^ | +a
z
^
k
Magnitude: a =j~ aj =
q
a
2
x
+a
2
y
+a
2
z
Dot product: ~ a
~
b =a
x
b
x
+a
y
b
y
+a
z
b
z
=ab cos
Cross product:
~ a
~
b
~ a
~
b

^ {
^ | ^
k
~ a
~
b = (a
y
b
z
a
z
b
y
)^ {+(a
z
b
x
a
x
b
z
)^ |+(a
x
b
y
a
y
b
x
)
^
k
j~ a
~
bj =ab sin
1.2: Kinematics
Average and Instantaneous Vel. and Accel.:
~ v
av
= ~ r=t; ~ v
inst
=d~ r=dt
~ a
av
= ~ v=t ~ a
inst
=d~ v=dt
Motion in a straight line with constant a:
v =u +at; s =ut +
1
2
at
2
; v
2
u
2
= 2as
Relative Velocity: ~ v
A=B
=~ v
A
~ v
B
Projectile Motion:
x
y
O
u sin
u cos
u

R
H
x =ut cos; y =ut sin
1
2
gt
2
y =x tan
g
2u
2
cos
2

x
2
T =
2u sin
g
; R =
u
2
sin 2
g
; H =
u
2
sin
2

2g
1.3: Newton's Laws and Friction
Linear momentum: ~ p =m~ v
Newton's rst law: inertial frame.
Newton's second law:
~
F =
d~ p
dt
;
~
F =m~ a
Newton's third law:
~
F
AB
=
~
F
BA
Frictional force: f
static, max
=
s
N; f
kinetic
=
k
N
Banking angle:
v
2
rg
= tan,
v
2
rg
=
+tan
1 tan
Centripetal force: F
c
=
mv
2
r
; a
c
=
v
2
r
Pseudo force:
~
F
pseudo
=m~ a
0
; F
centrifugal
=
mv
2
r
Minimum speed to complete vertical circle:
v
min, bottom
=
p
5gl; v
min, top
=
p
gl
Conical pendulum: T = 2
q
l cos
g
mg
T
l


1.4: Work, Power and Energy
Work: W =
~
F
~
S =FS cos; W =
R
~
F d
~
S
Kinetic energy: K =
1
2
mv
2
=
p
2
2m
Potential energy: F =@U=@x for conservative forces.
U
gravitational
=mgh; U
spring
=
1
2
kx
2
Work done by conservative forces is path indepen-
dent and depends only on initial and nal points:
H
~
F
conservative
 d~ r = 0.
Work-energy theorem: W = K
Mechanical energy: E =U +K. Conserved if forces are
conservative in nature.
Power P
av
=
W
t
; P
inst
=
~
F~ v

1.5: Centre of Mass and Collision
Centre of mass: x
cm
=
P
ximi
P
mi
; x
cm
=
R
xdm
R
dm
CM of few useful congurations:
1. m
1
, m
2
separated by r:
m
1
m
2
C
r
m
2
r
m
1
+m
2
m
1
r
m
1
+m
2
2. Triangle (CM Centroid) y
c
=
h
3
C
h
3
h
3. Semicircular ring: y
c
=
2r

C
2r

r
4. Semicircular disc: y
c
=
4r
3 C
4r
3
r
5. Hemispherical shell: y
c
=
r
2
C
r
r
2
6. Solid Hemisphere: y
c
=
3r
8
C
r
3r
8
7. Cone: the height of CM from the base is h=4 for
the solid cone and h=3 for the hollow cone.
Motion of the CM: M =
P
m
i
~ v
cm
=
P
m
i
~ v
i
M
; ~ p
cm
=M~ v
cm
; ~ a
cm
=
~
F
ext
M
Impulse:
~
J =
R
~
F dt = ~ p
Collision:
m
1
m
2
v
1
v
2
Before collision After collision
m
1
m
2
v
0
1
v
0
2
Momentum conservation: m
1
v
1
+m
2
v
2
=m
1
v
0
1
+m
2
v
0
2
Elastic Collision:
1
2
m
1
v
1
2
+
1
2
m
2
v
2
2
=
1
2
m
1
v
0
1
2
+
1
2
m
2
v
0
2
2
Coecient of restitution:
e =
(v
0
1
v
0
2
)
v
1
v
2
=

1; completely elastic
0; completely in-elastic
If v
2
= 0 and m
1
m
2
then v
0
1
=v
1
.
If v
2
= 0 and m
1
m
2
then v
0
2
= 2v
1
.
Elastic collision with m
1
=m
2
: v
0
1
=v
2
and v
0
2
=v
1
.
1.6: Rigid Body Dynamics
Angular velocity: !
av
=

t
; ! =
d
dt
; ~ v =~ !~ r
Angular Accel.:
av
=
!
t
;  =
d!
dt
; ~ a =~ ~ r
Rotation about an axis with constant :
! =!
0
+t;  =!t +
1
2
t
2
; !
2
!
0
2
= 2
Moment of Inertia: I =
P
i
m
i
r
i
2
; I =
R
r
2
dm
ring
mr
2
disk
1
2
mr
2
shell
2
3
mr
2
sphere
2
5
mr
2
rod
1
12
ml
2
hollow
mr
2
solid
1
2
mr
2
rectangle
m(a
2
+b
2
)
12
a
b
Theorem of Parallel Axes: I
k
=I
cm
+md
2
cm
I
k
d
Ic
Theorem of Perp. Axes: I
z
=I
x
+I
y
x
y
z
p
I=m
Angular Momentum:
~
L =~ r~ p;
~
L =I~ !
Torque: ~  =~ r
~
F; ~  =
d
~
L
dt
;  =I
O
x
y
P
~ r
~
F

Conservation of
~
L: ~
ext
= 0 =)
~
L = const.
Equilibrium condition:
P
~
F =
~
0;
P
~  =
~
0
Kinetic Energy: K
rot
=
1
2
I!
2
Dynamics:
~
cm
=I
cm
~ ;
~
F
ext
=m~ a
cm
; ~ p
cm
=m~ v
cm
K =
1
2
mv
cm
2
+
1
2
I
cm
!
2
;
~
L =I
cm
~ ! +~ r
cm
m~ v
cm
1.7: Gravitation
Gravitational force: F =G
m1m2
r
2
m
1
m
2 F F
r
Potential energy: U =
GMm
r
Gravitational acceleration: g =
GM
R
2
Variation of g with depth: g
inside
g

1
2h
R

Variation of g with height: g
outside
g

1
h
R

Eect of non-spherical earth shape on g:
g
at pole
>g
at equator
(*R
e
R
p
 21 km)
Eect of earth rotation on apparent weight:
mg
0

=mgm!
2
R cos
2

R
m!
2
R cos
mg
~ !

Orbital velocity of satellite: v
o
=
q
GM
R
Escape velocity: v
e
=
q
2GM
R

Kepler's laws:
vo
a
First: Elliptical orbit with sun at one of the focus.
Second: Areal velocity is constant. (* d
~
L=dt = 0).
Third: T
2
/a
3
. In circular orbit T
2
=
4
2
GM
a
3
.
1.8: Simple Harmonic Motion
Hooke's law: F =kx (for small elongation x.)
Acceleration: a =
d
2
x
dt
2
=
k
m
x =!
2
x
Time period: T =
2
!
= 2
p
m
k
Displacement: x =A sin(!t +)
Velocity: v =A! cos(!t +) =!
p
A
2
x
2
Potential energy: U =
1
2
kx
2
A 0 A
x
U
Kinetic energy K =
1
2
mv
2
A 0 A
x
K
Total energy: E =U +K =
1
2
m!
2
A
2
Simple pendulum: T = 2
q
l
g
l
Physical Pendulum: T = 2
q
I
mgl
Torsional Pendulum T = 2
q
I
k
Springs in series:
1
keq
=
1
k1
+
1
k2
k
1
k
2
Springs in parallel: k
eq
=k
1
+k
2
k
1
k
2
Superposition of two SHM's:
~
A
1
~
A
2
~
A
 
x
1
=A
1
sin!t; x
2
=A
2
sin(!t +)
x =x
1
+x
2
=A sin(!t +)
A =
q
A
1
2
+A
2
2
+ 2A
1
A
2
cos
tan =
A
2
sin
A
1
+A
2
cos
1.9: Properties of Matter
Modulus of rigidity: Y =
F=A
l=l
; B =V
P
V
;  =
F
A
Compressibility: K =
1
B
=
1
V
dV
dP
Poisson's ratio:  =
lateral strain
longitudinal strain
=
D=D
l=l
Elastic energy: U =
1
2
stress strain volume
Surface tension: S =F=l
Surface energy: U =SA
Excess pressure in bubble:
p
air
= 2S=R; p
soap
= 4S=R
Capillary rise: h =
2S cos
rg
Hydrostatic pressure: p =gh
Buoyant force: F
B
=Vg = Weight of displaced liquid
Equation of continuity: A
1
v
1
=A
2
v
2
v
1
v
2
Bernoulli's equation: p +
1
2
v
2
+gh = constant
Torricelli's theorem: v
eux
=
p
2gh
Viscous force: F =A
dv
dx
Stoke's law: F = 6rv
F
v
Poiseuilli's equation:
Volume
ow
time
=
pr
4
8l
l
r
Terminal velocity: v
t
=
2r
2
()g
9

Page 4

0.1: Physical Constants
Speed of light c 310
8
m=s
Planck constant h 6:6310
34
J s
hc 1242 eV-nm
Gravitation constant G 6:6710
11
m
3
kg
1
s
2
Boltzmann constant k 1:3810
23
J=K
Molar gas constant R 8:314 J=(mol K)
23
mol
1
Charge of electron e 1:60210
19
C
Permeability of vac-
uum
0 410
7
N=A
2
Permitivity of vacuum 0 8:8510
12
F=m
Coulomb constant
1
4
0
910
9
N m
2
=C
2
Mass of electron me 9:110
31
kg
Mass of proton mp 1:672610
27
kg
Mass of neutron mn 1:674910
27
kg
Atomic mass unit u 1:6610
27
kg
Atomic mass unit u 931:49 MeV=c
2
Stefan-Boltzmann
constant
 5:6710
8
W=(m
2
K
4
)
Rydberg constant R1 1:09710
7
m
1
Bohr magneton B 9:2710
24
J=T
10
m
Standard atmosphere atm 1:0132510
5
Pa
Wien displacement
constant
b 2:910
3
m K
1 MECHANICS
1.1: Vectors
Notation: ~ a =a
x
^ { +a
y
^ | +a
z
^
k
Magnitude: a =j~ aj =
q
a
2
x
+a
2
y
+a
2
z
Dot product: ~ a
~
b =a
x
b
x
+a
y
b
y
+a
z
b
z
=ab cos
Cross product:
~ a
~
b
~ a
~
b

^ {
^ | ^
k
~ a
~
b = (a
y
b
z
a
z
b
y
)^ {+(a
z
b
x
a
x
b
z
)^ |+(a
x
b
y
a
y
b
x
)
^
k
j~ a
~
bj =ab sin
1.2: Kinematics
Average and Instantaneous Vel. and Accel.:
~ v
av
= ~ r=t; ~ v
inst
=d~ r=dt
~ a
av
= ~ v=t ~ a
inst
=d~ v=dt
Motion in a straight line with constant a:
v =u +at; s =ut +
1
2
at
2
; v
2
u
2
= 2as
Relative Velocity: ~ v
A=B
=~ v
A
~ v
B
Projectile Motion:
x
y
O
u sin
u cos
u

R
H
x =ut cos; y =ut sin
1
2
gt
2
y =x tan
g
2u
2
cos
2

x
2
T =
2u sin
g
; R =
u
2
sin 2
g
; H =
u
2
sin
2

2g
1.3: Newton's Laws and Friction
Linear momentum: ~ p =m~ v
Newton's rst law: inertial frame.
Newton's second law:
~
F =
d~ p
dt
;
~
F =m~ a
Newton's third law:
~
F
AB
=
~
F
BA
Frictional force: f
static, max
=
s
N; f
kinetic
=
k
N
Banking angle:
v
2
rg
= tan,
v
2
rg
=
+tan
1 tan
Centripetal force: F
c
=
mv
2
r
; a
c
=
v
2
r
Pseudo force:
~
F
pseudo
=m~ a
0
; F
centrifugal
=
mv
2
r
Minimum speed to complete vertical circle:
v
min, bottom
=
p
5gl; v
min, top
=
p
gl
Conical pendulum: T = 2
q
l cos
g
mg
T
l


1.4: Work, Power and Energy
Work: W =
~
F
~
S =FS cos; W =
R
~
F d
~
S
Kinetic energy: K =
1
2
mv
2
=
p
2
2m
Potential energy: F =@U=@x for conservative forces.
U
gravitational
=mgh; U
spring
=
1
2
kx
2
Work done by conservative forces is path indepen-
dent and depends only on initial and nal points:
H
~
F
conservative
 d~ r = 0.
Work-energy theorem: W = K
Mechanical energy: E =U +K. Conserved if forces are
conservative in nature.
Power P
av
=
W
t
; P
inst
=
~
F~ v

1.5: Centre of Mass and Collision
Centre of mass: x
cm
=
P
ximi
P
mi
; x
cm
=
R
xdm
R
dm
CM of few useful congurations:
1. m
1
, m
2
separated by r:
m
1
m
2
C
r
m
2
r
m
1
+m
2
m
1
r
m
1
+m
2
2. Triangle (CM Centroid) y
c
=
h
3
C
h
3
h
3. Semicircular ring: y
c
=
2r

C
2r

r
4. Semicircular disc: y
c
=
4r
3 C
4r
3
r
5. Hemispherical shell: y
c
=
r
2
C
r
r
2
6. Solid Hemisphere: y
c
=
3r
8
C
r
3r
8
7. Cone: the height of CM from the base is h=4 for
the solid cone and h=3 for the hollow cone.
Motion of the CM: M =
P
m
i
~ v
cm
=
P
m
i
~ v
i
M
; ~ p
cm
=M~ v
cm
; ~ a
cm
=
~
F
ext
M
Impulse:
~
J =
R
~
F dt = ~ p
Collision:
m
1
m
2
v
1
v
2
Before collision After collision
m
1
m
2
v
0
1
v
0
2
Momentum conservation: m
1
v
1
+m
2
v
2
=m
1
v
0
1
+m
2
v
0
2
Elastic Collision:
1
2
m
1
v
1
2
+
1
2
m
2
v
2
2
=
1
2
m
1
v
0
1
2
+
1
2
m
2
v
0
2
2
Coecient of restitution:
e =
(v
0
1
v
0
2
)
v
1
v
2
=

1; completely elastic
0; completely in-elastic
If v
2
= 0 and m
1
m
2
then v
0
1
=v
1
.
If v
2
= 0 and m
1
m
2
then v
0
2
= 2v
1
.
Elastic collision with m
1
=m
2
: v
0
1
=v
2
and v
0
2
=v
1
.
1.6: Rigid Body Dynamics
Angular velocity: !
av
=

t
; ! =
d
dt
; ~ v =~ !~ r
Angular Accel.:
av
=
!
t
;  =
d!
dt
; ~ a =~ ~ r
Rotation about an axis with constant :
! =!
0
+t;  =!t +
1
2
t
2
; !
2
!
0
2
= 2
Moment of Inertia: I =
P
i
m
i
r
i
2
; I =
R
r
2
dm
ring
mr
2
disk
1
2
mr
2
shell
2
3
mr
2
sphere
2
5
mr
2
rod
1
12
ml
2
hollow
mr
2
solid
1
2
mr
2
rectangle
m(a
2
+b
2
)
12
a
b
Theorem of Parallel Axes: I
k
=I
cm
+md
2
cm
I
k
d
Ic
Theorem of Perp. Axes: I
z
=I
x
+I
y
x
y
z
p
I=m
Angular Momentum:
~
L =~ r~ p;
~
L =I~ !
Torque: ~  =~ r
~
F; ~  =
d
~
L
dt
;  =I
O
x
y
P
~ r
~
F

Conservation of
~
L: ~
ext
= 0 =)
~
L = const.
Equilibrium condition:
P
~
F =
~
0;
P
~  =
~
0
Kinetic Energy: K
rot
=
1
2
I!
2
Dynamics:
~
cm
=I
cm
~ ;
~
F
ext
=m~ a
cm
; ~ p
cm
=m~ v
cm
K =
1
2
mv
cm
2
+
1
2
I
cm
!
2
;
~
L =I
cm
~ ! +~ r
cm
m~ v
cm
1.7: Gravitation
Gravitational force: F =G
m1m2
r
2
m
1
m
2 F F
r
Potential energy: U =
GMm
r
Gravitational acceleration: g =
GM
R
2
Variation of g with depth: g
inside
g

1
2h
R

Variation of g with height: g
outside
g

1
h
R

Eect of non-spherical earth shape on g:
g
at pole
>g
at equator
(*R
e
R
p
 21 km)
Eect of earth rotation on apparent weight:
mg
0

=mgm!
2
R cos
2

R
m!
2
R cos
mg
~ !

Orbital velocity of satellite: v
o
=
q
GM
R
Escape velocity: v
e
=
q
2GM
R

Kepler's laws:
vo
a
First: Elliptical orbit with sun at one of the focus.
Second: Areal velocity is constant. (* d
~
L=dt = 0).
Third: T
2
/a
3
. In circular orbit T
2
=
4
2
GM
a
3
.
1.8: Simple Harmonic Motion
Hooke's law: F =kx (for small elongation x.)
Acceleration: a =
d
2
x
dt
2
=
k
m
x =!
2
x
Time period: T =
2
!
= 2
p
m
k
Displacement: x =A sin(!t +)
Velocity: v =A! cos(!t +) =!
p
A
2
x
2
Potential energy: U =
1
2
kx
2
A 0 A
x
U
Kinetic energy K =
1
2
mv
2
A 0 A
x
K
Total energy: E =U +K =
1
2
m!
2
A
2
Simple pendulum: T = 2
q
l
g
l
Physical Pendulum: T = 2
q
I
mgl
Torsional Pendulum T = 2
q
I
k
Springs in series:
1
keq
=
1
k1
+
1
k2
k
1
k
2
Springs in parallel: k
eq
=k
1
+k
2
k
1
k
2
Superposition of two SHM's:
~
A
1
~
A
2
~
A
 
x
1
=A
1
sin!t; x
2
=A
2
sin(!t +)
x =x
1
+x
2
=A sin(!t +)
A =
q
A
1
2
+A
2
2
+ 2A
1
A
2
cos
tan =
A
2
sin
A
1
+A
2
cos
1.9: Properties of Matter
Modulus of rigidity: Y =
F=A
l=l
; B =V
P
V
;  =
F
A
Compressibility: K =
1
B
=
1
V
dV
dP
Poisson's ratio:  =
lateral strain
longitudinal strain
=
D=D
l=l
Elastic energy: U =
1
2
stress strain volume
Surface tension: S =F=l
Surface energy: U =SA
Excess pressure in bubble:
p
air
= 2S=R; p
soap
= 4S=R
Capillary rise: h =
2S cos
rg
Hydrostatic pressure: p =gh
Buoyant force: F
B
=Vg = Weight of displaced liquid
Equation of continuity: A
1
v
1
=A
2
v
2
v
1
v
2
Bernoulli's equation: p +
1
2
v
2
+gh = constant
Torricelli's theorem: v
eux
=
p
2gh
Viscous force: F =A
dv
dx
Stoke's law: F = 6rv
F
v
Poiseuilli's equation:
Volume
ow
time
=
pr
4
8l
l
r
Terminal velocity: v
t
=
2r
2
()g
9

2 Waves
2.1: Waves Motion
General equation of wave:
@
2
y
@x
2
=
1
v
2
@
2
y
@t
2
.
Notation: Amplitude A, Frequency , Wavelength , Pe-
riod T , Angular Frequency !, Wave Number k,
T =
1

=
2
!
; v =; k =
2

Progressive wave travelling with speed v:
y =f(tx=v); +x; y =f(t +x=v); x
Progressive sine wave:

2

x
y
A
y =A sin(kx!t) =A sin(2 (x=t=T ))
2.2: Waves on a String
Speed of waves on a string with mass per unit length 
and tension T : v =
p
T=
Transmitted power: P
av
= 2
2
vA
2

2
Interference:
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx!t +)
y =y
1
+y
2
=A sin(kx!t +)
A =
q
A
1
2
+A
2
2
+ 2A
1
A
2
cos
tan =
A
2
sin
A
1
+A
2
cos
 =

2n; constructive;
(2n + 1); destructive:
Standing Waves:
2Acoskx
A N A N A
x
=4
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx +!t)
y =y
1
+y
2
= (2A coskx) sin!t
x =

n +
1
2


2
; nodes; n = 0; 1; 2;:::
n

2
; antinodes. n = 0; 1; 2;:::
String xed at both ends:
L
N
A N A
N
=2
1. Boundary conditions: y = 0 at x = 0 and at x =L
2. Allowed Freq.: L =n

2
;  =
n
2L
q
T

; n = 1; 2; 3;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
2L
q
T

4. 1
st
overtone/2
nd
harmonics: 
1
=
2
2L
q
T

5. 2
nd
overtone/3
rd
harmonics: 
2
=
3
2L
q
T

6. All harmonics are present.
String xed at one end:
L
N
A N
A
=2
1. Boundary conditions: y = 0 at x = 0
2. Allowed Freq.: L = (2n + 1)

4
;  =
2n+1
4L
q
T

; n =
0; 1; 2;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
4L
q
T

4. 1
st
overtone/3
rd
harmonics: 
1
=
3
4L
q
T

5. 2
nd
overtone/5
th
harmonics: 
2
=
5
4L
q
T

6. Only odd harmonics are present.
Sonometer: /
1
L
, /
p
T , /
1
p

.  =
n
2L
q
T

2.3: Sound Waves
Displacement wave: s =s
0
sin!(tx=v)
Pressure wave: p =p
0
cos!(tx=v); p
0
= (B!=v)s
0
Speed of sound waves:
v
liquid
=
s
B

; v
solid
=
s
Y

; v
gas
=
s

P

Intensity: I =
2
2
B
v
s
0
2

2
=
p0
2
v
2B
=
p0
2
2v
Standing longitudinal waves:
p
1
=p
0
sin!(tx=v); p
2
=p
0
sin!(t +x=v)
p =p
1
+p
2
= 2p
0
coskx sin!t
Closed organ pipe:
L
1. Boundary condition: y = 0 at x = 0
2. Allowed freq.: L = (2n + 1)

4
;  = (2n + 1)
v
4L
; n =
0; 1; 2;:::
3. Fundamental/1
st
harmonics: 
0
=
v
4L
4. 1
st
overtone/3
rd
harmonics: 
1
= 3
0
=
3v
4L

Page 5

0.1: Physical Constants
Speed of light c 310
8
m=s
Planck constant h 6:6310
34
J s
hc 1242 eV-nm
Gravitation constant G 6:6710
11
m
3
kg
1
s
2
Boltzmann constant k 1:3810
23
J=K
Molar gas constant R 8:314 J=(mol K)
23
mol
1
Charge of electron e 1:60210
19
C
Permeability of vac-
uum
0 410
7
N=A
2
Permitivity of vacuum 0 8:8510
12
F=m
Coulomb constant
1
4
0
910
9
N m
2
=C
2
Mass of electron me 9:110
31
kg
Mass of proton mp 1:672610
27
kg
Mass of neutron mn 1:674910
27
kg
Atomic mass unit u 1:6610
27
kg
Atomic mass unit u 931:49 MeV=c
2
Stefan-Boltzmann
constant
 5:6710
8
W=(m
2
K
4
)
Rydberg constant R1 1:09710
7
m
1
Bohr magneton B 9:2710
24
J=T
10
m
Standard atmosphere atm 1:0132510
5
Pa
Wien displacement
constant
b 2:910
3
m K
1 MECHANICS
1.1: Vectors
Notation: ~ a =a
x
^ { +a
y
^ | +a
z
^
k
Magnitude: a =j~ aj =
q
a
2
x
+a
2
y
+a
2
z
Dot product: ~ a
~
b =a
x
b
x
+a
y
b
y
+a
z
b
z
=ab cos
Cross product:
~ a
~
b
~ a
~
b

^ {
^ | ^
k
~ a
~
b = (a
y
b
z
a
z
b
y
)^ {+(a
z
b
x
a
x
b
z
)^ |+(a
x
b
y
a
y
b
x
)
^
k
j~ a
~
bj =ab sin
1.2: Kinematics
Average and Instantaneous Vel. and Accel.:
~ v
av
= ~ r=t; ~ v
inst
=d~ r=dt
~ a
av
= ~ v=t ~ a
inst
=d~ v=dt
Motion in a straight line with constant a:
v =u +at; s =ut +
1
2
at
2
; v
2
u
2
= 2as
Relative Velocity: ~ v
A=B
=~ v
A
~ v
B
Projectile Motion:
x
y
O
u sin
u cos
u

R
H
x =ut cos; y =ut sin
1
2
gt
2
y =x tan
g
2u
2
cos
2

x
2
T =
2u sin
g
; R =
u
2
sin 2
g
; H =
u
2
sin
2

2g
1.3: Newton's Laws and Friction
Linear momentum: ~ p =m~ v
Newton's rst law: inertial frame.
Newton's second law:
~
F =
d~ p
dt
;
~
F =m~ a
Newton's third law:
~
F
AB
=
~
F
BA
Frictional force: f
static, max
=
s
N; f
kinetic
=
k
N
Banking angle:
v
2
rg
= tan,
v
2
rg
=
+tan
1 tan
Centripetal force: F
c
=
mv
2
r
; a
c
=
v
2
r
Pseudo force:
~
F
pseudo
=m~ a
0
; F
centrifugal
=
mv
2
r
Minimum speed to complete vertical circle:
v
min, bottom
=
p
5gl; v
min, top
=
p
gl
Conical pendulum: T = 2
q
l cos
g
mg
T
l


1.4: Work, Power and Energy
Work: W =
~
F
~
S =FS cos; W =
R
~
F d
~
S
Kinetic energy: K =
1
2
mv
2
=
p
2
2m
Potential energy: F =@U=@x for conservative forces.
U
gravitational
=mgh; U
spring
=
1
2
kx
2
Work done by conservative forces is path indepen-
dent and depends only on initial and nal points:
H
~
F
conservative
 d~ r = 0.
Work-energy theorem: W = K
Mechanical energy: E =U +K. Conserved if forces are
conservative in nature.
Power P
av
=
W
t
; P
inst
=
~
F~ v

1.5: Centre of Mass and Collision
Centre of mass: x
cm
=
P
ximi
P
mi
; x
cm
=
R
xdm
R
dm
CM of few useful congurations:
1. m
1
, m
2
separated by r:
m
1
m
2
C
r
m
2
r
m
1
+m
2
m
1
r
m
1
+m
2
2. Triangle (CM Centroid) y
c
=
h
3
C
h
3
h
3. Semicircular ring: y
c
=
2r

C
2r

r
4. Semicircular disc: y
c
=
4r
3 C
4r
3
r
5. Hemispherical shell: y
c
=
r
2
C
r
r
2
6. Solid Hemisphere: y
c
=
3r
8
C
r
3r
8
7. Cone: the height of CM from the base is h=4 for
the solid cone and h=3 for the hollow cone.
Motion of the CM: M =
P
m
i
~ v
cm
=
P
m
i
~ v
i
M
; ~ p
cm
=M~ v
cm
; ~ a
cm
=
~
F
ext
M
Impulse:
~
J =
R
~
F dt = ~ p
Collision:
m
1
m
2
v
1
v
2
Before collision After collision
m
1
m
2
v
0
1
v
0
2
Momentum conservation: m
1
v
1
+m
2
v
2
=m
1
v
0
1
+m
2
v
0
2
Elastic Collision:
1
2
m
1
v
1
2
+
1
2
m
2
v
2
2
=
1
2
m
1
v
0
1
2
+
1
2
m
2
v
0
2
2
Coecient of restitution:
e =
(v
0
1
v
0
2
)
v
1
v
2
=

1; completely elastic
0; completely in-elastic
If v
2
= 0 and m
1
m
2
then v
0
1
=v
1
.
If v
2
= 0 and m
1
m
2
then v
0
2
= 2v
1
.
Elastic collision with m
1
=m
2
: v
0
1
=v
2
and v
0
2
=v
1
.
1.6: Rigid Body Dynamics
Angular velocity: !
av
=

t
; ! =
d
dt
; ~ v =~ !~ r
Angular Accel.:
av
=
!
t
;  =
d!
dt
; ~ a =~ ~ r
Rotation about an axis with constant :
! =!
0
+t;  =!t +
1
2
t
2
; !
2
!
0
2
= 2
Moment of Inertia: I =
P
i
m
i
r
i
2
; I =
R
r
2
dm
ring
mr
2
disk
1
2
mr
2
shell
2
3
mr
2
sphere
2
5
mr
2
rod
1
12
ml
2
hollow
mr
2
solid
1
2
mr
2
rectangle
m(a
2
+b
2
)
12
a
b
Theorem of Parallel Axes: I
k
=I
cm
+md
2
cm
I
k
d
Ic
Theorem of Perp. Axes: I
z
=I
x
+I
y
x
y
z
p
I=m
Angular Momentum:
~
L =~ r~ p;
~
L =I~ !
Torque: ~  =~ r
~
F; ~  =
d
~
L
dt
;  =I
O
x
y
P
~ r
~
F

Conservation of
~
L: ~
ext
= 0 =)
~
L = const.
Equilibrium condition:
P
~
F =
~
0;
P
~  =
~
0
Kinetic Energy: K
rot
=
1
2
I!
2
Dynamics:
~
cm
=I
cm
~ ;
~
F
ext
=m~ a
cm
; ~ p
cm
=m~ v
cm
K =
1
2
mv
cm
2
+
1
2
I
cm
!
2
;
~
L =I
cm
~ ! +~ r
cm
m~ v
cm
1.7: Gravitation
Gravitational force: F =G
m1m2
r
2
m
1
m
2 F F
r
Potential energy: U =
GMm
r
Gravitational acceleration: g =
GM
R
2
Variation of g with depth: g
inside
g

1
2h
R

Variation of g with height: g
outside
g

1
h
R

Eect of non-spherical earth shape on g:
g
at pole
>g
at equator
(*R
e
R
p
 21 km)
Eect of earth rotation on apparent weight:
mg
0

=mgm!
2
R cos
2

R
m!
2
R cos
mg
~ !

Orbital velocity of satellite: v
o
=
q
GM
R
Escape velocity: v
e
=
q
2GM
R

Kepler's laws:
vo
a
First: Elliptical orbit with sun at one of the focus.
Second: Areal velocity is constant. (* d
~
L=dt = 0).
Third: T
2
/a
3
. In circular orbit T
2
=
4
2
GM
a
3
.
1.8: Simple Harmonic Motion
Hooke's law: F =kx (for small elongation x.)
Acceleration: a =
d
2
x
dt
2
=
k
m
x =!
2
x
Time period: T =
2
!
= 2
p
m
k
Displacement: x =A sin(!t +)
Velocity: v =A! cos(!t +) =!
p
A
2
x
2
Potential energy: U =
1
2
kx
2
A 0 A
x
U
Kinetic energy K =
1
2
mv
2
A 0 A
x
K
Total energy: E =U +K =
1
2
m!
2
A
2
Simple pendulum: T = 2
q
l
g
l
Physical Pendulum: T = 2
q
I
mgl
Torsional Pendulum T = 2
q
I
k
Springs in series:
1
keq
=
1
k1
+
1
k2
k
1
k
2
Springs in parallel: k
eq
=k
1
+k
2
k
1
k
2
Superposition of two SHM's:
~
A
1
~
A
2
~
A
 
x
1
=A
1
sin!t; x
2
=A
2
sin(!t +)
x =x
1
+x
2
=A sin(!t +)
A =
q
A
1
2
+A
2
2
+ 2A
1
A
2
cos
tan =
A
2
sin
A
1
+A
2
cos
1.9: Properties of Matter
Modulus of rigidity: Y =
F=A
l=l
; B =V
P
V
;  =
F
A
Compressibility: K =
1
B
=
1
V
dV
dP
Poisson's ratio:  =
lateral strain
longitudinal strain
=
D=D
l=l
Elastic energy: U =
1
2
stress strain volume
Surface tension: S =F=l
Surface energy: U =SA
Excess pressure in bubble:
p
air
= 2S=R; p
soap
= 4S=R
Capillary rise: h =
2S cos
rg
Hydrostatic pressure: p =gh
Buoyant force: F
B
=Vg = Weight of displaced liquid
Equation of continuity: A
1
v
1
=A
2
v
2
v
1
v
2
Bernoulli's equation: p +
1
2
v
2
+gh = constant
Torricelli's theorem: v
eux
=
p
2gh
Viscous force: F =A
dv
dx
Stoke's law: F = 6rv
F
v
Poiseuilli's equation:
Volume
ow
time
=
pr
4
8l
l
r
Terminal velocity: v
t
=
2r
2
()g
9

2 Waves
2.1: Waves Motion
General equation of wave:
@
2
y
@x
2
=
1
v
2
@
2
y
@t
2
.
Notation: Amplitude A, Frequency , Wavelength , Pe-
riod T , Angular Frequency !, Wave Number k,
T =
1

=
2
!
; v =; k =
2

Progressive wave travelling with speed v:
y =f(tx=v); +x; y =f(t +x=v); x
Progressive sine wave:

2

x
y
A
y =A sin(kx!t) =A sin(2 (x=t=T ))
2.2: Waves on a String
Speed of waves on a string with mass per unit length 
and tension T : v =
p
T=
Transmitted power: P
av
= 2
2
vA
2

2
Interference:
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx!t +)
y =y
1
+y
2
=A sin(kx!t +)
A =
q
A
1
2
+A
2
2
+ 2A
1
A
2
cos
tan =
A
2
sin
A
1
+A
2
cos
 =

2n; constructive;
(2n + 1); destructive:
Standing Waves:
2Acoskx
A N A N A
x
=4
y
1
=A
1
sin(kx!t); y
2
=A
2
sin(kx +!t)
y =y
1
+y
2
= (2A coskx) sin!t
x =

n +
1
2


2
; nodes; n = 0; 1; 2;:::
n

2
; antinodes. n = 0; 1; 2;:::
String xed at both ends:
L
N
A N A
N
=2
1. Boundary conditions: y = 0 at x = 0 and at x =L
2. Allowed Freq.: L =n

2
;  =
n
2L
q
T

; n = 1; 2; 3;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
2L
q
T

4. 1
st
overtone/2
nd
harmonics: 
1
=
2
2L
q
T

5. 2
nd
overtone/3
rd
harmonics: 
2
=
3
2L
q
T

6. All harmonics are present.
String xed at one end:
L
N
A N
A
=2
1. Boundary conditions: y = 0 at x = 0
2. Allowed Freq.: L = (2n + 1)

4
;  =
2n+1
4L
q
T

; n =
0; 1; 2;:::.
3. Fundamental/1
st
harmonics: 
0
=
1
4L
q
T

4. 1
st
overtone/3
rd
harmonics: 
1
=
3
4L
q
T

5. 2
nd
overtone/5
th
harmonics: 
2
=
5
4L
q
T

6. Only odd harmonics are present.
Sonometer: /
1
L
, /
p
T , /
1
p

.  =
n
2L
q
T

2.3: Sound Waves
Displacement wave: s =s
0
sin!(tx=v)
Pressure wave: p =p
0
cos!(tx=v); p
0
= (B!=v)s
0
Speed of sound waves:
v
liquid
=
s
B

; v
solid
=
s
Y

; v
gas
=
s

P

Intensity: I =
2
2
B
v
s
0
2

2
=
p0
2
v
2B
=
p0
2
2v
Standing longitudinal waves:
p
1
=p
0
sin!(tx=v); p
2
=p
0
sin!(t +x=v)
p =p
1
+p
2
= 2p
0
coskx sin!t
Closed organ pipe:
L
1. Boundary condition: y = 0 at x = 0
2. Allowed freq.: L = (2n + 1)

4
;  = (2n + 1)
v
4L
; n =
0; 1; 2;:::
3. Fundamental/1
st
harmonics: 
0
=
v
4L
4. 1
st
overtone/3
rd
harmonics: 
1
= 3
0
=
3v
4L

5. 2
nd
overtone/5
th
harmonics: 
2
= 5
0
=
5v
4L
6. Only odd harmonics are present.
Open organ pipe:
L
A
N
A
N
A
1. Boundary condition: y = 0 at x = 0
Allowed freq.: L =n

2
;  =n
v
4L
; n = 1; 2;:::
2. Fundamental/1
st
harmonics: 
0
=
v
2L
3. 1
st
overtone/2
nd
harmonics: 
1
= 2
0
=
2v
2L
4. 2
nd
overtone/3
rd
harmonics: 
2
= 3
0
=
3v
2L
5. All harmonics are present.
Resonance column:
l1 +d
l2 +d
l
1
+d =

2
; l
2
+d =
3
4
; v = 2(l
2
l
1
)
Beats: two waves of almost equal frequencies !
1
!
2
p
1
=p
0
sin!
1
(tx=v); p
2
=p
0
sin!
2
(tx=v)
p =p
1
+p
2
= 2p
0
cos !(tx=v) sin!(tx=v)
! = (!
1
+!
2
)=2; ! =!
1
!
2
(beats freq.)
Doppler Eect:
 =
v +u
o
vu
s

0
where, v is the speed of sound in the medium, u
0
is
the speed of the observer w.r.t. the medium, consid-
ered positive when it moves towards the source and
negative when it moves away from the source, and u
s
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 =E
0
sin!(t
x
v
); I =I
0
Spherical Wave: E =
aE0
r
sin!(t
r
v
); I =
I0
r
2
Young's double slit experiment
Path dierence: x =
dy
D
S1
P
S2
d
y
D

Phase dierence:  =
2

x
Interference Conditions: for integer n,
 =

2n; constructive;
(2n + 1); destructive;
x =

n; constructive;

n +
1
2

; destructive
Intensity:
I =I
1
+I
2
+ 2
p
I
1
I
2
cos;
I
max
=

p
I
1
+
p
I
2

2
; I
min
=

p
I
1

p
I
2

2
I
1
=I
2
:I = 4I
0
cos
2 
2
; I
max
= 4I
0
; I
min
= 0
Fringe width: w =
D
d
Optical path: x
0
=x
Interference of waves transmitted through thin lm:
x = 2d =

n; constructive;

n +
1
2

; destructive:
Diraction from a single slit: 
b
y
y
D
For Minima: n =b sinb(y=D)
Resolution: sin =
1:22
b
Law of Malus: I =I
0
cos
2

I0 I


```
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