Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
The Gateway to Success
IkkB’kkyk
GURUKUL
(The Temple of Education
...
Basic Mathematics used in Physics
*
Roots of ax2 + bx + c = 0 are x
b
a
*
c
a
cot
hypotenuse
base
co sec
sin
Binomial theorem
a
a 2 b2
cosec
n(n 1) 2 n(n 1)(n 2) 3
(1 x) n 1 nx
x
x
...
2
6
b
cos
sec
hypotenuse
perpendicular
2
1
cos
1
tan
sin(A B) sin A cos B cos A sin B
co s(A B) cos A cos B sin A sin B
Logarithm
tan(A B)
log m = log m – log n
P a
q b
then
tan A tan B
1 tan A tan B
sin2A=2sinAcosA
cos2A = cos2A – sin2A = 1 – 2sin2A = 2cos2A–1
n
log mn = n log m
logem = 2
...
3010
log 3 = 0
...
a + (n – 1)d
here d = common difference
Sum of n terms Sn n 2a (n 1)d
2
Note :
(i) 1 2 3 4 5
...
n 2
*
n(n 1)(2n 1)
6
Geometrical progression-GP
a, ar, ar2, ar3,
...
3o
sin
perpendicular
hypotenuse
Sine law
cos
base
hypotenuse
*
cosine law
cos A
b 2 c2 a 2
c2 a 2 b 2
a 2 b2 c 2
,cos B
,cos C
2bc
2ca
2ab
GURUKUL ikB”kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
By help of differentiation, if y is given, we can find
and by help of integration, if
Average of a varying quantity
x2
4
3
K
U
Integration
*
3
Volume of a sphere r (r = radius)
du
dv
v u
u
dy
dx
dx
y
v
dx
v2
n
L
U
Volume of a rectangular slab = length × breadth × height =
abt
Volume of a cube = (side)3
dy df g x d g x
y f g x
dx
dg(x)
dx
*
Area of a parallelogram = base × height
Area of curved surface of cylinder 2 r
(r = radius and = length)
Area of whole surface of cylinder 2r(r )
( = length)
Area of ellipse ab (a & b are semi major and semi minor axis respectively)
Surface area of a cube = 6(side)2
Total surface area of a cone r 2 r
dy
dx
is given, we can find y
...
The maximum and minimum values of function
A cos Bsin are A 2 B2 and A 2 B2 respectively
(a + b)2 = a2 + b2 + 2ab
(a – b)2 = a2+b2 – 2ab
2
2
(a + b)(a – b) = |a – b |
(a + b)2 = a3 + b3 + 3ab(a + b)
3
3
3
(a + b) = a – b – 3ab(a – b)
x1
Formulae for determinantion of area
Area of a square = (side)2
Area of rectangle = length × breadth
1
2
Area of triangle base × height
Area of a trapezoid 1 (distance between parallel sides)
2
× (sum of parallel sides)
Area enclosed by a circle
(r = radius)
r2
Surface area of a sphere 4 r
2
dy
dx
(r = radius)
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Unit and Dimension
*
*
*
Fundamental or base quantities :
The quantities which do not depend upon other
quantities for their complete definition are known
as fundamental or base quantities
...
g
...
Derived quantities :
The quantities which can be expressed in terms
of the fundamental quantites are known as derived quantities e
...
Speed (=distance/time), volume, acceleration,
dorce, pressure, etc
...
Physical Quantity = Numerical Value × Unit
R
U
G
K
U
L
U
* Supplementary Units
Radian (rad) - for measurement of plane angle
Steradian (sr)- for measurement of solid angle
* Dimensional Formula
Relation which express physical quantities in
terms of appropriate powers of fundamental
units
...
It can only be checked
...
The formulae containing exponential, trigonometrical and logarithmic
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Formulae containing more tha one term which are
1
2
2
added or subtracted like s ut at also can’t
t
be derived
...
If dimensions are given, physical quantity may
not be unique as many physical quantities ave
the same dimensions
...
SI PREFIXES
The magnitudes of physical quantities vary over
a wide range
...
R
U
K
U
L
U
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Dimensions of differential coefficients d
n
y y
n
n
dx x
Dimensions of integrals ydx yx
We can’t add or subtract two physical quantities
of different dimensions
...
R
U
K
U
L
U
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Vectors
*
*
Angle made with x-axis cos
Vector Quantities
A physical quantity which requires magnitude
and a particular direction, when it is expressed
...
R min A B for 180o
L
U
r x i yj r cos i sin j
Parallelogram Law of Addition of Two Vectors
If two vectors are represented
Examples
by two adjacent sides of a par1
...
away from their common point
then their sum (i
...
resultant
1
3
vector) is given by the diagonal of the parallelo- sol
...
2
...
tan
R
U
Bsin
Asin
and
A Bcos
B A cos
Vector subtraction
R A B R A ( B)
If A = B the R 2A sin
K
U
G
R A 2 B2 2ABcos , tan
*
A
A
Bsin
tan
A Bcos
*
Ay
Angle made with z-axis cos A z
R A 2 B 2 2ABcos
*
cos
Ax
Ax
2
A
Ax A 2 A2
y
z
Bsin
A Bcos B
2
Addition of More than Two Vectors (Law of
Polygon)
If some vectors are represented by sides of a polygon in same order, then their resultant vector is
represented by the closing side of polygon in the
opposite order
...
r 1 cos135o i sin135o j
1
( i j)
2
* Scalar product (Dot Product)
A
...
B ABcos Anglebetween two vectors cos 1
AB
If A A x i A y j Az k & B Bx i By j Bz k then
A
...
B
2
AB
A x A 2 A 2 B2 B2 B 2
y
z
x
y
z
i
...
j 1, k
...
j 0, i
...
k 0
Component of vector b along vector a,b|| b
...
a a
* Cross Product (Vector product)
*
Rectangular component of a 3-D vector
A Ax i Ay j Azk
A B ABsin n where n is a vector perpen
dicular to A & B or their plane and its direction
given by right hand thumb rule
...
: 0551-2200338
G-6
The Gateway to Success
* For Parallel vectors
i
j
A B Ax
Bx
Ay
By
* For perpendicular vectors
k
* For coplanar vectors
Examples of dot products :
Work, W F
...
A (A B)
...
B
...
dt
dt
dt
d
dA dB
AB
B A
dt
dt
dt
*
K
U
R
U
G
r r2 r1 x 2 i y 2 j z 2 k x1 i y1 j z1k
x 2 x1 i y 2 y1 j z 2 z1 k
Magnitude
x 2 x1
2
L
U
B magnetic field, A Area
Potential energy of dipole in
where
P dipole moment
uniform field, U p
...
A BA cos where
F force,d displacement
Power, P F
...
A EA cos where
Differentiation
AB 0
A
...
(B C) 0
2
y2 y1 z 2 z1
2
Examples of cross products :
,
Torque r F where r position vector,
F force
Angular momentum J p where r position
r
vector, p linear momentum
Linear velocity v r where r position vector, angular velocity
Torque on dipole placed in electric field p E
where p dipole moment, E electric field
KEY POINTS :
* Tensor : A quantity that has different values in
different directions is called tensor
...
Moment of Inertia
In fact tensors are merely a generalisation of scalars and vectors; a scalar is a zero rank tensor,
and a vector is a first rank tensor
...
* A unit vector has no unit
...
* A scalar or a vector can never be divided by a
vector
...
: 0551-2200338
G-7
The Gateway to Success
Change in velocity
4
...
Displacement vector or displacement is the minimum distance (AB) and directed from initial
position to final position
...
x
v
t
t
2
2
2
dx
dx vdt x 2 x1 vdt
dt
x1
t1
t1
Change in position
= displacement
= Area between velocity
curve and time axis, from
t1 to t2
...
Uniform motion
2
...
Uniformly accelerated
If distance > | displacement | this implies
with u 0 at t = 0
(a) atleast at one point in path, velocity is zero
...
Uniformly accelerated
tion
motion with u 0
Acceleration positive indicates velocity increases
and s = s0 at t = 0
ad speed may increase or decrease
Speed increase if acceleration and velocity both 5
...
e
...
Uniformly retarded
then accelerated in
opposite direction
_________________________________________
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
s
a
sn u 2n 1
2
In scalar form (for one dimensional motion) :
*
Sn u
1 2
1 2
uv
s
t ut at vt at
2
2
2
v 2 u 2 2as
relative to ground v m v v R
...
e
...
r
...
If water is also
flowing with velocity v R then velocity of man
u v 1 2 1 2
r r2 r1 s
t ut at vt at
2
2
2
v u at
tan
* If the swimming is in the direction opposite the
the flow of water of along the upstream them
a
2n 1
2
L
U
RELATIVE MOTION
There is no meaning of motion without reference
or observer
...
ure i
...
v and v R not collinear then use the vecGenerally velocity or displacemnt of the particle
tor algebra v m v v R (assuming v > vR)
w
...
t
...
If we describe the
motion of a particle w
...
t
...
r
...
ground then velocity of par-
R
U
K
U
ticle w
...
t
...
and
velocity of particle w
...
t
...
and the velocity of moving
object (w
...
t ground) is the reference velocity
vref
...
v actual v relative v reference
Relative velocity of Rain w
...
t
...
vsin v R sin
vR
v
represented by OA
...
The relative velocity of
rain
w
...
t
...
Note: If vR > v then for minimum drifting sin
* For minimum time
v rm v 2 v 2 2v r v m cos90o v r2 v 2
r
m
m
If is the angle which v rm makes with the verti-
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Sn = 1 : 3 : 5 :
...
* If a body thrown pwards crosses a
point in time t1 & t2 respectively then
height of point h ½gt1t 2
R
U
Maximum height H 1 g(t1 t 2 ) 2
8
*
At any instant : v x u cos , v y u sin gt
For projectile motion :
* A body crosses two points at same height in time
t1 & t2 the points are at distance x + y from starting point then
(a) x + y = R
(b) t1 + t2 = T
(c) h ½ gt1t 2
(d) Average velocity from A to B is u cos
* If a person can throw a ball to a maximum distance ‘x’ then the maximum height to which he
can throw the ball will be (x / 2)
Velocity of particle at time t :
L
U
v v x i v y j u x i (u y gt) j u cos i (u sin gt) j
If angle of velocity v from horizontal is , then
tan
vy
vx
K
U
u y gt
ux
u sin gt
gt
tan
u cos
u cos
At highest point : v y 0, v x u cos
T
Time of flight :
2u y
g
2u sin
g
Horizontal range :
R (u cos ) T
2u 2 sin cos u 2 sin 2 2u x u y
g
g
g
A body is thrown upward, downward & horizontally with same speed takes
time t1, t2 & t3 respectively to
reach the ground then
It is same for and (90o– ) and maximum for =45o
t 3 t1t 2 & height from
H 1
tan
R 4
G
where the particle was
1
thrown is H gt1 t 2
2
*
PROJECTILE MOTION
Horizontal Motion :
Maximm height
H
u2
y
u 2 sin 2 1 2
gT
2g
2g
8
gx2
x
Equation of trajectory y xtan 2u2 cos2 xtan1 R
* Horizontal projection form some height
Time of flight
T
2h
g
u cos u x
2h
g
ax 0
Horizontal range
x u x t (u cos )t
Angle of velocity at any instnat with horizontal
Vertical Motion :
v y u y gt where u y u sin ;
R uT u
gt
tan 1
u
1
1
y u y t gt 2 u sin t gt 2
2
2
Net acceleration a a x i a y j g j
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
* The x-t graph for a particle undergoing rectilinear motion, cannot be as shown in figure because
infinitesimal changes in velocity are physically
possible only in infinitesimal time
...
motion (put in above)
* In free fall, the initial velocity of a body may not
be zero
...
* Average velocity of abody may be equal to its
instantaneous velocity
...
* The path of one projectile as seen from another
projectile is a straight line as relative accelera2u 2u sin
Time of flight
T 2t H
tion of one projectile w
...
t
...
Rangle on inclined plane R OA 2u
2
sin cos
2
R
U
G
K
U
2
u 2 sin 2
Maximum height H u
2a
2g cos
Rangle on inclined plane R OA 2u
Maximum range
R max
2
cos sin
g cos 2
u2
at angle
g 1 sin
4 2
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Newton’s Laws of Motion and Friction
If constnat force acts for an interval t then :
*
Impulse = F t
Impulse - Momentum theorem
Impulse of a force is equal to the change of mo-
*
*
*
*
*
R
U
G
inertia mass
*
Force
A push or pull that one object exerts on another
...
Gravitational force 2
...
Strong nuclear force 4
...
Ex
...
(i) Normal forces (ii) Frictional force
(c) Attachment to Another Body :
Tension (T) in a string and spring force (F = kx)
comes in this group
...
Inertia : Inertia is the property of the body due
to which body opposes the change of it’s state
...
Newton’s second law
dp d
dv dm
F
(mv) m v
dt dt
dt
dt
L
U
Note : Spring force is non impulsive in nature
...
If the lower spring is cut, find acceleration of the
blocks, immediately after cutting the spring
...
Initial stretches
x upper
3mg
mg
and x lower
k
k
On cutting the lower spring, by
virtue of non-impulsive nature
of spring the stretch in upper
spring remains same immediately
after cutting the spring
...
* Spring Force (According to Hooke’s law) :
In equilibrium F = kx (k is spring constnat)
K
U
For constnat mass system F ma
Momentum : It is the product of the mass and
velocity of a body i
...
momentum p mv
SI Unit : kg m s–1
Dimensions : [M L T–1]
Impulse : Impulse = product of force with time
t2
mentum F t p
* Motion of bodies in contact
When two bodies of masses m1 and m2 are kept
on the frictionless surface and a force F is applied on one body, then the force with which one
body presses the other at the point of contact is
called force of contact
...
(i) When the force F acts on the body with mass
m1 as shown in fig
...
: 0551-2200338
G-12
The Gateway to Success
erence is called an inertial frame of reference
...
* Non-inertial frame of reference : A accelerating frame of reference is called a non-inertial
frame of reference
...
* Pseudo force : The force on a body due to acceleration of non-inertial frame is called fictitious
or apparent or pseudo force and is given by
F m1 m 2 a
F ma 0 , where a 0 is acceleration of non-iner-
For body m 2 = f 1 = m 2 a action of m 1
tial frame with respect to an inertial frame and m
is mass of the particle or body
...
the direction of pseudo force must be op* Pulley system
posite to the direction of acceleration of the nonA single fixed pulley changes the direction of
inertial frame
...
with respect to an inertial frame of reference we
apply only the real forces (forces which are actuSOME CASES OF PULLEY
ally acting on the mass)
...
for mass m2, T – m2 g = m2 a
* Main in a lift
m1 m 2
(a) If the lift moving with constnat velocity v upnet pulling force
g
Acceleration a
wards or downwards in this case there is no ac m1 m2 totalmass to be pulled
celerated motion hence no pseudo force experi2m1m 2
2 Pr oduct of masses
g
g
Tension T
enced by observer inside the lift
...
4m1m 2 g
Reaction at the suspension of pulley R 2T
(b) If the lift is accelerated upward with constant
m1 m 2
acceleration a
...
r
...
observed inside the lift are
Case - II
(i) Weight W = Mg downward
(ii) Fictitious force F0 = Ma downward
...
m1 m 2
m1 m 2
Then w
...
t
...
which is either at rest or in uniform motion along
So
apparent
weight
the straight line
...
: 0551-2200338
G-13
The Gateway to Success
Special Case :
If a=g then W'=0 (condition of weightlessness)
...
N N
(d) If lift accelerates downward with acceleration
a>g
...
Apparent weight
W'=M(g-a) is negative, i
...
, the man will be accelerated upward and will stay at the ceiling of
* Angle of repose : The maximum angle of an inthe lift
...
FRICTION
Friction is the force of two surfaces in contact,
or the force of a medium acting on a moving
object
...
e
...
)
Frictional forces arise due to molecular interactions
...
* Cause of Friction: Friction is arises on account
of strong atomic or molecular forces fo attraction between the two surfaces at the poit of ac- * For smooth surface 0
R
tual contact
...
xn
No
...
of strings
Method II : Method of virtual work : The sum
of scalar products of forces applied by connecting links of constnat length and displacement of
corresponding contact points equal to zero
...
r1 0 F1
...
a1 0
Static friction coefficient
s
F
, 0 f s s N, fs Fapplied
N
fs max s N limiting friction
*
k
Sliding friction coefficient h N , fk k N v relative
*
Angle of Friction ( )
F
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
aeroplane displaces air & at low altitude density
of air is high
...
* Pulling a lawn roller is easier than pushing it * While walkign on ice, one should take small steps
because pushing increase the apparent weight and
to avoid slipping
...
increases the normal reaction and that ensure
* A moongphaliwala sells his moongphali using a
smaller friction
...
He gain more * A man in a closed cabin (lift) falling freely does
profit if the elevator is accelerating up becuase
not experience gravity as inertial and gravitathe apparent weight of an object increases in an
tional mass have equivalence
...
R
U
K
U
L
U
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Circular Motion
*
2
* Magnitude of net acceleration : a
v 2 dv
2
a 2 a1
C
r dt
2
Definition of Circualr Motion
When a particle moves in a plane such that its * Maximum speed of in circular motion
...
Radius Vector :
where angle of friction tan 1 s ; angle of
The vector joing the centre of the circle and the
banking
center of the particle performing circular motion
is called radius vector
...
It is directed outwards
...
of revolutions described by particle per sec
...
Its unit is revolutions per secound
(r
...
s
...
p
...
)
* Circular motion in vertical plane
Time period(T) :
It is time taken by particle ot complete one revos
max
s
lution
...
Condition to complete vertical circle u 5gR
*
If u 5gR then Tension at C is equal to 0
and tension at A is equal to 6mg
Velocity at B :
vB 3gR
Velocity at C : vC gR
t
n or f = frequency
*
*
v
r
v r
Relation between and v
* In vector form velocity
Acceleration
From A to B: T mg cos
From B to C: T
mv 2
R
mv 2
mg cos
R
dv d
d dr
a
r
r
r v a1 a C
dt dt
dt
dt
*
Tangential acceleration : a1
dv
r
dt
dv
a1 component of a along v a
...
Condition for pendulum motion (oscillating
condition)
u 2gR (in between A to B)
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Because T is given by T mg cos
mv 2
R
...
Condition for leaving path : 2gR u 5gR
KEY POINTS
* Average angular velocity is a scalar physical
quantity wherease instantaneous angualr velocity is a vector physical quantiy
...
d1 d2 d 2 d1 But 1 2 2 1
* Relative Angular Velocity
Relativ angular velocity of a particle 'A' w
...
t
...
r
...
B
...
Tension will be zero i between B to C & the angle
where T = 0
cos
u 2 2gR
3gR
is from vertical line
...
R
U
K
U
L
U
That means it is the rate at which position vector
of 'A' w
...
t
...
r
...
B pe rp end icu arl t o line A B
sep eration betw ee n A an d B
here VAB VA sin 1 v B sin 2
AB
v A sin 1 v B sin 2
r
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
* Work done is completely recoverable
...
dr Fdr cos
Non-conservative Forces
* Work done depends upon path
...
* For constnat force W F
...
g
...
W dW Fdx Area between F-x curve
* Work done against a non-conservative force may
and x-axis
be dissipated as heat energy
Calculation of work done from force-displace- * Work done is not recoverable
...
7
...
Total work done,
Potential energy
x
x
r
* The energy which a body has by virtue of its poW dW Fdx Area of P1P2 NM fdx
sition or configuration in a conservative force
x
r
x
field
...
scalar quantity, yet its value may be positive,
* Potential energy is defined only for conservative
negative or even zero
force field
...
v
2
2
L
U
K
K
U
U U U
F U grad(U)
i
j
k
x
y
z
* If force varies only with one dimension (along
x-axis) then
x
F
2
dU
U Fdx
dx
x1
* Potential energy may be positive or negative
If F is a conservative force then V F 0 (i
...
curl of F is zero)
Conservative Forces
(i) Potential energy is positive, if force field is
* Work done does not depend upon path
repulsive in nature
* Work done in a roud trip is zero
...
are conservaattractive in nature
tive forces
* When only a conservative force acts within a * If r (separation between body and force censystem, the kinetic energy and potential energy
tre), U , force field is attractive or vice-versa
...
However, their sum,
the mechanical energy of the system, doesn’t * If r , U , force field is repulsive in nature
...
: 0551-2200338
G-18
The Gateway to Success
*
* Work energy theorem : W KE
Change in kinetic energy = work done by all force
Potential energy curve and equilibrium
* For conservative force F(x) dU
dx
*
It is a curve which shows change in potential
energy with position of a particle
...
U U min ,
*
dU
d2U
0 and
positive
dx
dx 2
R
U
change in potential energy U F(x)x
* Law of conservation of Mechanical energy
Total mechanical (kinetic + potential) energy of
a system remains constant if only conservative
forces are acting on the system of particle of the
work done by all other forces is zero
...
K
U
* Mass and energy are equivalent and are related
by E = mc2
* Power
* Power is a scalar quantity with dimenstion M1L2 T 3
...
* Average power Pav W / t
Unstable equilibrium :
When a particle is slightly displaced from equi
dW F
...
v
* Instantaneous power
dt
dt
rium position then it is said to be in unstable
equilirbium
G
At point E: slope
dU
dx
is positive so F is negative
At point G: slope
dU
dx
is negative so F is positive
At point B: It is the point of unstable equilibrium
U Umax ,
*
dU
d2 U
0 and
negative
dx
dx 2
Neutral equilibrium :
When a particle is slightly displaced from
equilirbium position and no force acts on it then
equilibrium is said to be neutral equilibrium
...
v v 2
dt
dt
P
d
dt
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
*
Comets move around the sun in elliptical orbits
...
*
Work done by static friction may be positive because static friction may atcs along the direction
of motion of an object
...
R
U
K
U
L
U
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Centre of Mass and Collision
*
*
Centre of mass : For a system of particles centre of mass is that point at which its total mass is
supposed to be concentrated
...
m n then
m r m 2 r2 m3 r3
...
M
co ordinates of centre of mass :
x cm
*
1
1
1
m1 x1 , ycm m1 y1 and z cm m1z1
M
M
M
K
U
Centre of mass of continuous distribution of
particles
R CM
R
U
1
r dm
M
L
U
G
1
1
xcm xdm, ycm ydm
M
M
and
x, y, z are the co-ordinate of the COM of the dm
mass
...
The centre of mass after removal of a part of a body
Original mass (M) = mass of the removed part (m)
= [original mass (M)] + 1 – mass of the removed
part (m)]
The formula changes to :
x CM
Mx mx
My my
Mz mz
; yCM
; yCM
Mm
M m
M m
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
dt
m1 m 2
...
d
v CM 1 1
dt
m1 m 2
...
d mvCM
From Newton’s second law Fext
...
0 then MvCM constant
*
If no external force acts on a system the velocity
of its centre of mass remains constnat, i
...
velocity of centre of mass is unaffected by internal
forces
...
velocity of separation along line of impact v 2 v1
velocity of approach along line of impact u1 u 2
Value of e is 1 for elastic collision, 0 for perfectly inelastic collision and 0
(i)
(ii) Kinetic energy is not conserved
...
(ii)
1
2
By solving eq
...
Thus, if two
The event or the process, in which two bodies
bodies of equal masses undergo elastic collision
either coming in contact with each other or dy to
in one dimension, then after the collision, the
mutual interaction at distance apart, affect each
bodies will exchange their velocities
...
to other
...
same velocity and the body B will move with
* The particles need not come in contact with each
velocity double that of A
...
then v2 = 0, v1 = –u1
...
same velocity just in opposite direction while the
body B should practically remains at rest
...
: 0551-2200338
G-22
The Gateway to Success
m1u1 cos 1 m 2 u 2 cos 2 m1v1 cos 1 m 2 v 2 cos 2
&
m 2 u 2 sin 2 m1u1 sin 1 m 2 v 2 sin 2 m1 v1 sin 1
* Sum of mass moments about contre of mass is
zero
...
e
...
e
...
m1 u1 sin 1 m1 v1 sin 1 & m 2 u 2 sin 2 m 2 v 2 sin 2
By using Newton’s experimental law along he
v cos v cos
2
2
1
1
line of impact e u cos u cos
1
1
2
2
Rocket propulsion :
dm
Thrust force on the rocket vr dt
Velocity of rocket at any instant
R
U
m
v u gt v r n 0
m
KEY POINTS
* A quick collision between two bodies is more
violent then slow collision, even when initial and
final velocities are equal because the rate of
change of momentum determines that the impulsive force small or large
...
* For a system, conservation of linear momentum
is equivalent o Newton’s third law of motion
...
: 0551-2200338
G-23
The Gateway to Success
9
...
For constant angular acceleration
0 t, 0 t
*
1 2 2
t , 0 2 2, n 0 2n 1
2
2
Moment of Inertia
A tensor but for fixed axis it is a scalar
For discrete distribution of mass
I m1r12 m 2 r22
...
Theorem of perpendicular axes Iz = Ix + Iy
Rod
*
Radius of gyration
Rectangular Lamina
k
G
L
U
* Solid cylinder
K
U
I
M
Theorems regarding moment of inertia
*
Disc:
* Solid & Hollow sphere
* Rolling motion
Total kinetic energy =
1
1
Mv 2 CM I cm (t) 2
2
2
Total angular momentum Mv CM R I cm
* Pure rolling (or rolling without slipping) on
stationary surface
Condition : v cm R
In accelerated motion a cm R
If vcm R then rolling with forward slipping
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
* When a sphere is rolls on a horizontal table, it
Pure rolling motion on an inclined plane
shows down and eventually stops because when
the sphere rolls on the table, both the sphere and
the surface deform near the contact
...
g sin 0
* The spokes near the top of a rolling bicycle wheel
Acceleration a
1 k2 / R2
are more blurred than those near the bottom of
the wheel because the spokes near the top of
tan
Minimum frictional coefficient min
2
2
wheel are moving laster than those near the bot1 R / k
tom of the wheel
...
Change in angular momentum L t
* The relative angular velocity between any two
Work done by a torque W
...
* All particles of a rigid body, which do not lie on
an axis of rotation move on circular paths with
centres at an axis of rotation
...
r
...
ground
...
The
sediment that they carry increases the time of
rotation of the earth about its own axis because
the angular momentum of the earth about its rotation axis is conserved
...
R
U
K
U
L
U
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Gravitation
*
*
Newton’s law of gravitation
Force of attraction between two point masses
2h
If h << R ; g h g s 1
R
GM(R d)
d
g s 1
At depth d g d
Gm1m 2
3
R
F
R
r2
Effect of rotation on g :g g 2 R cos 2
Directed along the line joining of point masses
...
energy is conserved
* Gravitational potential
* It is a central force field angular momenDue to a point mass at a distance V GM
tum is conserved
r
Gravitational field due to spherical shell
* Gravitational potential due to spherical shell
GM
Outside the shell E g 2 , where r > R
r
GM
, where r = R
R2
Inside the shell Eg = 0, where r < R
On the surface E g
K
U
L
U
Outside the shell
*
R
U
[Note : Direction always towards the centre of
the sphere]
Gravitational field due to solid sphere
G
GM
Outside the sphere E g 2 , where r > R
r
On the surface E g
GM
, where r = R
R2
Inside the sphere E g
V
GM
,rR
r
Inside/on the surface the shell V GM , r R
R
* Potential due to solid sphere
Outside the sphere
V
GM
, rR
r
On the surface
V
GM
,rR
R
Inside the sphere
V
GM 3R 2 r 2
2R 3
, rR
GM
, where r < R
R3
* Potential on the axis of a
thin ring at a distance x V
*
GM
Acceleration due to gravity g 2
R
At height h g h
GM
R2 x2
* Escape velocity from a planet
of mass M and radius R
ve
GM
(R h) 2
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
v0
For nearby satellite
*
Time period of satellite T
*
2r 2r 3/2
v
GM
Energies of a satellite
Potential energy
U
Kinetic energy
K
GMm
r
1
GMm
mv 2
2
2r
GMm
Mechanical energy E U K
2r
Binding energy
*
BE E
GMm
2r
KEY POINTS
* At the centre of earth, a body has centre of amass,
but not centre of gravity
...
* You does not experience gravitational force in
daily life due to objects of same sizer as value of
G is very small
...
L
U
* Space rockets are usually launched in equatorial
line form West to East because g is minimum at
equator and earth rotates from West to East about
its axis
...
Path of a planet is elliptical with the sun at a focus
* Kepler’s second law or constancy of areal velocIInd Law of areas
ity is a consequence of conservation of angular
dA
L
momentum
...
: 0551-2200338
G-27
The Gateway to Success
tion, then this process is known as one oscillation
...
Simple Harmonic Motion
*
*
*
*
Periodic Motion
Any motion which repeats itself aftr regular interval of time (i
...
time period) is called periodic
motion or harmonic motion
...
(i) Motion of planets around the sun
(ii) Motion of the pendulum of wall clock
...
of time
...
* Phase
Ex
...
Phase of a vibrating particle at any instant is the
(ii) Oscillation of the mass suspended from
state of the vibrating particle regarding its disspring
...
every periodic motion is not oscillatory
...
H
...
)
In the equation x A sin(t ),(t ) is the
Simple harmonic motion is the simplest form of
phase of the particle
...
The phase angle at time t = 0 is known as initial
Some Basic Terms in SHM
phase or epoch
...
known as its mean position
...
e
...
known as restoring force
...
define as amplitude
...
(F x)
...
R
U
K
U
L
U
G
2 1
where is angular
n
frequency and n is frequency
...
: 0551-2200338
G-28
The Gateway to Success
*
* Differential equation of SHM
d
0 cos (t )
dt
Angular velocity
Linear SHM :
d 2x
A2 sin(t ) 2 x
dt 2
*
Acceleration a
*
Angular acceleration
*
Kinetic energy K 1 mv 2 1 m2 A 2 cos 2 (t )
*
Potential energy U 1 kx 2 1 m2 A 2 sin 2 (t )
2
2
*
Total energy E K U 1 m2 A 2 = constant
2
d2x
2 x 0
dt 2
d 2
2
Angular SHM : 2 0
dt
* Spring block system
d 2
02 sin(t ) 2
dt 2
2
2
T 2
m
k
*
T 2
T 2
L
U
=reduced mass
k
m
k
where
m1m 2
m1 m 2
* When spring mass is not negligible :
K
U
* Series combination of springs
R
U
*
1
1
1
1
m
where
Note :
T 2
k eff k1 k 2 k 3
k eff
(i) Total energy of a particle in S
...
M
...
(ii) Total energy depends upon mass, amplitude * Parallel combination of springs
and frequency of vibration of the particle executm
ing S
...
M
...
E
...
E
...
E
...
E
...
: 0551-2200338
G-29
The Gateway to Success
the radius of the earth R, then T 2
1
1 1
g
R
* SHM of gas-piston system
Here elastic force is developed due to bulk
elesticity of the gas
...
SHM of a particle in a tunnel inside the earth
...
: 0551-2200338
G-30
The Gateway to Success
KEY POINTS
* SHM is the projection of uniform circular motion along one of the diameters of the circle
...
* For a system executing SHM, the mechanical
energy remains constnat
...
* The frequency of oscillation of potential energy
and kinetic energy is twice as that of displacement or velocity or acceleration of a particle executing S
...
M
...
: 0551-2200338
G-31
The Gateway to Success
12
...
Area of cross sec tion A
of shear ()
AA r and Arc AA
r
where = angle of twist,
There are three types of stress :* Longitudinal Stress
(a) Tensile Stress :
So r
= angle of shear
* Stress – Strain Graph
(b) Compressive stress :
Volume Stress
*
Tangential Stress or Shear Stress
Strain
*
Longitudinal strain
*
*
G
Change in length of the body L
Initial length of the body
L
Volume strain
R
U
Change in size of the body
Original size of the body
*
Change in volume of the body V
original volume of the body
V
K
U
Y
displacement of upper face
or
L
L dis tan ce between two faces
Longitudinal stress
F
Longitudinal strain A
* If L is the length of wire, r is radius and l is the
increase in length of the wire by suspending a
weight Mg at its one end then Young’s modulus
of elestricity of the material of wire
Mg / r MgL
Y
2
/ L
r 2
* Increment in length due to own weight
MgL gL2
2AY
2Y
* Bulk modulus of elasticity
R
Shear strain :
tan
L
U
* Hooke’s Law within elastic limit Stress strain
* Young's modulus of elasticity
*
Volume stress
F/ A
P
Volumestrain V V
V V
* Bulk modulus of an ideal gas is process dependence
* For isothermal process PV = constant
PdV VdP 0 P
*
dP
So bulk modulus = P
dV / V
For adiabatic process PV' = constant
PV 1dV V dP 0
PdV VdP 0 P
dP
;
dV / V
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
e
...
Plasticity increases with temperature
...
Lead is not much clastic at room temperature but when cooled in liquid notrogen exhibit highly elastic behaviour
...
This steel is
called 'INVAR steel'
...
The inter molecular attraction force inside wire effectively
increase by impurity due to this external force
can easily opposed
...
: 0551-2200338
G-33
The Gateway to Success
13
...
M1 M 2 M 3
1 2
3
ma 0 a 0
mg
g
If P1 and P2 are pressures at point 1 & 2 then
L
U
P1 P2 g h1 h 2 g tan a 0
(iii)Free surface of liquid in case of rotating cylinder
R
U
1V1 2 V2 3 V3
V1 V2 V3
...
Pressure
area
* A liquid exerts equal pressures in all directions
...
The difference of pressure
container without being diminished in magnibetween two points separated by a depth h
tude
...
)
(i) Liquid placed in elevator : When elevator accelerates upward with acceleration a0 then pres- * Atmospheric pressure : Force exerted by air
sure in the fluid, at depth h may be given by,
column on unit cross-section area of sea level
P h[g a 0 ]
called atmospheric pressure (Po)
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
F
101
...
*
Po 1
...
Atmospheric pressure varies from place to place
* Rotatory - Equilibrium in Floatation : for roand at a particular place from time to time
...
Excess Pressure (P – P ) measured with the hekp
atm
of pressure measuring instrument called Gauge
pressure
...
R
U
L
U
* Relative density of body
K
U
Density of body
Density of water
G
Gauge pressure is always measured with hekp
of ''manometer''
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Surface Tension
Surface tension is basically a property of liquid
...
This property of liquid is called surface tension
...
(b) Adhesive force
The force acting between different types of molecules or molecules of different substance is
called adhesive force
...
This
distance is nearly 10–9 m
...
Molecular range depends on the nature of the
substance
Properties of surface tension
* Surface tension is a scalar quantity
...
* Surface tenstion is always produced due to cohesive force
...
* When surface area of liquid is increased, molecules from the interior of the liquid rise to the
surface
...
Dependency of Surface Tension
* On Cohesive Force : Those factors which increases the cohesive force between molecules
increase the surface tenstion and those which
decrease the cohesive force between molecules
decrease the surface tenstion
...
e
...
, on dissolving ionic
salts in small quantities in a liquid, its surface
R
U
G
(a)
(b)
*
*
*
tension increases
...
g
...
Surface tenstion of water is more than (alcohol
+ water) mixture
...
At critical temperature and boiling point
it becomes zero
...
On Contamination
The dust particles or lubricating materials on the
liquid surface decreases its surface tension
...
K
U
L
U
Definition of surface tenstion
The force acting per unit length of an imaginary
line drawn on the free liquid surface at right
angles to the line and in the plane of liquid surface, is defined as surface tenstion
...
: 0551-2200338
G-36
The Gateway to Success
Work done = Change in surface energy
* Effect of impurities on angle of contact
(a) Solute impurities increase surface tenstion,
1 1
4R 3T 4R 2 T n1/3 1
r R
*
so cos c decreases and angle of contact c
increases
...
* Effect of Water Proofing Agent
Angle of contact increases due to water profing
agent
...
4T
R
ANGLE OF CONTACT ( c)
The angle enclosed between the tangent plane at
the liquid surface and the tangent plane at the
liquid surface and the tangent plane at the solid
surface at the point of contact inside the liquid is
defined as the angle of contact
...
* Angle of contact
* Capillary rise h
2T cos
rg
1
r
L
U
h
*
Zurin’s law
*
Jeager’s method T
*
rg
H hd
2
The height 'h' is measured from the bottom
of the meniscus
...
Iff correction of
this is applied then the formuls will be
K
U
90o concave shape, Liquid rise up
* Angle of contact
1
rg h r
90o convex shape, Liquid falls
3
T
* Angle of contact
2 cos
o
plane shape, Liquid neither rise nor falls
90
* When two soap bubbles are
rr
r 1 2 r1 r2
* Effect of Temperature on angle of contact
in contact then radius of
r1 r2
On increasing temperature surface tenstion decurvature of the common surface
1
creases, thus cos , increases cos c * When two soap bubbles are
T
r r12 r22
combining to form a new
and c decrease
...
* Force required to separate
2AT
F
two plates
d
R
U
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Hydrodynamics
*
*
*
*
*
*
*
Steady and Unsteady Flow : Steady flow is
defined as that type of flow in which the fluid
characteristics like velocity, pressure and density
at a point do not change with time
...
This line or path
is called a streamline
...
Compressible and incompressible Flow : In
compressible flow the density of fluid varies from
point to point i
...
the density is not constant for
the fluid whereas in incompressible flow the density of the fluid remains constnat throughout
...
Equation of continuity A1 v1 = A2 v2
Based on conservation of mass
R
U
Bernoulli’s theorem : P 1 v 2 gh co ns tan t
2
* Pressure Energy
Pressure energy per unit volume
Pr essure Energy
P
volume
* For horizontal flow in venturimeter
1 2
1
2gh
P1 v1 P2 v2 v1 A 2
2
2
2
A1 A 2
2
* Rate of flow :
Volume of water flowing per second
Q A1 v1 A1A 2
2gh
A A2
2
2
1
L
U
* Velocity of efflux v 2gh
K
U
* Horizontal range R 2 h(H h)
G
Based on energy conservation
*
Kinetic Energy
kinetic energy per unit volume
Kinetic Energy 1 m 2 1 2
v v
volume
2V
2
Potential Energy
Potential energy per unit volume
*
Potential Energy m
gh gh
volume
V
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Viscosity
*
17
...
E
...
(b) The viscosity of gases is the result of
diffusion of gas molecules from one moving
layer to other moving layer
...
So, the viscosity also increases
...
15
X LFP
Thus, the viscosity of gases increases with
100 0 212 32 373
...
15 UFP LFP
the rise of temperature
...
(b) The viscosity of gases is practically independent
* Old thermometry
of pressure
...
e
...
T0
X
* Modern thermometry 273
...
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Scale reading : Due to linear expansion / contraction, scale reading will be lesser / more than
actual value
...
Thermal Stress
In solids, liquids and gases
L
U
Thermal strain
volume exp ansion V V0 1 T
As Young's modulus Y
K
U
F/ A
;
/
So thermal stress YA
Thermal expansion in liquids (Only volume expansion)
[For isotropic solids : : : 1 : 2 : 3 ]
Thermal expansion of an isotropic object may
be imagined as a photographic enlargement
...
Bi-metallic strip(used as thermostat or auto-cut
in electric heating circuits)
a
Apparent increases in volume
Initial volume Temperature rise
r
real increasein volume
Initial volume Temperature rise
r a vessel
Change in volume of liquid w
...
t vessel
V V0 r 3 T
Expansion in enclosed volume
Increase in height of liquid level in tube when
bulb was initially completely filled
...
Simple pendulum :
T 2
T 1
T 1/ 2
g
T
2
h
apparent change in volume of liquid
area of tube
V0 L 3 T
A 0 1 2 T
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Aquatic life is able to survive in very cold countries as the lake bottom remains unfrozen at the
temperature around 4oC
...
Therefore water equivalent of a body is te quantity of water, whose heat capacity is the same as
the heat capacity of the body
...
V
1
* Coefficient of volume expansion v
* Latent Heat (Hidden heat) : The amount of heat
V0 T T
that has to supplied to (or removed from) a body
[PV = nRT at constnat pressure V T V T ]
for its complete change of state (from solid to
V
T
liquid, liquid to gas etc) is called latent heat of
P
1
the body
...
e
...
KEY POINTS :
* Liquids usually expand more than solids because * Principle of calorimetry : Heat lost = heat
gained
the intermolecular forces in liquids are weaker
than in solids
...
is plotted between temperature and time, the
* Water expands both when heated or cooled from
graph is called heating curve
...
CALORIMETRY
R
U
K
U
L
U
G
1cal 4
...
2 J
Q
T
Amount of heat required to raise the temp
...
*
Thermal capacity of a body
*
Specific heat capacity
*
*
Q
(m mass)
m T
Amount of heat required to raise the temperature of unit mass of a body through 1oC (or 1K)
Specific heat (or thermal capacity)
1
slope of curve
Latent heat length of horizontal line
...
It is called
than unity
...
Water equivalent of a body is * The steam at 100oC causes more severe burn to
the mass of water which when given same
human body than the water at 100oC because
amount of heat as to the body, changes the temsteam has greater internal energy than water due
Molar heat capacity
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Growth of Ice on Ponds
Heat is energy in transit which is transferred from
Thus taken by ice to grow a thickness from
hot body to cold body
...
5 oC to 15
...
Clausins & clapeyron equation (effect of pressure on boiling point of liquids & melting point
of solids related with latent heat)
dP
L
[K = thermal conductivity of ice, =density of ice]
dT T V2 V1
In condiction, heat is transferred from one point
ot another without the actual motion of heated
particles
...
In radiation, intervening medium is not affected and heat is transferred
without any material medium
...
(a) Reflection (b) Absorption (c) Transmission
K
U
L
U
From energy conservation
Qr
Q
Q
* Absorptive Coefficient : a a
Q
Q
* Transmittive Coefficient : t t
Q
r = 1 and a = 0, t = 0 Perfect reflector
a = 1 and r = 0, t = 0 Ideal absorber
(ideal black body)
t = 1 and a = 0, r = 0 Perfect transmitter
(daithermanons)
*
Reflective Coefficient :
r
Qr
Reflection power (r) 100 %
Q
Qa
Absorption power (a) 100 %
Q
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
89 10 3
mK = Wein’s constant
...
If surrounding has temperature T 0 then
E net e r T 4 T0 4
*
*
*
*
*
*
L
U
Kirchhoff's law : the ratio of emissive power to
Area under e graph e d e T 4
absorptive power is same for all surfaces at the
0
same temperature and is equal to the emissive Solar constant
power of a perfectly black body at that temperaThe Sun emits radiant enture
...
The solar
a A 1
a
radiant energy received per
Therefore a good absorber is a good emitter
...
placed at the mean distance of the Earth (in the
Absorptive Power (a) : Absorptive power of a
absence of atmospher) is called solar constnat
...
4r 2
4r 2
r
For ideal black body, absorptive power = 1
where RS = radius of sun
Emissive power(e) : For a given surface it is
r = average distance between sun & earth
defined as the radiant energy emitted per second
Note: S = 2 cal cm–2min = 1
...
T = temperature of sun 5800 K
Newton’s law of cooling :
If temperature difference is small
KEY POINTS :
d
Rate of colling 0
* Stainless steel cooking pans are preferred with
dt
extra copper bottom because thermal conductiv 0 1 0 e kt
ity of copper is more than steel
...
* Animals curl into a ball when they feel very cold
1 2
k 1 2 0 [where k = constant]
to reduce the surface area of the body
...
: 0551-2200338
G-43
The Gateway to Success
*
natural movement of heated fluid is not possible
Metals have high thermal conductivity because
metals have free electrons
...
Basic postulates of Kinetic theory of gases
* Every gas consists of extremely small particles
known as molecules
...
* The molecules of a gas are identical, spherical,
rigid arenperfectly elestic point masses
...
* The speed of gas molecules lie between zero and
infinity (very high speed)
* The numbr of molecules moving with most probable speed is maximum
...
These collision are perfectly elastic
...
e
...
)
Assumptions regarding force :
* No attractive or repulsive force acts between gas
molecules
...
Assumptions regarding pressure :
* Molecules constantly collide with the walls of
container due to which their momentum changes
...
Consequently pressure is
exerted by gas molecules on the walls of container
...
R
U
G
Kinetic interpretation of pressure : PV 1 mNv2
rms
3
[m = mass of a molecule, N = no
...
of molecules N1 = N2
...
V
K
U
L
U
diffusion
1
Dalton’s law : P = P1 + P2 +
...
8a
Critical temperature TC 27 Rb
The maximum temperature below which a gas
can be liquefied by pressure alone
...
1
1
3
2
2
PV mNVrms & PV RT mVrms kT
3
2
2
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
For monoatomic gas (He, Ar etc) f = 3 (only translational)
For diatomic gas (H2, O2 etc) f = 5 (3 translational + 2 rotational)
At higher temperature, diatomic molecules have
two degree of freedom due to vibrational motion
(one for KE + one for PE)
At higher temperature diatomic gas has f = 7
Maxwell’s Law of equipartition of energy:
Kinetic energy associated with each degree of
freedom of particles of an ideal gas is equal to
1
kT
2
*
Mean free path :
Average distance between two consecutive collisions m
2d 2 n
where d = diameter of molecule, n = molecular
density =
N
V
For mixture of non-reacting gases
Molecular weight
M Wmix
Specific heat at constant V CV
1M W1 2 M W2
...
1CV1 2 C V2
...
mix
L
U
Specific heat at constant P CP
1CP1 2 C P2
...
mix
Average KE of a particle having f degree of freedom
1
f
kT
2
3
kT
2
*
Translational KE of molecule =
*
3
Translational KE of a mole = RT
2
*
Internal energy of an ideal gas : U RT
R
U
1
2
Specific heats (Cp and Cv) :
G
dQ
*
Molar specific heat of a gas C dT
*
dQ
dU
CV
V cons tan t
dT
dT
*
dQ
CP
dP 0 CV R Mayer 's equation
dT
K
U
mix
KEY POINTS :
* Kinetic
energy
CPmix
C Vmix
per
1C P1 2 CP2
...
unit
volume
1 mN 2
3
EV
v rms P
2 V
2
* At absolute zero, the motion of all molecules of
the gas stops
...
* For any general process
(a) Internal energy change U nCV dT
(b) Heat supplied to a gas Q nCdT
where C for any polytropic process
PVx = constant is C CV
R
1 x
(c) Work done for any process W PV
It can be calculated as area under P-V curve
(d) Work done Q U
nR
dT
1 x
For any polytropic process PVx = constant
THERMODYNAMICS
* Zeroth law of thermodynamics : If two systems are each in thermal equilibrium with a third,
they are also in thermal equilibrium with each
other
...
: 0551-2200338
G-45
The Gateway to Success
*
First law of thermodynamics : Heat supplied
(Q) to a system is equal to algebraic sum of
change in internal energy (U) of the system and
mechanical work (W) doen by the system
*
dP
P
x
dV
V
For differential change
Q
Work done by working subs tan ce W Q1 Q 2
1 2
Heat sup plied
Q1
Q1
Q1
Sign Convention
Heat absorbed by the system positive
Heat rejected by the system negative
Increase in internal energy (i
...
rise in tempera- For carnot cycle
Q 2 T2
Q
T
ture) positive
so 1 2 1 2
Decrease in internal enrgy (i
...
fall in temperaQ1 T1
Q1
T1
ture) negative
For refrigerator Coefficient of performance
Work done by the system positive
Q
Q2
T2
2
Work done on the system negative
W
*
*
For cyclic process
U 0 Q W
For isochoric process
V = constant P T & W 0
Q U C V T
*
R
U
For isobaric process
P = constant V T
Q C P T, U C V T
W P V2 V1 RT
*
PV constant
or T V 1 constant
In this process Q = 0 and
L
U
Q1 Q 2
K
U
G
For adiabatic process
or T P1 constant
T1 T2
Bulk modulus of gases : B
P
V
V
Isothermal bulk modulus of elasticity,
P
BIT V
V T cons tan t
Adiabatic bulk modulus of elasticity,
P
BAD V
BAD BIT
V
KEY POINTS
* Work done is least for monoatomic gas (adiabatic process) in shown expansion
PV P V
W U C V T1 T2 1 1 2 2
1
*
)
Efficiency of a cycle
Q W U [Here W PdV: U nC V T]
*
Slope of P-V diagram
(also known as indicator diagram at any point
For isothermal Process T = constant
or T 0 PV constant
In this process U C v T 0
V2
P1
RT n
V1
P2
So, Q W RT n
*
For any general polytropic process PVx = constant
* Molar heat capacity
* Work done by gas
W
C CV
R
1 x
nR T1 T2
x 1
P1V1 P2 V2
* Air quickly leaking out of a balloon becomes
cooler as the leaking air undergoes adiabatic expansion
...
: 0551-2200338
G-46
The Gateway to Success
*
*
First law of thermodynamics does not forbid flow
of heat from lower temperature to hither temperature
...
18
...
CLASSIFICATION OF WAVES
*
CARNOT ENGINE
It is a hypothetical engine with maximum possible efficiency
Process 1 2 & 3 4 are isothermal
Process 2 3 & 4 1 are adiabatic
R
U
G
K
U
L
U
* A mechanical wave will be transverse or longitudinal depends on the nature of medium and
mode of excitation
...
* In gases and liquids, mechanical waves are always longitudinal because fluids cannot sustain
shear
...
These waves are
called as ripples as they are combination of transverse & longitudinal
...
* In longitudinal wave motion, oscillatory motion
of the medium particles produce regions of compression (high pressure) and rarefaction (log pressure)
...
: 0551-2200338
2
G-47
The Gateway to Success
wave propagation constant
2
*
Differential equation :
1
P 2 A2 Sv
2
2
y 1 y
x 2 v 2 t 2
dx
(where S = Area of cross-section]
Power
t kx cons tan t
*
dx
dt k
v
dy
A cos t kx
dt
Particle velocity v P
2 y
2 A sin t kx 2 y
t 2
For particle 1 : vp
For particle 1 : vp
For particle 1 : vp
For particle 1 : vp
*
KEY POINTS
* A wave can be represented by function
y f (kx t) because it satisfy the differential
Particle asseleration :
ap
T
where = mass/length and T = tension
in the string
...
t
k
2
2
* A pulse whose wave function is given by
and ap
and ap
and ap
K
U
4
y
[(2x 5t) 2] is propagates in –x direction as
R
U
G
Relation between phase difference, path difference & time difference
2
this wave function is of the form y f (kx t)
which represent a wave travelling in –x direction
...
WAVE FRONT
* Spherical wave front (source point source)
Energy to Wave Motion
*
KE
1 m 2 1 2 1 2 2
v p vp A cos2 t kx
volume 2 volume
2
2
*
PE
1
1
dy
v 2 2 A 2 cos2 t kx
volume 2
2
dx
*
TE
2 A 2 cos2 t kx
volume
*
Energy density [i
...
Average total energy / vol-
* Cylindrical wave front (source linear source)
2
* Plane wave front (source point / linear
source at very large distance)
1
2 2
ume] u 2 A
*
Power : P = (energy density) (volume / time)
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
r
r
1
r
A
r
sin t k
...
r
y A sin t kx ,
1
1
y A sin t kx
2
2
y y y A sin t kx
1
where A A A 2A A cos
and
2
1
2
1
2
A sin
tan
A A cos
2
2
R
U
0
IA
So
I I I 2 I I cos
2
2
1
K
U
G
As
1
0
0
1
2
2
1
1
2
r
1
2v
A
v v
* Amplitude of transmitted wave A t
2
2
v v
* Amplitude of reflected wave A
A
v v
0
L
U
* The frequency of the wave remain unchanged
INTERFERENCE OF WAVES
*
min
Reflection and Refraction (transmission) of waves
y r, t
*
min
2
y r, t
*
max
max
INTENSITY OF WAVE
2
1
* If v2 > v1 i
...
medium-2 is rarer
Ar > 0 no phase change in reflected wave
* If v2 < v1 i
...
medium-1 is rarer
Ar < 0 There is a phase change of in reflected wave always remains unchanged
...
e
...
* the transmitted wave is never inverted
...
0
I
max
*
I I
1
2
2
*
For destructive interferece [Minimum intensity]
2n 1 or path diffeence 2n 1
0
where n = 0,1,2,3,
...
: 0551-2200338
G-49
The Gateway to Success
Transverse stationary waves in stretched string
* [fixed at both ends] [fixed end Node & free
end Antinode]
*
Beats :
When two sound waves of nearly equal (but not
exactly equal)frequencies travel in same direction, at a given point due to their super position,
intensity alternatively increases and decreases
periodically
...
Beat frequency = difference of frequencies of two
interfering waves
...
*
Let
two
waves
are
R
U
y Asin t kx ;
1
L
U
Fixed at one end
K
U
y A sin t kx by principle of superposition
2
G
y y y 2A cos kx sin t Equation of station1
*
2
ary wave
As this equation satisfies the wave equation
y 1 y
,
dx
v t
2
2
2
*
*
2
2
it represent a wave
...
Nodes amplitude is minimum:
cos kx 0 x
*
3 5
, , ,
...
2
2
[p : number of loops]
Sound Waves :
Velocity of sound in a medium of elasticity E
and density is
The nodes divide the medium into segments
(loops)
...
As nodes are permanently at rest, so no energy
can be transmitted across them, i
...
energy of one
region (segment) is confined in that region
...
: 0551-2200338
G-50
The Gateway to Success
*
Newton’s formula : Sound propagation is isoP
thermal B P v
*
Laplace correction : Sound propagation is adiabatic B v P v
*
*
vP
KEY POINTS
* With rise in temperature, velocity of sound in a
gas increases as v
Only odd harmonics are present
Maximum possible wavelength = 4
*
Frequency of mth overtone (2m 1) v
4
* Open end organ pipe
RT
M
W
*
*
With rise in humidity velocity of sound increases
due to presence of water in air
...
*
*
L
U
All harmonics are present
Maximum possible wavelength is 2
Displacement and pressure wave
v
* Frequency of mth overtone (m 1)
A sound wave can be descried either in terms of
2
the longitudinal displacement suffered by the * End correction :
particles of the medium (called displacement
Due to finite momentum of air molecules in orwave) or in terms of the excess pressure genergan pipes reflection takes place not exactly at open
ated due to compression and rarefaction (called
end but some what above it, so antinode is not
pressure wave)
...
Displacement wave y A sin t kx
In closed organ pipe f v
where e = 0
...
g
...
R
U
0
K
U
G
0
1
1
KEY POINTS
* The pressure wave is 90o out of phase w
...
t displacement wave, i
...
displacement will be maximum when pressure is minimum and vice-versa
...
: 0551-2200338
G-51
The Gateway to Success
Intensity of sound in decibels
If v0, vs <<
I
Sound level, SL 10 log I
0
–12
2
Where I0 = threshold of human ear = 10 W/m
*
Mach Number
v v
v
s
n
speed of source
speed of sound
* Doppler’s effect in light :
Characteristics of sound
* Loudness Sensation received by the ear due to
intensity of sound
...
* Quality (or Timbre) Sensation received by the
ear due to waveform of sound
...
A moving observer will encounter more or lesser
number of wavefronts depending on whether he
is approaching or receding the source
...
e
...
The apparent change in frequency or pitch due
to relative motion of source and observer along
the line of sight is called Doppler Effect
...
r
...
observer
observed wavelength
vv
vv
n
n
vv vv
n
0
0
0
s
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Wave nature of light & Wave Optics
* Intensity width of slit amplitude
2
2
Huygen’s Wave Theory :
2
I1 I 2
a1 a 2
I max
I1 W1 a1
2
Huygen’s in 1678 assumed that a body emits light
2
I 2 W2 a 2
I min
a1 a 2
I1 I 2
in the form of waves
...
* Distance of nth bright fringe x nD
n
The locus of all the particles of the medium vid
brating in the same phase at a given instant is
Path difference n
called a wavefront
...
* Each point on a wave front is a source of new
(2m 1)D
disturbace, called secondary wavelets
...
* The forward envelope of the secondary wavelets
at any instant gives the new wavefront
...
R
U
G
K
U
*
L
U
Path difference (2m 1)
* Fringe width
where m = 0,1,2,3,
...
Incoherent sources :
Two sources are said to be incoherent if they have
different frequency and initial phase difference
is not constant w
...
t
...
*
Interference : YDSE
* Resultant intensity for coherent sources
I I1 I 2 2 I1I 2 cos 0
If a transparent sheets of referactive index and
thickness t is introduced in one of the paths of
interfering waves, optical path will becomes ' t '
instead of 't'
...
In Fresnel’s Biprism fringe width
(a b)
2a( 1)
Resultant intensity for incoherent sources
I I1 I 2
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Electrostatics
* The law of conservation of energy holds good in
Electric Charge :
the phenomenon of interferece
...
There are
locus of a point which moves such a way that its
two kinds of charges-positive and negative
...
I
...
Charge is quantized, conserved,
constant
...
* The interference fringes for two coherent point
Coulomb’s law :
sources are hyperboloids with axis S1S2
...
Moving charges may result in
coherent sources are infinitely close to each other,
magnetic interaction
...
d
* If maximum number of maximas / minimas are
asked in the question, use the fact that value of Principle Of Superposition
Force on a point charge due to many charges is
sin / cos can’t be greater than 1
...
DIFFRACTION
Note : The force due to one charge is not affected
* In Fraunhofer diffraction
by the presence of other charges
...
The direction of electric field
is direction of force experienced by a positively
2
Angular width of central maxima W0
charged particle and the magnitude of the field
a
(electric field intensity) is the force experienced
2
2
by a unit charge
...
Where I0 = Intensity of central maxima
q
* Electric field intensity due to charge Q
F
1 Q
E Lim
r
q0 0 q
4 0 r 2
0
R
U
K
U
L
U
G
Null point for two charges
(a)
and x
Q1
Q1 Q 2
r
For same nature charges Null point will lie in
beween the charges
...
: 0551-2200338
G-54
The Gateway to Success
(b)
x
& |Q1| > |Q 2| &
Q2
Q1 Q 2
r
For point charge : V K q
r
then null point will lie beyond the charge & closer
to smaller magnitude charge
...
dr
x
y
z
Q1Q 2
For several point charges : V K q i
V
E grad V V, E
;
r
(i) Two charges must be of like nature
...
x
It is the work done against the field to take a unit
posotive charge from infinity (reference point)
to the given point
L
U
* Electric potential energy of two charges :
2
Equilibrium of symmetric geometrical point
charged system
U
1
q1q 2
4 0 r
* Electric dipole
K
U
Value of Q at centre for which system to be in
state of equilibrium
R
U
(i) For equilateral triangle Q q
3
(ii) For square Q
q 2 2 1
G
4
Equilibrium of suspended point charge system
For equilibrium position
T cos mg & T sin Fe
tan
*
kQ 2
x2
Electric dipole moment p = qd
Torque on dipole placed in uniform electric
field p E
Work done in rotating dipole placed in uni
form electric field
Potential energy of dipole placed in an uni
form field U p
...
* Potential
Fe
kQ2
2
mg x mg
0
W d 00 PE sin d pE cos 0 cos
V
1 p cos
40 r 2
1
If whole set up is taken into an artificial satellite
* Electric field E 4
0
* Direction
geff 0
tan
p 1 3cos 2
r3
E0 1
tan
Er 2
Electric field at axial point (or End-on)
*
Electric
V
*
potential
work
W/q
ch arg e
p
Electric potential Vp E
...
: 0551-2200338
G-55
The Gateway to Success
Equipotential Surface & Equipotential Region
Concept of solid angle :
In an electric field the focus of points of equal
Flux of charge q passing through the circle of
potential is called an equipotential surface
...
The region where E = 0,
Potential of the whole region must remain constant as no worl is done in displacement of charge
in it
...
Mutual Potential Energy Or Interaction Energy
“The work to be done to integrate the charge system”
...
2
For r R :E
K
U
Total energy of a system U cell U mutual
Electric flux : E
...
A EA cos
where angle between E & area vector (A)
...
(ii) If E is not uniform throughout the area A, then
E
...
ds
0
R
U
L
U
* For a conducting sphere
G
1 q
1 q
, V
4 0 r 2
40 r
For r R :E 0, V
1 q
40 R
* For a non-conducting sphere
For r R :E
1 q
1 q
, V
4 0 r 2
40 r
For r R:E
1 qr
1 q(3R 2 r 2 )
, V
3
40 R
4 0
2R 3
(Applicable only to closed surface)
q en
Net flux emerging out of a closed surface is
0
q
E
...
does not depend on the
(i) Shape and size of the closed surface
(ii) The charges located outside the closed surface
...
: 0551-2200338
G-56
The Gateway to Success
* For conducting long sheet of surface charge
density
*
For a conducting/non conducting spherical
E
0
1
2
* Energy density in electric field u E 0 E 2
Electric lines of force
Electric lines of electrostatic field have following properties
shell
1 q
1 q
, V
For r R :E
2
4 0 r
40 r
For r R :E 0, V
1 q
40 R
(i) Imaginary
(ii) Can never cross each other
(iii)Can never be closed loops
(iv)The number of lines originating or terminating
on a charge is proportional to the magnitude f
L
U
charge
...
(v) Lines of force ends or starts normally at the surface of force
...
(vii) Lines of force per unit area normal to the area
at a point represents magnitude of intensity,
crowded lines represent strong field while distant lines weak field
...
So
a positive charge free to move follow the line of
R
Electric field will be maximum at x
force
...
q
No tangential component of electric field on such
For r R :E
2 0 r
surfaces
...
* Charge density at convex sharp points on a concharged conductor E
0
ductor is greater
...
2
P
* For a conductor of any shape E (just outside)
20
0
For non-conducting long sheet of surface * Potential difference between two points in an
electric field does not depend on the path between
charge density
E
them
...
: 0551-2200338
G-57
The Gateway to Success
*
*
*
*
*
*
Potential at a point due to positive charge is posi- * A charged particle is free to move in an electric
tive & due to negative charge is negative
...
It may or may not move along an electric
Positive charge flows from higher to lower (i
...
line of force because initial conditions affect the
in the direction of electric field) potetial and negamotion of charged particle
...
e
...
humid days because water is a good conductor
of electricity
...
When a charged isolated conducting sphere is
connected to an uncharged small conducting
sphere then potential (and charge) remains almost same on the larger sphere while smaller is
charged
...
e depends
only on r) behaves as if its charge is concentrated
at the centre (for outside points)
...
The particle such as photon or neutrino which
have no (rest) mass can never has a charge because charge cannot exist without mass
...
A spherical body behaves like a point charge for
outside points because a finit charged body may
behave like a point charge if it produces an inverse square field
...
charged sphere
R
U
L
U
G
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
Capacitor & Capacitance
t
effectivel reduced by t irrespective of the
r
CAPACITOR & CAPACITANCE :
A capacitor consists of two conductors carrying
position of the dielectric slab
...
(iii) Composite Medium :
The capacitance C of any capacitor is the ratio
of the charge Q on either conductor to the poten-
tial difference V between them C Q
...
Capacitance Of An Isolated Spherical ConducCylindrical Capacitors :
tor
It consists of two co-axial cyclinders of radii a &
C 40r R in a medium C 40 R in air
b, the outer conductor is earthed
...
tween the cylinders is r
...
Here capacitance of region belength is C
tween the two shells is C1 and that outside the
n b
a
shell is C2
...
ba
K
U
PARALLEL PLATE CAPACITOR :
(i) Uniform Di-Electric Medium : If two parallel
plates each of area A & separated by a distance d
are charged with equal & opposite charge Q, then
the system is called a parallel plate capacitor &
G
0r A
its capacitance is given by, C
in a
d
medium; C 0 A with air as medium
...
(ii) Medium Partly Air :
C
0 A
1
dt
r
L
U
Concept of variation of parameters :
As capacitance of a parallel plate capacitor is
0 kA
, if either of k, A or d varies in the
d
region between the plates, we choose a small dC
in between the plates and for total capacitance
of system
...
When a di-electric slab of
thickness t & relative permittivity r is introduced between the plates of an air capacitor, then the distance between between the plates is
(ii) Capacitors in Parallel : When one plate of each
capacitor is connected to the positive terminal
GURUKUL ikB’kkyk : Gandhi Ashram Gali Golghar-Gorakhpur Ph No
...
The capacitors have the same poten1/ RC
)
tial difference, V but the charge on each one is * Charging of a capacitor : q q 0 (1 e
different (if the capacitors are unequal)
...
C1 C 2 C3
...
Energy stored in a charged capacitor :
Capacitance C, charge Q & potential difference
V; then energy stored is
* Discharging of a capacitor : q q 0 e 1/ RC
1
1
1 Q2
U CV 2 QV
2
2
2 C
K
U
This energy si stored in the electrostatic field set
up in the di-electric medium between the conducting plates of the capacitor :
Heat produced in switching in capacitive circuit:
Due to charge flow always some amout of heat
is produced when a switch is closed in a circuit
which can be obtained by energy conservation
as –
Heat = Work done by battery – Energy absorbed
by capacitor
...
* The energy of an uncharged condenser = 0
...
(ie
...
When tow charged conductors of capacitance C1 * When a capacitance is charged by a battery, both
& C2 at potential V1 & V2 respectively are conthe plates received charged equal in magnitude,
nected by a conducting wire, the charge flows
no matter sizes of plates are identical or not befrom higher potential conductor to lower potencause the charge distribution on the plates of a
tial conductor, until the potential of the two concapacitor is in accordance with charge conservadensers becomes equal
...
(V) after sharing of charges ;
* On filling the space between the plates of a parnet ch arg e
q1 q 2 C1V1 C2 V2
allel plate air capacitor with a dielectric, capacV
net capaci tan ce C1 C 2
C1 C 2
ity of the capacitor isincreased because the same
amout of charge can be stored at a reduced pocharges after sharing q1 C1V & q 2 C 2 V
...
this process energy is lost in the connecting wire
* The potential of a grounded object is taken to be
as heat
...
large
...
: 0551-2200338
G-60