Units And Measurements
Quick Revision
UNITS AND MEASUREMENTS
TERMINOLOGY
VALUES
d
= Plane Angle
ds
= Arc Length
r
= Radius
d
= Solid Angle
D
= Distance between two planet
b
= Diameter Of Planet
a = Absolute error
amean = Mean absolute error
Z, A & B = Physical Quantities
10) Absolute error : The magnitude of the
difference between the individual
measurement and the true value of the quantity
is called absolute error.
11) Mean absolute error : The arithmetic mean of
all the absolute errors is called mean absolute
error.
12) Relative error : The ratio of mean absolute error
in the measurement of a physical quantity to its
most profable value is called relative error.
13) Percentage error : The relative error multiplied
by 100 is called the percentage error.
DEFINITIONS
DEFINITIONS
1)
System of Units : A complete set of units both
fundamental and derived for all kinds of 1)
physical quantities is called system of units.
2)
Unit : The reference standard used for the
measurement of a physical quantity is called a
unit.
3)
Fundamental Quantities : The physical
quantities which do not depend on any other
physical quantities for their measurements are
known as fundamental quantities.
2)
Derived Quantities : The physical quantities
which depends on one or more fundamental
quantities for their measurements are known
as derived quantities.
4)
5)
Parallax Method : The method used to measure
large distances are called Parallax Method.
6)
Dimensional Analysis : The dimensions of a
physical quantity are the powers to which
fundamental units must be raised in order to 3)
obtain the unit of the given physical quantity.
7)
Order of Magnitude : The value of its
magnitude rounded off to the nearest integral
power of 10.
8)
Significant Figures : Figure which is of some
significance but it does not necessarily denote
a certainty.
9)
Error : The difference between the true value
and measured value of physical quantity is
called error.
FORMULAE
FORMULAE
Plane Angle : d 
ds
r

dA
r2
Solid Angle : d
Parallax Method : D 
d
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1
Units And Measurements
4)
Error Analysis : a1, a2, a3, ...... an values obtained Combination of Errors:
in measurement.
1) Error of sum or difference
Z=A+B
a1  a2  ........an
amean 
Z  A  B
n
2) Error of product
a1  a1  amean
Z = AB
a2  a2  amean
Z A B


Z
A
B
an  an  amean
a1  a2  .......  an
amean 
3)
Z
n
Relative error 
amean
amean
a
Percentage error  mean
amean
Error of Division
4)
100
A
B
Z A B


Z
A
B
Error in case of raised power
Z
A p Bq
Cr
Z
A
B
C
p
q
r
Z
A
B
C
IMPORTANT
DIMENSIONAL
ANALYSIS
Dimension
[M0L0T–1]
1
2
–2
1
–1
–2
1
1
–1
1
0
–2
[M L T ]
[M L T ]
[M L T ]
[M0L1T–2]
[M1L1T–2]
[M1L2T–1]
[M L T ]
Quantity
Frequency, angular frequency, angular velocity, velocity gradient and
decay constant
Work, internal energy, potential energy, kinetic energy,
torque, moment of force
Pressure, stress, Young's modulus, bulk modulus, modulus of rigidity,
energy density
Momentum, impulse
Acceleration due to gravity, gravitational field intensity
Thrust, force, weight, energy gradient
Angular momentum and Planck's constant
Surface tension, Surface energy (energy per unit area)
Strain, refractive index, relative density, angle, solid angle,
distance gradient, relative permittivity (dielectric constant), relative
[M L T ]
permeability Poisson's ratio etc.
0 2 –2
[M L T ]
Latent heat and gravitational potential
2 –2  1
[ML T θ ] Thermal capacity, Boltzmann's constant and entropy
0
0
0
0
0
1
[M L T ]
l / g , m / k , R / g , where l  length
g = acceleration due to gravity, m = mass, k = spring constant,
R = Radius of earth
[M0L0T1]
L/R, LC , RC where L = inductance, R = resistance, C = capacitance
2
–2
[ML T ]
V2
q2
t , VIt , qV , LI 2 , , CV 2
R
C
where I = current, t = time, q = charge,
L = inductance, C = capacitance, R = resistance
I 2 Rt ,
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Scalars And Vectors
Quick Revision
SCALARS AND VECTORS
TERMINOLOGY
VALUES
FORMULAE
FORMULAE
2)

A
ˆ 
Unit vectors : A
A
  
Law of Triangle : R  A  B
3)
    
Law of Polygon : R  A  B  C  D
Scalar Quantity : A physical quantity which can
be completely described by its magnitude only
4)
is known as scalar quantity.
  
Law of Parallelogram : R  A  B


A & B : Are two vectors


: Angle between A and B

R : Resultant of vectors.

: Angle made by vector with x-axis

: Angle made by vector with y-axis
1)
: Angle made by vector with y-axis
i
: unit vector along x axis
j
: unit vector along y axis
k
: unit vector along z axis
DEFINITIONS
DEFINITIONS
1)
2)
Vector Quantity : A physical quantity which has
magnitude and direction and obeys all the laws
of vector algebra is called vector quantity.
3)
Parallel Vector : Those vectors which have the
same directions are called as parallel vectors.
4)
Equal Vector : Vectors which have equal
magnitude and same direction are called equal
vectors.
5)
Anti-parallel Vectors : Those vectors wich have
the opposite directions are called as Antiparallel vectors.
6)
Opposite Vectors : Vectors have equal
magnitude but opposite directions are called as
opposite vectors.
7)
Unit Vectors : Vectors whose magnitude is one
is called a unit vector.
8)
Rectangular components of vector : When a
vector is splitted into components which are
right angle to each other then the components
are called rectangular components of vectors.
9)
Dot Product : The dot product of two vectros
can be defined as the product of their
magnitudes with cosine angle between them.
R  A 2  B 2  2 AB cos
tan  
A sin
B  A cos
Cases
1) If  0, R  A  B
2) If
5)
 90, R  A2  B2
3) If  180, R  A  B
Substraction of vectors :
R  A 2  B 2  2 AB cos
10) Cross Product : the cross product of two vectors
can be defined as the product of their
magnitudes with sine angle between them.
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3
Scalars And Vectors
6)
Resolution of vectors
a) Two dimensions
6) In case of orthogonal vectors
i j j kk i0
7) Scalar product of a vector by itself
 
A A  A2
8) Incase of unit vector
i i  j j  k k 1
9) Interms of components
 
A B  ( Ax i  Ay j  Az k ) ( Bx i  By j  Bz k )
b) Three dimensions

A  Ax i  A y j  A z k
 
A B  Ax Bx  Ay By  Az Bz
A A A A
2
x
2
y
2
z
10) Projection of vector
A
cos   x
A
cos  
cos 
Ay
A
Az
A
cos 2   cos 2   cos 2  1
7)
sin 2   sin 2   sin 2  2
 
Dot Product : A B  AB cos
8)
 


A B
Projection of B on to A  B cos 
A
 


A B
Projection of A on to B  A cos 
B
Cross product :
  
CA B

C  C  AB sin
Key points
Key points
 
1) If Q  0, A B  AB
 
If Q  90, A B  0
 
If Q  180, A B   AB
2) Angle between two vectors
 
A B
cos 
AB
3) It is commulative
   
A BB A
4) It is Distributive
  
   
A (B  C )  A B  B C
5) It is associative
   
       
( A  B(C  D )  A C  A D  B C  B D
1) If

 
 0, C  A B  0
If

 
 90, C  A B  AB
If

 
 180, C  A B  0
2) Angle between the vectors
 
A B
sin 
AB
3) It is anti-commutative
   
A BB A
4) It is distributive
      
A (B  C)A B  A C
5) It is associative
 
 
       
( A  B) (C  D)  A C  A D  B C  B D
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4
Scalars And Vectors
6) Incase of orthogonal vector
i jk
j k i
k ij
j i  k
10) If two vectors are parallel

A  Ax i  A j j  A z k

B  Bx i  B j j  B z k
Ax Ay Az
then B  B  B
x
y
z
k j  i
i k  j
7) Vector product of a vector by itself
 
A A0
8) Incase of unit vector
i i j jk k0
9) In terms of components
 

A B  ( Ax i  Ay j  Az k ) ( Bx i  By j  Bz k )
i
 
A B  Ax
j
Ay
k
Az
Bx
By
Bz
 i[ Ay Bz  Az By ]  j[ Ax Bz  Az Bx ]
 k[ Ax By  Ay Bx ]
  
 

11) A , B and C are coplanar then A ( B C )  0
 
 
12) Angle between ( A  B) and ( A B) is 90°.
13) Formuale to find area


i) If A and B are two sides of triangle
then its area 
1  
AB
2


ii) If A and B are two adjacent sides of
 
parallelogram then its area  A  B


iii) If A and B are diagonals of a
paralelogram then its area 
1  
AB
2
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Motion In A Straight Line
Quick Revision
MOTION IN A STRAIGHT LINE
VALUES
TERMINOLOGY
S
:
v
:
u
:
t
:
a
:
g
:
vAB :
vBA :
|v| :
vo :
1)
Displacement
Velocity
Initial velocity
Time taken
Acceleration
Acceleration due to gravity
Velocity of ‘A’ with respect to ‘B’
Velocity of ‘B’ with respect to ‘A’
Speed
Initial velocity or velocity at t = 0
11) Acceleration : The rate of change of velocity of
an object with time is called acceleration. In S.I
unit, it is the change in velocity in one second.
13) Free fall: When a body released near the earth’s
surface, it accelerates downwards towards
earth. In the absence of air resistance its velocity
continuously increases. The motion of the body
is called free fall.
FORMULAE E
FORMULA
1)
DEFINITIONS
DEFINITIONS
Distance : It is the length of actual path
traversed by a body between its initial and final
position. Distance is a scalar quantity.
2) Displacement : It is the shortest length between
initial and final position of body. It is a vector
quantity.
3) Speed : The rate of change of position of an
object with time in any direction is called its
speed. It is equal to distance travelled per unit
time.
4) Average speed : For an object moving with
variable speed, the average speed is total
distance travelled per unit time.
5) Instantaneous speed : The speed of an object
at any particular instant of time or at particular
point of path is called instantaneous speed.
6) Velocity : The rate of change of position of
object with time in a given direction is called
velocity. It is equal to displacement per unit
time.
7) Average velocity : Average velocity is called at
the ratio of total displacement to the total time
interval of the body.
8) Instantaneous velocity : The velocity of an
object at a particular instant of time or at a
particular point on its path is called
instantaneous velocity.
9) Uniform motion: An object is said to be in
uniform motion if it covers equal distance in
equal intervals of time however small these
intervals may be.
10. Non-uniform motion: A body is said to be in
non uniform motion if it covers unequal
intervals in equal periods of time. In this
motion, its velocity changes with time.
Speed 
Distance
Time
4)
Displacement
Time

d(s )
Velocity 
dt
Speed = |Velocity|
5)
Acceleration 
6)

 dv
a
dt
2)
3)
7)
8)
9)
Velocity 
vu
t
  

s1  s2  s3  ......sn
Average velocity 
t1  t 2  t3  ....tn
v = u + at
v2 – u2 = 2as
1 2
10) s  ut  at
2
a
11) sn th = u + ( 2n - 1)
2
12) vAB = vA – vB
13) vBA = vB – vA
14) Relative velocity 
Relative change in sepration
Time
For motion under gravity
I)
v = u – gt
II) v2 – u2 = –2gs
III)
1
s  ut  gt 2
2
IV)
g
sn th = u  ( 2n - 1)
2
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6
Motion In A Straight Line
GRAPHS GRAPH
DISPLACEMENT–TIME
I)
II)
v
s
t
III)
t
Uniformly increasing velocity
v
Object at Rest
s
II)
*
t
Uniformly decreasing velocity
Relative Velocity
Graphs
GRAPHS
I)
s
t
Object moving with uniform velocity
B
A
s
III)
t
vA = vB
Relative velocity is zero
t
Object moving with uniform +ve acceleration
s
IV)
II)
B
s
A
t
t
vB > vA
Object is moving with decreasing velocity
VELOCITY–TIME
GRAPHS GRAPH
I)
III)
vB – vA = +ve
s
A
v
B
P
t
t
Object moving with constant velocity
vA > vB
vA – vB = +ve
P = Point of overtaking
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Motion In A Plane
Quick Revision
MOTION IN A PLANE
TERMINOLOGY
VALUES

r : Displacement Vector


r1 or r0 : Initial position vector

r2 : Final position vector

v : Velocity vector
vx : x-component of velocity
vy : y-component of velocity

a : Acceleration vector
ax : x-component of acceleration vector
vy : y-component of acceleration vector
R : Range of Projectile
T : Time of light
u
: Initial velocity
t
: Half time of projectile or Instantaneous time
m : slope
1)
2)
3)
4)
6)

dv

lim t 0
dt
7)
Speed | v| vx2  vy2
8)

a  ax iˆ  a y jˆ
9)
ax 

a
10) ay 
2)
3)
4)
5)
dv y
dt
dv y ˆ
 dv
j
11) a  x iˆ 
dt
dt
  
12) v  v0  at
 

13) r  r0  r
 
DEFINITIONS
DEFINITIONS
   v  vo 
Projectile : A projectile is the name given to any 14) r  r0  
t
 2 
body which since thrown its space with some
initial velocity moves thereafter under the effect 15) v 2  v 2  2 a.(r  r )
o
o
of gravity alone, without being propelled by
vy
any engine or fuel. The path followed is called
m

tan
T

16)
trajectory.
vx
Assumptions used in projectile motion:
ANGULAR
PROJECTION
ANGULAR PROJECTILE
a) There is no air resistance on the projectile.
b) The effect due to earth’s curvature is negligible.
u2 sin 2T
c) The effect due to rotation of earth is zero. 1) R 
g
d) Acceleration due to gravity is constant in all
points of motion.
2u u
Time of flight : It is the total time which the 2) R  2u sin T .u cos T  y x
g
g
projectile remains in the flight.
Horizontal Range : It is the horizontal distance
2u sin T 2uy
covered by the projectile during its time of

3) T 
flight.
g
g
Horizontal component : It is the component of
velocity which remain constant and is parallel
T u sin T uy
t



4)
to surface of earth.
2
g
g
FORMULAE
FORMULAE
1)
dvx
dt
  
r  r2  r1

r  xiˆ  yjˆ

v  v x iˆ  v y jˆ

 r
v
t


dr
v 
lim t 0
dt
u2 sin 2 T
2g
5)
H
6)
H
7)
T2
uy2
2g
2H
g
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8
Motion In A Plane
8)
x
y  x tan T  1  
R

9)
y  x tanT 
3)
Y
gx 2
2u2 cos2 T
HORIZONTALPROJECTION
PROJECTILE
HORIZONTAL
x
u
1)
t
2)
T
3)
v  v x2  v y2
4)
v y  2 gh
5)
v y  gt
6)
Rv
7)
t
2h
g
4)
vy
u
x
R
2h
g
tan E 
1)
Vertical distance with time.
O

gt
u
T
t
Horizontal distance with time.
GRAPHS
GRAPHS
Y
5)
r
r0
vy
v sin
r
O
X

ro = Initial Position

 r = Displacement vector

r = Final position
2)
ux
t
O
-v sin
Vertical velocity with time
t
Horizontal velocity in projectile w.r.t. time
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9
Circular Motion
Quick Revision
VALUES
TERMINOLOGY
:
Angular Displacement [rad]
:
Initial Angular Velocity [rad/s]
f :
Final Angular Velocity [rad/s]
 :
V :
r
:
acp :
at :
Fcp :
Wcp :
Pcp :
Ft :
s
:
:
Angular Acceleration [rad/s2]
Linear Velocity [m/s]
Radius [m]
Centripetal Acceleration [m/s2]
Tangential Acceleration [m/s2]
Centripetal Force [N]
Work Done by Centripetal Force [ J ]
Power by centripetal Force [W]
Tangential force [N]
Displacement [m]
Coefficient of Friction
i
b
:
Angle of Banking
m
q
:
:
Mass [kg]
Charge [coulomb]
CIRCULAR MOTION
8)
Centrifugal force : The pseudo force in circular
motion which acts along radius and directed
away from the center of circle is called as
centrifugal force.
9)
Angle of Banking : The angle made by the
surface of road with the horizontal surface of
road is called angle of banking.
10) Banking of Road : The process of raising outer
edge of road over its inner edge through certain
angle is called as Banking of road.
FORMULAE
FORMULAE
1)
DEFINITIONS
DEFINITIONS
1)
2)
3)
4)
5)
6)
7)
Circular Motion : Movement of an object along
the circumference of a circle or rotation along
a circular path is calloed as circular motion.
Uniform Circular Motion : Periodic motion of
a particle movin g along circumference of a 2)
circle with constant angular speed is called as
uniform circular motion.
Non-uniform Circular Motion : Motion of a
particle moving along circumference of a circle
with variable angular speed is called as Nonuniform circular motion.
Angular Displacement ( ) : The angle
described by radius vector in a given time at the
center of circle is called as angular displacement.
Angular velocity [  ] : The time rate change of
limiting angular displacement is called angular
velocity.
Angular Acceleration [ ] : The time rate change
of an angular velocity is called angular
acceleration.
Centripetal Force : Force acting on a particle
performing circular motion which is along
radius of circle and directed towards.
Kinematics of Circular Motion
1
 i t  t 2
2
2f  i2  2
nth   

(2n  1)
2
relative 
Vrelative
rrelative

VB  V A
rB  rA
Uniform Circular Motion

d
0
dt
d
0
dt
at  r
acp  2 r 
V2
 V
r


V2
 
acp  2 r  
rˆ   V
r
Fcp 
mv 2
 m2 r  mv
r
Wcp  0
Pcp  0
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10
Circular Motion
3)
Non-uniform Circular Motion

 
 d 2  1


dt
t
d
dt
at  r


 
v2
acp  2 r   rˆ   v
r
  
at   r



anet  acp  at


Fcp  macp


Ft  mat
For condition of slack
TA  0 , V A  gl
VB  5 gl
For any condition
TB  TA  6 mg
6)
Wcp = 0
Wt  Ft S
qvB 
Pcp = 0
Pt 
Motion of a Charged aprticle in a magnetic field
mv 2
r
Wt
 Ft V
t
p  Pcp  Pt
4)
Banking of Road
Horizontal curve road
V
rg
Banked road
V
If
7)
tan 
rg(  tan )
(1  tan )

0
tan 
V2
rg
5)
V2
rg
Verticle Circular Motion
V2
rg
g
h
T  2
Pendulum in a car
tan 
Conical Pendulum
cos
g
r
Tension  mg 1   
h
8)
2
Change in vector quantities
 
Change in velocity  V  2V sin  
2
 
Change in radial vector  V  2r sin  
2
 
Change in momentum  P  2mv sin  
2
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11
Circular Motion
9)
Death Well :
V
rg
12) Reaction of road on car
1) car on a concave bridge
R  mg cos 
mv 2
r
2) car on a convex bridge
10) Hemispherical Vessel :

g
R cos
11) Inverted Cone :
V  gh
R  mg cos 
mv2
r
13) Motion of a block on frictionless hemisphere
2
h r
3
cos 
h
r
2
 cos 1  
r
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12
Newton’s Laws Of Motion
Quick Revision
NEWTON’S LAWS OF MOTION
TERMINOLOGY
VALUES
F
:
Force [N]
m
:
Mass [kg]
dv
:
Change in velocity [m/s]
p
:
Linear momentum [kg m/s]
I
:
Impulse [N-s]
dp
:
Change in momentum [N-s]
W
:
Weight [N]
T
:
Tension [N]
L
:
Length [m]
a
:
Acceleration [m/s2]
:
FORMULAE
FORMULAE
2)

 dp
dV

F
m
 ma
dt
dt


p  mV
3)
I  Ft  mV  mu
1)
I  F dt
Area under F-t graph
will give impulse.
4)
W  mg
Angle
5)
Fpseudo  maframe
m0 :
Initial mass of rocket [kg]
6)
Law of conservation of momentum
dm
:
dt
Rate of fuel consumption [kg/s]
Vr
:
Velocity of gases relative to rocket
N
:
Normal reaction [N]
m1u1  m1 v1
7)
i) For two bodies
DEFINITIONS
DEFINITIONS
1)
Force : Force is a push or pull which tries or
change the state of rest or uniform motion of
a body.
2)
Weight : Force given by earth towards its centre
on an object is called weight.
3)
Reaction : If a body is pressed against a rigid
support, the body experienced a force which is
perpendicular to surface in contact is called
reaction or normal reaction.
4)
Impulse : When a large force acting for a short
interval of time is called impulse.
6)
Intertial frame of reference : A non-accelerating
frame of reference is called inertial frame of
reference.
7)
Pseudo Force : Those force which do not
actually act on the particles but appear to be
acting on the particles due to accelerated
motion of frame of reference are called pseudo
force.
a
F
m1  m2
T
m1 F
m1  m2
ii) For three bodies
a
Linear Momentum : The quantity of motion
present in a body is called as Linear
Momentum.
5)
Motion of connected bodies
8)
F
m1  m2  m3
T1 
m1 F
m1  m2  m3
T2 
( m1  m2 )F
m1  m2  m3
Rope on a horizontal surface

l
T  1  F
L

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13
Newton’s Laws Of Motion
9)
Pulleys
Case - V
Case - I
a
( m1  m2 sin )
g
( m1  m2 )
T
( m1 m2 )(1  sin )
g
( m1  m2 )
m1  m2  m
T = mg
a=0
Case - II
m1  m2
a
( m1  m2 )
g
( m1  m2 )
T
2m1 m2
g
( m1  m2 )
Case - III
a
( m2  m3  m1 )
g
( m1  m2  m3 )
Case - VI
a
( m1 sin   m2 sin )
g
( m1  m2 )
T
m1 m2 (sin   sin )
g
( m1  m2 )
10) Tension in lift wire
i) Lift is stable : T  mg
ii) Lift moving up : T  m( g  a)
iii) Lift moving down : T  m( g  a)
11) Apparent weight
i) Lift is stable : N = Mg
ii) Lift moving up
N  M ( g  a)
iii) Lift moving down
N  M ( g  a)
12) Rocket propulsion
acceleration of rocket  
Vr dm
m dt
dm
dt
If gravitational force is considered acceleration
Thrust on rocket  ma  Vr
Case - IV
a
m2 g
m1  m2
T
m1m2
g
( m1  m2 )
of rocket  
Vr dm
g
m dt
 dm 
m  m0  
t
 dt 
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14
Friction
FRICTION
Quick Revision
FORMULAE
FORMULAE
VALUES
TERMINOLOGY
F
:
Force (N)
µ
:
Coefficient of friction
N
:
Normal reaction (N)
µk
:
Coefficient of kinetic friction
µs
:
Coefficient of static friction
m
:
Mass (kg)
W
:
Weight (N)
T
:
Angle of repose
v
:
Velocity (m/s)
s
:
Distance (m)
t
:
Time (s)
fs
:
Static friction
fk
:
Kinetic friction
DEFINITIONS
DEFINITIONS
1)
1)
Friction:
fs = µsN
fk = µkN
2)
Angle of friction:
= tan–1(µ)
R
N
F
f
3)
Angle of repose:
= tan–1(µs)
Friction:
The resistive force which will act when two
bodies tries to slide is called friction.
2)
Static friction:
The frictional force acting between two surfaces
at rest is called as static friction.
3)
4)
Kinetic friction:
The frictional force acting between two surfaces 4)
in relative motion is called kinetic friction.
Pull is easier than push:
Rolling friction:
FPush 
The frictional force acting when object perform
rolling motion is called rolling friction.
5)
Limiting friction:
µmg
cos  µ sin
F
The maximum force of static friction upto
which body does not move is called limiting
friction.
6)
Angle of friction:
The angle which the resultant contact force
makes with the normal reaction is called angle
of friction.
7)
Angle of repose:
FPull 
µmg
cos  µ sin
F
The maximum angle of inclination of a plane
with the horizontal at which the object placed
on it just begin to slide down is called as angle
of repose.
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15
Friction
5)
Motion of an insect in the rough bowl:
8)

1 
h  R1

µ2  1 

Friction on an inclined surface:
i) Object moving up
a  g[sin  µ cos ]
R
h
6)
ii) Object moving down
a  g[sin  µ cos ]
Maximum length (Y) hung from table:
Y
µL
µ1
L–Y
9)
Y
Minimum force to move:
Fmin 
µmg
µ2  1
Fmin
7)
Stopping of block due to friction:
v2
s
2µg
t
v
µg
a  µg
10) f Versus FApplied:
f
Fapplied
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16
Work, Energy And Power
Quick Revision
WORK, ENERGY AND POWER
VALUES
TERMINOLOGY
W
:
Work done
P
:
Power
s
:
Displacement
ds
:
Small change in displacement
K.E.
:
Kinetic energy
T.E.
:
Total energy
T.M.E. :
Total mechanical energy
P.E.
:
Potential energy
p
:
Momentum
KEF
:
Final kinetic energy
KEI
:
Initial kinetic energy
U
:
Potential energy
U
x
:
Partial derivative of energy w.r.t. x
U
y
:
Partial derivative of energy w.r.t. y
U
z
:
5)
Work done by net force acting on a body is equal
to change in kinetic energy of body.
6)
6)
7)
8)
Power:
It is the rate of doing work. In other words, if
work done or energy consumed per unit time.
9)
Watt:
Watt is S.I. unit of power. The power of an agent
is one watt if it does work at the rate of 1 joule
per second.
Work:
10) Kilowatt-hour:
Kilowatt-hour is the commercial unit of energy.
One kilowatt hour is the electrical energy
consumed by an appliance of 1000 watt in
1 hour.
Positive work:
The force acting on a body has component in
the direction of displacement. The work done
is called positive work.
Conservation of mechanical energy:
This principle states that when only
conservative forces are acting on body then its
net mechanical energy (potential energy +
kinetic energy) remains constant.
DEFINITIONS
DEFINITIONS
2)
Non-conservative forces:
If the amount of work done in moving an object
against a force from one point to another point
depends along the path along which the body
moves, then such a force is called nonconservative force.
Partial derivative of energy w.r.t. z
Work is said to be done whenever a force acts
on a body and body moves through some
distance in the direction of force.
Conservative forces:
A force is conservative if work done by force
in displacing a particle from one point to
another is independent on the path followed by
particle and depends only on the initial and end
points.
00
1)
Work energy principle:
FORMULAE
FORMULAE
1)
3)
Negative work:
4)
2)
If the force acting on a body has the component
opposite to direction of displacement then work
done is called negative work done.
3)
Zero work:
When force or displacement for cos , either of
4)
them is zero then work done is zero.
 
W F.s
W | F ||s|cos

s2
 
W  F .ds

s1
K.E. =
p2
1
mv 2 
2
2m
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17
Work, Energy And Power
5)
p  2m( K.E.)
 
12) P  F . v
6)
1
1
W  mv 2  mu 2
2
2
 dv
13) F  
ds
7)
  1
Fnet .s  m( v 2  u2 )   K .E.
2
14) Fx 
 U
x
8)
P.E. of spring =
Fy 
 U
y
9)
U
1 2
kx
2
1 2 1
F2
kx  Fx 
2
2
2k
10)  P.E.  mgh2  mgh1  mgh
W dW

11) P 
t
dt
 U
z
15) For equilibrium,
Fz 
F 0
dU
0
ds
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18
Centre Of Mass And Collision
Quick Revision
CENTRE OF MASS AND COLLISION
TERMINOLOGY
VALUES
Fnet
m

rcom
6)
: Net force on a particle (Fext + Fint)
: Total mass
: Position of center of mass
xcom
: Coordinates of Centre of mass on x-axis.
ycom
: Coordinates of Centre of mass on y-axis.
zcom
: Coordinates of Centre of mass on z-axis. 7)
m1, m2 : Mass of particles
 
r1 , r2
: Position of individual particle
d1
d2
R
h
A
V
Vcom
acom
: Distance of centre of mass m1.
8)
: Distance of centre of mass m2.
: Linear mass density.
: Radius of circle, semicircle, disc, sphere,
hemisphere and ring.
: Height of triangle, hollowand solid cone. 1)
: Density of material
: Area
: Volume
2)
: Velocity of centre of mass
: Acceleration of centre of mass
3)
DEFINITIONS
DEFINITIONS
1)
2)
3)
4)
5)
Particle :
Is defined as an object whose mass in finite but
size and structure is neglected.
System :
Is a collection of very large no of particle, having
finite size and structure.
Internal force :
The mutual force exerted by particles of system
on one another is called as internal force.

 Fnet (Internal)  0 .... always on system
4)
5)
6)
7)
1)
2)
 ex. - Intermolecular force
- Friction
- Explosion
3)
- Electrostatic
- Gravitation
4)
External force :
The outside force exerted on a system by
external agent is called external force.


 Fnet (Internal)  ma
Centre of Mass :
Is a point where whole mass of a body is
supposed to be concentrated is called centre of
mass.
Centre of gravity :
A point where whole weight of body is act or
supposed to be concentrated is called centre of
gravity.
OR
Is a point at which resultant of gravitational
force of all the particles of a body act.
Velocity of centre of mass :
Taking time defivative of position vector of
centre of mass will get the velocity vector of
centre of mass.
Acceleration of centre of mass :
Taking time derivative of velocity vector of
centre of mass will get the acceleration of centre
of masss.
CONCEPT
CONCEPT
Two particle system : Centre of mass divide the
distance between particles in inverse ratio of
there masses.
Centre of mass is closer to a massive body.
Applied force is in line with centre of mass then
body will travel in translational motion.
For small body centre of mass and centre of
gravity both one same.
Try to place particles on co-ordinates system,
so that we get maximum no. of zero.
Try to find symmetry.
Centre of mas of a two particle always lie on
line joining of these two particle.
Internal force is action reaction pair.
FORMULAE
FORMULAE


m r  m2 r2  .......

rcom  1 1
Weighted average
m1  m2  ......
xcom 
m1 x1  m2 x2  .......
m1  m2  m3  ......
m1 y1  m2 y2  .......
m1  m2  ......
Centre of mass of two particle system
ycom 
 d1 

m1d
m2 d
,  d2 
m1  m2
m1  m2
d1 m2

d2 m1
or m1d1  m2 d2
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19
Centre Of Mass And Collision
5)
Centre of mass of a continious body



m1r1  m2 r2  ..........mnrn

rcom 
m1  m2  ..................mn
xcom 
x dm
dm
, ycom 
y dm
dm
Trick : For regular shape body
7)
Motion of centre of mass
a) Velocity of centre of mass



m1V1  m2 V2  ........
Vcom 
m1  m2  ........


P  MVcom
b) Acceleration of centre of mass


m1a1  m2 a2  ......

acom 
m1  m2  ........


Fnet  Macom



Fnet  Fexternal  Finternal
6)
Centre of mass a remaining portion
a) For solid body

Finternal  0
Cases:

If Fexternal  0


m1r1  m2 r2

rcom 
m1  m2
xcom 
m1 x1  m2 x2
m1  m2
m1 y1  m2 y2
m1  m2
b) For two dimensional body (lamina)
ycom 


m r  m2 r2

rcom  1 1
m1  m2
m x  m2 x2
A1 x1  A2 x2
xcom  1 1
or xcom 
m1  m2
A1  A2
ycom 
Then
m1 y1  m2 y2
A1 y1  A2 y2
or ycom 
m1  m2
A1  A2
Irrespective of the individual acceleration of
particle.

a) Vcom  const :
If initially Vcom = constant then it will remain
always constant.

b) Vcom  0
If initially Vcom = 0 then it will remain always
zero.
ex. - Particles is at rest initially
Initial velocity is zero
Body is at rest.
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20
Rotational Motion
Quick Revision

VALUES
TERMINOLOGY
ROTATIONAL MOTION
7)
: Angular velocity [rad/s]
2 : Final angular velocity [rad/s]
1 : Initial angular velocity [rad/s]
T
f

m
I
r
I0
Ic
Iz
Ix
Iy
m1
m2
F
r1
k
h
v
1)
2)
3)
4)
5)
6)
1)
: Time period [s]
: Frequency [1/s]
: Angular acceleration [rad/s2]
: Mass [kg]
: Moment of inertia [kg m2]
: Perpendicular distance [m]
: Moment of inertia of an object through point O.
: Moment of inertia of an object through centre
of mass.
: Moment of inertia of an object through z axis.
: Moment of inertia of an object through x axis
: Moment of inertia of an object through y axis
: Mass of first particle [kg]
: Mass of second particle [kg]
: Torque [N-m]
: Force [N]
2)
: Perpendicular distance [m]
: Angle
: Radius of gravition [m]
3)
: height [m]
: Velocity [m/s]
Rolling Motion : When a body perform
translational and rotational motion is called as
rolling motion.
FORMULAE
FORMULAE
Fundamental of rotational motion

2

T
f
1
T
 1
t
2  2
t
Comparison of linear and rotational motion
Linear motion
Rotational Motion

V  u  at
2  1  t
V 2  u2  2as
22  12  2
1
1
S  ut  at 2
 1t  t 2
2
2
Moment of inertia of a particle
I  mr 2
Theorem of parallel axis
I 0  I c  Mh 2
DEFINITIONS
DEFINITIONS
Rotation Motion : The change in the orientation
of body during its motion is called rotational
motion.
Moment of Inertia : The property of body due
to which it oposees any change in its state of
rest or of uniform rotation is called moment of
ienrtia.
Radius of Gyration : The distance from an axis
of rotation where entire mass of the body 4)
supposed to be concentrated and the value of
moment of inertia is same that due to actual
distribution of masses of body is called radius
of gyration.
Torque : If a pivoted, hinged or suspended body
tends to rotate under the action of force it is said
to be acted on by a torque.
Angular Momentum : The moment of linear
momentum of body with respect to any axis of
rotation is called as angular momentum.
Kinetic Energy of Rotation : The energy which
a body has by virtue of it rotational motion is
called as kinetic energy of rotation.
2
X
a
d
CM
X’
a’
Theorem of perpendicular axis
Iz  Ix  I y
Z
Y
O
X’
Z’
X
Y’
Formula for moment of inertia of regular
bodies
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21
Rotational Motion
Shape Of
Body
Axis Of Rotation
Figure
(1)
1.Passes through the centre
Circular Ring
& perpendicular to plane
M = Mass
R = Radius 2.About it’s diameter in
MR
(3)
Hollow
Cylinder
M = Mass
R = Radius
L = Length
2
R
2
(1/2)MR
it’s own plane
(2)
Circular Disc
M = Mass
R = Radius
Moment Of Radius Of
Inertia (I)
Gyration
R/ 2
3.A b ou t a t a n g en t i a l a xi s
perpendicular to its own plane
2MR
4.About a tangential axis in its
own plane
3
2
MR
2
3
1.Passing through the centre and
perpendicular to the plane
MR 2
2
R
2.About diameter
MR 2
4
R
2
3.About a tangential axis lying
its own plane
5
MR 2
4
5
R
2
4.A b ou t a t a n g en t i a l a xi s
perpendicular to its own plane
3
MR 2
2
3
R
2
1.About its geometrical axis
MR 2
R
2
2R
2
R
2
l
2.About an axis passing through
its CM and perpendicular to its
length
3.About an axis perpendicular to
its length and passing through
one end of the cylinder
cm
 MR 2 Ml 2 



 2
12 

R2
2

l2
12
l
 MR 2 Ml 2 



 2
3 

R2 l 2

2
3
l
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22
Rotational Motion
(4)
Solid
Cylinder
M = Mass
R = Radius
L = Length
1.About its geometrical axis
2.About an axis passing through
its CM and perpendicular to its
length
MR 2
R
C
2
2
2
C
MR
Ml

4
12
R2 l 2

4
12
R2
l2
R2 l 2

2
3
2
l
3.About an axis perpendicular to
its length and passing through
one end of the cylinder
M
4

3
(5)
Annular disk
1.Passing through centre and
perpendicular to the plane
R1
R1
M
R2
2
[ R12  R22 ]
R12  R22
2
R2
M [ R12  R22 ]
R1  R2
4
2
M[ R12  R22 ]
R12  R 22
2
R2
M = Mass
R1 = Internal
Radius
R2 = Outer
Radius
2.About its diameter
(6)
1.About its geometrical axis or
Hollow
about the axis which is passing
Cylinder R1
through centre
= Internal
Radius
R2 = Outer
Radius 2.Passing through centre of mass
and perpendicular to its length
M = Mass
L = Length
(7)
Solid Sphere 1.About its axis OR diameter
whi ch i s passi ng through
M = Mass R =
which is
Radius
2.About tangential axis
R1
R1
R2
2
2
cm
M
2
2
L
( R  R2 )
 1
12
4
2
5
MR
2
7
MR 2
2
2
2
L2 R12  R22

12
4
2
R
5
7
5
R
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23
Rotational Motion
(8)
Thin Spherical 1 .Pa s s in g t h ro ug h a xis or
Shell
diameter
(Hollow Sphere)
M = Mass
R = Radius
(Thickness
negligible)
2
MR 2
2
R
3
5
MR2
3
5
2 [ R5  r5 ]
M
5 [ R3  r3 ]
2 (R 5  r 5 )
3
2.About tangential axis
3
R
(9)
Solid Sphere
With Cavity
r = Internal
Radius
R = Outer
Radius
About passing through centre
OR about diameter
5 (R 3  r 3 )
M = Mass
(10)
1.Passing through centre of mass
Thin Rod
and perpendicular to length
[Thickness is
negligible
w.r.t. length]
ML
12
L
2.Passing through its one end and
perpendicular to axis
(11)
Rectangular
Plate
a = Length
b = Width
M = Mass
ML2
3
1.About an axis passing through
CM and perpendicular to side
a in its plane
cm
b
Ma 2
12
L
2 3
L
3
a
2 3
a
2.About an axis passing through
CM and perpendicular to side
b in its plane
3.About an axis passing through
CM
cm
Mb
12
b
2
b
2 3
a
cm
b
M (a 2  b 2 )
12
a2  b2
12
a
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24
Rotational Motion
(12) Triangular
Prism
a = (Side of base
and height)
Passing through centre of mass
and perpendicular triangular
face
Ma 2
6
a
6
a
(13)
Cone
R = Radius
h = Height
About the line joining of top of
the cone and mid-point of base
3
MR 2
10
3
10
R
R
Moment of inertia of some special bodies
(a)
Moment of inertia of square
plate
I1  I 3  I4 
I5 
Ma2
I5
12
cm
6
a
I3
(b)
Moment of inertia of cube
I2 
I4
I5
I1
I2
I1 
I1
a
Ma 2
Ma2
6
2Ma2
a
3
a
a
IAB
(c)
I n a tri angle, M.I. w il l be
maximum relative to smallest
side
If
AC > BC > AB,
IA C < I BC < I AB
A
B
C I
AC
I BC
(d)
(e)
I n t ria n g le, M. I . wi ll b e
maximum rela tive to tha t
perpen dicula r a xis whi ch
passes through least angle
I2
If
2
< 2 < 3,
I1 > I 2 > I 3
1
I3
3
1
I1
Greater the mass away from
axis of rotation, more will be
M.I.
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25
Rotational Motion
10)
5)
Radius of gyration [k]
6)
I  mk 2
Moment of inertia of two point masses
r  r1  r2
m1
m1r1  m2 r2
r1 
r2 
r1
CM
m2
Rolling motion
i) Kinetic energy in rolling motion
K.E.rolling 
r2

m2 r
m1  m2
1
k2
MV 2 1  2
2
R
ii) Rolling : motion on an inclined plane
m1r
m1  m2
a
I  m1r12  m2 r22
I
7)
1 2 1
I   MV 2
2
2
m1 m2
r 2  mr r 2
m1  m2
V
Torque
  
r F
 rF sin
t
 F r1
Moment of couple = Fr
 m
Rolling body
g sin
k2
1 2
R
VCM
Height
S
h
2 gh
k2
1 2
R
1
sin
Inclined plane
2h 
k2 
1



g 
R2 
iii) Pure rolling
V  R
F
r
F
8)
Angular momentum [L]
  
Lr p
iv) Energy distribution in rolling motion
L  rmv sin
L  mvr sin
Law of conservation of angular momentum if
there is no external torque.
I 11  I 2 2
9)
Kinetic energy of rotation
K.E. 
1 2 1 2 V2
I   mk
2
2
R2
Body
K2
E trans
2
Erotation
R

1
K 2 / R2
E trans

Etotal
1
K2
1 2
R
Erotation
E total

K 2 / R2
1
1
1
2
2
2
3
1
3
Ring
1
1
Disc
1
2
2
Solid Sphere
2
5
5
5
2
2
7
7
Spherical Shell
2
3
3
2
3
5
2
5
Solid cylinder
1
2
2
2
3
1
3
Hollow Cylinder
1
1
1
2
1
2
K2
R2
1
k 2 L2
 mV 2 2 
2
2I
R
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26
Gravitation
Quick Revision
GRAVITATION
VALUES
TERMINOLOGY
F
: Force due to gravitation
G
 20

: Universal gravitational constant 
10 11 
 3

M
: Mass of earth (6 × 1024 kg)
m
: Mass of object
g
: Acceleration due to gravity
DEFINITIONS
DEFINITIONS
1)
Law states that every particle on planet or
universe attract every other partical by a force
which is directly preportional to product of
their masses and inversely proportional to
square of distance between them.
2)
g’h : Acceleration due to gravity at height h above
h
: Height above the earth surface
d
: Depth below the earth surface
R
: Radius of earth (6400 km)
E
: Gravitational field.
V
: Graivtational potential
U
: Gravitational potential energy.
W
: Work done
Ve
: Escae velocity
Vc
: Orbital or critical velocity.
C
: Speed of light (3 × 108 m/s)

: Angular velocity
3)
4)
OR
A gravitational field is defined as a sphere of
influence around a mass in which gravitational
force has been experienced.
5)
a
: Semi-major axis.
b
: Semi-minor axis.
VA
 dA 
: Areal velocity 

 dt 
L
: Angular momentum.
6)
g’eq : Acceleration due to gravity due to at a equator
Rp
: Radius of earth at pole.
Req : Radius of earth at equator.
Gravitational potential energy :
GPE of a body is defined as amount of
workdone required to bring a mass from
infinity to that point.
7)
Escape velocity :
The minimum velocity with which a body
projected to just overcome the gravitational pull
of planet.
8)
Orbital velocity (critical velocity) :
The velocity of sattelite required to put this
satellite in to a circular orbit around the planet.
g’R : Acceleration due to gravity due to rotation.
g’P : Acceleration due to gravity due to at a pole
Gravitational potential :
Gravitational potential at a point in a
gravitational field of a body is defined as
amount of wokdone required to bring a mass
from infinity to that point per unit mass.
VH : Horizontal velocity
: Time period of satellite or planet.
Gravitational field :
The gravitational force of attraction per unit
mass is called as gravitational field.
K.E. : Kinetic energy
T
Gravity or Gravitational pull :
The force of attraction due to earth on a body
is called as gravity or gravitational force or
gravitational pull.
B.E. : Binding energy
VP : Velocity of projection
Acceleration due to Gravity :
Acceleration produced in body due to gravity
or gravitational pull is called as acceleration due
to gravity.
: Density of object
g’d : Acceleration due to gravity at depth d
Newton’s law of gravitation :
9)
Binding energy :
The minimum amount energy required to
remove a satellite from earth’s gravitational
influence.
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27
Gravitation
FORMULAE
FORMULAE
Gm1 m2
1)
F
2)
 Gm1m2
F
( rˆ )
r2
Null point
r

F12
2
r

2h 
then g 'h  g  1  
R


F21
m1
m2
 g 'd 

d
4
G( R  d) or g  1   .. At depth ‘d’
R
3

g ' R  g  2 R cos 2
r
x
3)
a) if h <<<< R
... Due to rotation of Earth
= Latitude
m2
1
m1
 g 'p 
Gravitational force between spherical shell and
point mass.
6)
GM
GM
g 'eq  2
2 ,
Rp
Req
Gravitational field : E 
F
m
i) For point mass
x
xR
R
F0
4)
F
xr
GMm
R2
F
GMm
R2
 GM
E  2 ( rˆ)
r
ii) For spherical shell
Gravitational force between solid sphere and
point mass.
x

E0
M
Mx

4 3 4 3
R
x
3
3
x R
F
5)
R
GMm
R3
xR
GM
E 2
R
xr
GM
E 2
r
iii) For solid sphere
xR
F
GMm
R2
xr
F
GMm
R2
Acceleration due to gravity
g
GM
R2
 g 'h 
... Surface
GM
R2
g
or
.... height ‘h’
( R  h )2
( R  h) 2
x
R
GMx
E 3
R
xR
GM
E 2
R
xr
GM
E 2
r
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28
Gravitation
7)
Gravitational Potential  V 
iii) Relation between GP and GPE
U = Vm
iv) U on the earth surface
U = – mgR
v) U at height (h << R)
U = mgh
10) Escape velocity
W
m
i) For point mass :
Ve 
ii) For spherical shell :
2GM
.... from surface
R
2GM
... from height ‘h’ abovee
Rh
11) Orbital velocity
V 'e 
x R
GM
VR 
R
xR
GM
VQ 
R
xr
GM
VP 
r
8)
xR
W R
m
GM
VQ  
R
VQ 
GM
...... r = R + h
r
T  2
r3
GM
GMm
2r
GMm
Potential Enegy 
r
GMm
Total Energy 
2r
Binding Energy = – Total Energy
KE  TE  2 PE  BE
12) Areal Velocity :
dA
L
VA 

dt 2 m
Kinetic Energy 
iii) For solid sphere :
x R
W  WR x
VR   R
m
GM
x2

2 2
2R
R
VC 
xr
W r
r
GM
VP  
r
VP 
Relation between V and E
dv
E
dr
GRAPHS
GRAPHS
i) Acceleration due to gravity Vs. distance from
center
V   E dr
9)
Gravitational potential energy
GMm
i) U  W r  
...... point mass
r
ii) For solid sphere (earth)
Ux  
GMm
x2
3 2
2R
R
ii) E vs. Position Vector
a) Point mass
..... inside earth
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29
Gravitation
b) Spherical Shell
b) Spherical shell
c) Solid Sphere
c) Solid sphere
iii) V vs. position (Distance)
iv)Energy
a) Point mass
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30
Mechanical Properties Of Solids
Quick Revision
MECHANICAL PROPERTIES OF SOLIDS
TERMINOLOGY
VALUES
5)
Rigid body:
F
A
:
:
External force
Area of cross section
l
:
Change in length
L
:
V :
Original length
Change in volume
V
:
Original volume
Y
K
:
:
:
Shear strain
Young’s modulus
Bulk modulus

E
:
:
Shear modulus
Modulus of elasticity

P
C

:
:
:
:
Poisson’s ratio
Breaking stress
Compressibility
Angle of twist

:
Angle of shear
W
U
Work done
Elastic potential energy
Final density
Initial density
Isothermal elasticity
11) Longitudical strain:
Kt
:
:
:
:
:
K
v
:
:
Adiabatic elasticity
Heat capacity
12) Volume strain (Bulk strain):
f
i
DEFINITIONS
DEFINITIONS
1)
Elasticity:
2)
The property of matter by virtue of which body
regains its original size and shape after removal
of deforming force is called as elasticity.
Deforming force:
The force which produces deformation is called
as deforming force.
3)
4)
The body which is having regular shape.
6)
Stress:
Internal restoring force (external force) per unit
area.
7)
Longitudinal stress:
When applied force / deforming force produces
change in length of body is called as
longitudinal force.
8)
Volume stress:
When deforming force produces change in its
volume of body is called as volume stress (also
called as change in pressure).
9)
Shear stress:
When applied force produces change in its
shape only of body is called shear stress.
10) Strain:
The ratio of change in dimension by original
dimension is called strain.
Longitudinal strain or tensile strain is the ratio
of change in length to the original length.
It is the ratio of change in volume to the original
volume.
13) Shear strain:
Ratio of change in shape by original shape.
OR
The applied force produces change in shape
(cube to parallelopiped) the strain is called as
shear strain.
14) Elastic limit:
Deformation:
The change in size, shape or both in a body
The maximum deforming force up to which
arising due to external force called as
body regains its original size and shape is called
deformation.
as elastic limit.
Plasticity:
15) Hooke’s law:
The property of matter to undergo a permanent
It states that within elastic limit stress is directly
deformation after removal of deforming force
proportional to strain.
is called as plasticity.
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31
Mechanical Properties Of Solids
FORMULAE
FORMULAE
16) Modulus of elasticity:
It is a slope of stress-strain graph within elastic
limit.
1)
F
Stress = A
C /S
17) Young’s modulus:
= Longitudinal stress
Within elastic limit, it is the ratio of longitudinal
(tensile) stress to longitudinal strain.
18) Bulk modulus:
2)
Within elastic limit, it is the ratio of volume
stress to the volume strain.
3)
It measures the resistance offered by solid,
liquid and gas to change its volume.
4)
Volume stress =
Shear stress 
F
 P
Total area
Tangential force Ft

A
A
l
L
Longitudinal strain =
19) Shear modulus (Modulus of rigidity):
Within elastic limit, it is the ratio of shear stress
5)
to shear strain.
Volume strain = 
It measures the resistance offered by solids to
change in its shape.
6)
20) Compressibility:
Shear strain  
Stress
Strain
8)
Y
FL
AL
9)
Breaking stress  P 
21) Poisson’s ratio:
22) Lateral strain:
Strain developed in the direction perpendicular
to the applied deforming force.
23) Longitudinal strain:
x
L
E
The reciprocal of bulk modulus of elasticity is
7)
called compressibility.
Within elastic limit, it is the ratio of lateral strain
to longitudinal strain.
V
V
10) K 
Breaking force
C/ S Area
P
V / V
Strain developed in the direction of applied
1
deforming force.
11) C 
K
24) Strain energy:
 i [1  C P]
f
It is defined as elastic potential energy stored 12)
by wire during elongation or compression by
13) Kt = Pressure
deforming force.
14) K = v × Pressure
Area of stress-strain graph gives work done or
elastic potential energy stored in stretched wire
F
FL
per unit volume.
15) Shear modulus,   t or t
A
Ax
Molecules having mininum potential energy
when they are in stable equibrium position, for
r
any other position potential energy increases. 16)   L  Angle of shear
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32
Mechanical Properties Of Solids
GRAPHS
GRAPHS
17) Relation between Y, K and :
1)
Hooke’s law:
9 3 1
 
Y  K
r / r
18)  
L / L
Stress
Elastic limit
Proportional limit
Value 0  0.5
19) Relation between Y, K,  and :
Strain
2)
Stress–Strain Graph:
Y  3K(1  2)
Y  2(1  )
Elastic region
Yield point
1
× Force× Elongation
2
W U 
1
× Stress× Strain× Volume
2
W U 
1
× Stress 2 × Volume
2Y
W U 
1
× Y Strain 2 × Volume
2
21) Elongation due to self weight
L 
Mg L
2 AY
Stress
20) Work done = Change in potential energy
W U 
Breaking strength
B
E
Elastic limit
P
C
Rupture point
Proportional limit
Strain
B &C
Distance
Small then material is brittle
Large then material is ductile
Very very small then elastomer
3)
Elastomer:
 Stress is not directly proportional to strain
Does not obey Hooke’s law.
Stress
22) Breaking stress due to self weight
P
Mg
A
Strain
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33
Mechanical Properties Of Fluids
Quick Revision
MECHANICAL PROPERTIES OF FLUIDS
TERMINOLOGY
VALUES
dv/dt : Velocity Gradient
A
: Area

: Coefficient of Viscosity
: Shearinag stress
Fv
: Viscous force
W
: Weight
FB
: Buoyancy force
Vt
: Terminal velocity
: Density of ball (spherical object)

: Density of fluid
r
: Radius of sphere
g
: Acceleration due to gravity
Re
: Reynold’s number
P
: Pressure
Q
: volumetric flow (Discharge)
L
: Length
Vsub : Submerged volume
H
: Total height of tank
h
: Height of hole
DEFINITIONS
DEFINITIONS
1)
2)
3)
4)
FORMULAE
FORMULAE
1)
Fv  A
2)

2000
Fv
dv

A
dt
3)
1 poise = 0.1 N.s/m2
4)
Fv  6rv
5)
FB  Vsub g
6)
Re 
Vd

7)
Vt 
2r 2 (   ) g
g
8)
Viscosity :
The characteristic of fluid by virtue of which 9)
relative motion between different layers is
opposed is known as viscosity.
10)
Viscosity is internal friction of a fludi in motion.
Critical Velocity :
The maximum velocity up to which fluid
11)
motion is steady is called critical velocity.
Laminar flow :
Flow in which one liquid particle never cross
a path of other liquid particle.
Reynold’s number :
It can be defined as a ratio of inertia force to
viscous force.
Re
dv
dt
= density of liquid
Q = AV
Velocity of efflux V  2 gh
Range of efflux R  2 h( H  h)
Q  A1 A2
2gh
A  A22
2
1
GRAPHS
GRAPHS
... Laminar flow
3000  Re  2000
... transition flow
Re  3000
... Turbulent flow
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34
Surface Tension
Quick Revision
SURFACE TENSION
TERMINOLOGY
VALUES
FORMULAE
FORMULAE
T or  or s : Surface Tension
1)
F
:
Force due to surface tension
l
:
Length of object in contact with liquid 2)
surface (for circular object it is
circumference)
W
:
Work done
A
:
Total change in surface area.
Q
:
Increase or decrease in temperature
J
:
Mechanical equivalent of heat.
V
:
Volume
d or
:
Density
S
:
P
Surface Tension (T) =
F
l
Name of object
i) Wire
ii) Ring
Length
2l
iii) Circular plate
2 r
iv) Hollow disc
2r1  2 r2
v) Square plate
vi) Square frame
4l
8l
W  T A
Name of object
Surface Area
Spacific heat
Droplet
4 r 2
:
Pressure difference
Bubble
2 4 r 2
:
Angle of contact
h
:
Rise in capillary
Air bubble inside liquid 4r 2
Formation of smaller droplet from bigger
t
:
Thickness between two plate.
3)
4)
5)
W  4 T[nr 2  R 2 ]
DEFINITIONS
DEFINITIONS
1)
Surface Tension :
Surface Energy :
The potential energy stored in surface film per
unit surface area is called as surface energy per
unit area.
3)
W  3VT
 
6)
Capillarity :
The phenomena of rise or fall or a liquid inside 7)
a capillary tube when it is dipped in the liquid
is called capillaryity.
5)
... Temperature Decreased
Formation of bigger droplet from smaller
W  4 T [nr 2  R2 ]
... Energy released
Or
 3VT
 
1 1

r R
3T 1 1

JSd r R
... Temperature increased
Excess pressure ( P )
P 
2T
... For droplet and Air bubble inside
r
liquid
P 
4T
... For bubble
r
Sphere of Influence :
An imaginary sphere around a molecule in
which intermolecular force has been
experienced is called as sphere of influence.
1 1

r R
3T 1 1

JSd r R
Angle of Contact :
When liquid is in contact with solid, the angle
between tangent drawn to the free surface of
liquid and the surface of solid at the point of
contact measured inside the liquid.
4)
... Energy absorbed
Or
Surface tension of liquid is measured by the
force acting per unit length on either side of an
imaginary line drawn on the free surface of
liquid.
2)
2 2r
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35
Surface Tension
8)
10) Formation of Double bubble :
1
1
1


R R1 R2
2T cos
h
rg
1
or hr = constant or h1r1  h2 r2
r
h
h
r
(tube) (meniscus radius) (Angle of contact)
R
R1 R2
R2  R1
11) Force required to pull two plate (JEE concept)
2TA
F
t
GRAPHS
GRAPHS
Rise in capillary Vs radius of tube
9)
Formation of single bubble or droplet from two
bubble or droplet in isothermal condition
c  a 2  b2
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36
Thermal Properties Of Matter
Quick Revision
THERMAL PROPERTIES OF MATTER
TERMINOLOGY
VALUES
TC :
Tk :
x0 :
x100 :
xt :
l
:
V :
P :
R :
 :
Temperature in °C
Temperature in °K
Thermometric property at 0°C
Thermometric property at 100°C
Thermometric property at t°C
Length
Volume
Pressure
Resistance
Linear expansion coefficient

:
Coeficient of areal expansion
:
Coeficient of volume expansion
:
T :
Y :
vapp :
W :
J
:
Q :
S
:
 :
C :
M :
L
:
LF :
K :
A :
i
:
R :
E :
U :
a
:
 :
e
:
S
:
Density
Time period
Young’s modulus
Apparent coefficient of volume expansion
workdone
Mechanical equivalent of heat (4.2 J/cal)
Heat
Specific heat capacity
Change in temperature
Molar heat capacity
Molecular weight
Latent heat
Latent heat of fusion
Coefficient of thermal conductivity
Area of cross-section
Heat current
Thermal resistance
Emissive power
Energy radiator
Absorptive power
Stefans constant (5.67 × 10–8 W/m2k4)
Emmissivity of the surface
Solar constant
DEFINITIONS
DEFINITIONS
1)
2)
3)
4)
5)
6)
7)
8)
8)
9)
Heat :
is energy in transit which is transfered from one
body to other due to temperature difference
between them.
Heat Capacity :
The heat required to raise the temperature of
body by 1°C is called heat capacity.
Water equivalent :
Water equivalent of a body is the mass of water
having the same heat capacity as a given body
Latent heat :
The amount of heat required to change the state
of unit mass of a substance at a constant
temperature is called latent heat.
Thermal conductivity is a measure of the ability
of a substance to conduct heat through it.
Black body :
A body which absorb all the radiation falling
on it is caused black body.
Emissive power :
Emissive is the energy radiated per unit area
per unit time per unit solid angle along the
normal to the area.
Absorptive power :
Absorptive power is a fraction of the incident
radiation that is absorbed by the body.
FORMULAE
FORMULAE
tt  t0
100 C
t100  t0
1)
t
2)
C0
F  32
K  273.15


100  0 212  32 373.15  273.15
3)
Liquid thermometer : t 
4)
V  V0
Gas thermometer : t  V  V
Temperature :
It is defined as degree of coldness or hotness
of a body and it is measured by thermometer.
Zeroth law of Thermodynamics :
If two bodies x and y are are in equlibrium and
x and z are in equilibrium then y and z are in 5)
equilibrium.
l  l0
100 C
l100  l0
100
t
100 C
0
P  P0
100 C
P100  P0
Resistance thermometer : t 
R  R0
100 C
R100  R0
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37
Thermal Properties Of Matter
6)
Thermal expansion
14) E 
b) Area expansion A2  A1 [1   ]
15) u   T 4
c) Volume expansion V2  V1 [1   ]
d) pendulum clock time period
16) Rate of cooling 
1
T2  T1 1  
2
e) Density
2

1
Newtons law 
Thermal stress t  
8)
Vapp  V app 
where
app
K
 1
17)
2
Wien’s law
m
4eAT03
ms
T = b = constant
b = 0.288 cm-k
9) W = JQ
10)
 ms ,  nc ,  mL
1)
11) Heat lost = Heat gained
While solving problems, when temperature
change is involved, use  ms or nc ,
when state change is involved, use  mL
KA( 1 


t
L
dT eA 4

(T  T04 )
dt
ms
dT
 K (T  T0 )
dt
[1   ]
7)
12)
U
At 
a) Linear expansion L2  L1[1   ]
GRAPHS
GRAPHS
Heating Curve
)
d
d
  KA
dt
dx
 

t
L
 iR
KA
2)
13) Keq
1
1
1
a) Series K  K  K .........
eq
1
2
b) Parallel Keq  K1  K2  K3 ......
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38
Thermodynamics
Quick Revision
THERMODYNAMICS
TERMINOLOGY
VALUES
n
c
cp
:
:
:
cv
:
R
T
Q
w
u
u

:
:
:
:
:
:
:
Number of moles of gas.
Molar specific heat capacity.
Molar specific heat capacity at constant
pressure.
Molar specific heat capacity at constant
6)
volume.
Universal gas constant.
Absolute temperature of gas.
Heat energy supplied to the gas.
7)
Work done by the gas.
Internal energy of gas.
Change in internal energy of gas.
8)
Efficiency
:

cp 
Adiabatic exponent   c 
v 

P
V
f
:
:
:
Pressure exerted by gas.
Volume occupied by gas.
Degree of freedom.
k
:
F.L.T. :
1)
2)
3)
4)
5)

R 

Boltzmann constant  k 
NA 

First law of thermodynamics.
DEFINITIONS
DEFINITIONS
Heat ( ) :
It is the energy which is transfered from a
system to surroundings (or) vice versa due to
temperature difference between system and
surroundings.
 It is a macroscopic quantity..
 Path dependent.
Work :
Work is the energy that is transmitted from one
system to other by a force moving its points of
application.
 It is a macroscopic quantity..
 Path dependent.
Internal energy :
The total kinetic energy of gas and gas
molecule. u T
Isothermal process :
A thermodynamic process in which the
temperature of the system remains constant
throughout.
Adiabatic process :
If system is completely isolated from
surroundings so that no heat flows ‘in’ or ‘out’,
then any change that the system undergoes is
called an adiabatic process.
Iso-baric process :
A process taking place at constant pressure
throughout.
Isochoric (Isometric) process :
Thermodynamic process in which volume of
the system remains constant throughout.
Cyclic process :
The process in which the initial and final states
of gas after traversing a cycle are same.
9)
Second laws of thermodynamics :
It states that it is impossible for a self acting
machine unaided by any external agency, to
transfer heat from a body at lower temperature
to a body at higher temperature. It is decided
from this law that, the efficiency of any heat
engine can never be 100%.
10) Heat engine : It is a device which converts heat
energy into mechanical energy.
11) Refrigerator :
Refrigerator is a heat engine running in
backward direction i.e., working substance
takes heat from cold body and gives out to
hotter body with the help of external agency.
12) Carnot’s theorem :
It states that no heat engine can have efficiency
greater than carnot’s engine working between
same hot and cold reservair.
13) Reversible process :
A process which can proud in opposite
direction in such a way, that the system passes
through the same states as in the direct process
and finally the system and surroundings
acquire the initial conditions.
14) Irreversible process :
The process which cannot be traced back in the
opposite direction.
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39
Thermodynamics
FORMULAE
FORMULAE
1)
6)
First law of thermodynamics
 u  
 nc T  heat energy supplied to gas.
u  nGT  change in internal energy of gas.

1
2
dV  work doen by the gas.
Isothemal process :
a) condition T = constant throughout
b) State equation pv = constant
supplied
7)
( P1V1  P2 V2 )
c) F.L.T.  u  ncv T  0 ; ( T  0)
8)

3)
Iso-baric process :
a) Condition p = constant throughout
b) State equation
4)
c) F.L.T.   ut
d) Work done by the gas   P( V )  R( T )
Iso-choric process : (Isometric process)
a) Condition : v = cconstant throughout
b) State equation
5)
V V 
V
= constant.  1  2 
T
 T1 T2 
P P 
P
 constant   1  2 
T
 T1 T2 
c) F.L.T.   u;(   0)
d) Work done by the gas    0 ;
( v = constant)
Adiabatic process :
a) Condition   0
b) State equation  PV = constant
TV 1 = constant
P 1 T = constant
c) F.L.T.  u    0 ; (
 0)
d) work done by the gas
PV PV
R(T1  T2 )
 1 1 2 2 
1
1
Molar specific heat capacity for a polytropic
process with state equation PVh = constant is
R
given by C  CV 
1 n
Area covered by P-V graph with volume axis
gives work done by the gas.
P
u 
d) Work done by the gas   RT ln  2 
 u1 
P 
 RT ln  1 
 P2 

e) cycle 
v1
2)
Cyclic process :
a) Initial and final states are same.
b) u  0
c) Work done by gas = Area inside cycle.
 = +ve for clockwise cycle and –ve for anticlockwise cycle.
d) net   ; ( net  supplied  released )
Work
V
9)
Bulk modulus of gas Bisothermal = P
Badiabatic = P
10) Efficiency of heat engine :
c/p

efficiency 

i/p
1


 1
1
2
1
for carnot engine ' ' is
maximum is given by
T
max  1  2  1  2
T1
1
11) Refrigerator
Co-efficient of performance

1


1

2

1
(Hot Reservoir)
Source
T1
T2  T1
(Cold Reservoir)
Sink
Engine
T1
T2
Q1
Q2
W = Q1 – Q2
12) Relation between "  " and "  "
1 


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40
Thermodynamics
GRAPHS
GRAPHS
1)
5)
Iso-thermal process :
What does the slope of P-V graph gives ..?
a) for isothermal process graph
dP
P

dV
V
b) for adiabatic graph
slope 
slope 
2)
dP
P
  
dV
V 
6)
Iso-choric process :
7)
Iso-baric process :
Adiabatic process :
3)
Iso-thermal and Adiabatic comparison :
4)
Comparison of mono, dia and polyatomic gases
for adiabatic process
8)
Carnot’s cycle :
T1  T2  TH (temperature of hot reservior)
T3  T4  TC (temperature of cold reservior)
cycle  1 
more
 more slope
1

1  monoatomic  
5
3
2  diatomic gas  
7
5
3  polyatomic gas  
2
4
3

TC
TH
3
1  2  isothermal expansion
2  3  adiabatic expansion
3  4  isothermal compression
4  1  adiabatic compression
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41
Kinetic Theory Of Gases
Quick Revision
KINETIC THEORY OF GASES
TERMINOLOGY
VALUES
P
:
Pressure exerted by the gas
V
n
:
:
Volume occupied by the gas
Number of moles of the gas
R
:
Universal gas constant
T
:
Absolute temperature of gas
M
m
:
:
Molar mass of the gas
Mass of gas sample
µ
K
:
:
Molecular mass of the gas
Boltzmann constant
C
:
Specific heat capacity
Cp
:
Specific heat capacity at constant pressure
Cv
U
N
NA
:
:
:
:
:
Specific heat capacity at constant volume
Density of gas
Number of molecules
Avagadro’s number
Adiabatic exponent (Cp / Cv)
:
:
:
:
Degree of freedom
Speed of gas molecules
Heat energy
Internal energy of gas
f
v
Q
U
6)
7)
8)
2)
3)
9)
Gas:
10)
Type of matter that does not have any fixed
shape or volume.
11)
Ideal gas:
Gas in which, size of molecule and force of
interaction between molecules is considered
12)
zero.
Real gas:
The gas that shows deviation from ideal gas
13)
behaviour is called a real gas.
4)
5)
V
 Constant
T
Dalton’s law of partial pressure:
Partial pressure of a gas is the pressure which
it would exert if contained alone in the given
confined space.
P = P1+P2+P3
P = Total pressure of mixture of gases
P1+P2+P3 = Partial pressure of individual gases
in mixture
Graham’s law of diffusion:
Graham’s law of diffusion states that, rate of
diffusion of gas varies inversely as the square
root of density of gas.
r
DEFINITIONS
DEFINITIONS
1)
Charle’s law:
It states that, volume of given mass of gas
varies directly proportional to its absolute
temperature, given its pressure is constant.
Avagadro’s number (NA):
It is the number of carbon atoms contained in
12 gms of C-12 carbon.
14)
23
NA = 6.023 × 10
Boyle’s law:
It states that the volume of a given amount of
15)
gas varies inversely as its pressure, provided its
temperature is kept constant.
PV = Constant
1
Avagadro’s law:
It states that under similar conditions of
pressure and temperature equal volumes of all
gases contain equal number of molecules.
PV = nRT
Root mean square speed (vrms):
It is the square root of the mean of squares of
individual speeds of the molecules of gas.
Average speed:
It is the arithmetic mean of speed of the
molecules of a gas.
Most probable speed:
It is the speed possessed by maximum number
of molecules of a gas sample.
Degree of freedom:
Number of possible independent ways in which
the position and configuration of the system
may change.
Law of equipartition of energy:
In a gas sample, in thermal equilibrium, the
total internal energy of the gas is divided
equally among all the degree of freedom.
Gram specific heat capacity (c):
Amount of heat energy required by unit mass
of gas to rise its temperaure by 1°C (or) 1 K.
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42
Kinetic Theory Of Gases
16) Molar specific heat capacity (c):
6)
Amount of heat energy required by 1 mole of
gas in order to rise its temperature by 1°C or
1 K.
17) Heat capacity:
7)
Heat energy consumed by gas sample to rise is
temperature by 1°C or 1 K.
18) Adiabatic exponent ():
It is the ratio of Cp to Cv of a gas.
8)
Cp

Cv
FORMULAE
FORMULAE
1)
Ideal gas equation:
a) PV = nRT
RT
b) P 
M
3)
4)
3 RT
3P
3KT


M
µ
Average speed,
8 RT
8P
8 KT


M

µ
Most probable speed,
2 RT
2P
2KT


M
µ
In a given gas sample vrms  vavg  vmp
vrms : vavg : vmp  3 :
Energy per mole per degree of freedom is
1
RT
2
Internal energy (U):
f
KT
2
f
nRT
2
8
: 2

f
nRT  nC v T
2
11) For a given gas,
a) C v  f
R
2
b) Cp – Cv = R
(If Cp & Cv are molar specific heat capacities)
c) C p  Cv 
R
M
(If Cp & Cv are gram specific heat capacities)
Cp
d) C 
v
e)
vavg 
5)
1
KT
2
U 
where a, b are Vander waal’s constants
Pressure exerted by a gas
2
P e
3
where e is translational KE per unit volume of
gas.
1 2
v
and P 
;
= Density of gas
3 rms
R.M.S. speed,
vmp 
Energy per molecule per degree of freedom is
10) Change in internal energy of a gas sample is,
n2 a
( V  nb)  nRT
V2
vrms 
3
nRT
2
For n moles of gas 
d) PV = NKT
Real gas equation:
P
K .E.T 
For one molecule 
KT
c) P 
µ
2)
9)
Translational K.E. of a gas,
 1
f) Cv 
2
f
R
R
; Cp 
1
1
12) Degree of freedom (f) (Excluding vibrational
energies):
i) Monoatomic = 3, (3 Translational)
ii) Diatomic (or) polylinear = 5, (3 Translational
+ 2 Rotational)
iii)Poly non-linear = 6, (3 Translational
+ 3 Rotational)
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43
Kinetic Theory Of Gases
GRAPHS
GRAPHS
13) For a mixture of gases:
C p (mix.) 
Cv (mix.) 
mix

n1C p1  n2C p2  .....
Boyle’s law:
n1  n2  .....
n1Cv1  n2Cv2  .....
P
n1  n2  .....
1
; T = constant
V
C p(mix)
Cv (mix)
2)
Absolute temperature of mixture,
Tmix. 
1)
f1n1T1  f2 n2T2  .....
f1n1  f2 n2  .....
Charle’s law:
V
P = constant
14) Heat energy supplied to a gas
Q  mC ab T
Cab
or
Q  nC molar T
T
= gram sp. heat capacity
Cmolar = molar sp. heat capacity
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44
Simple Harmonic Motion
Quick Revision
SIMPLE HARMONIC MOTION
TERMINOLOGY
VALUES
T

x
4)
:
:
Time period [s]
Angular frequency [rad’s]
:
Displacement of particle from mean 5)
position [m]
Velocity [m/s]
6)
Acceleration [m/s2]
Spring constant [N/m]
7)
Mass of block [kg]
V
:
a
k
:
:
m
:
Ms :
Mass of spring [kg]
F
A
Force [N]
Amplitude [m]
:
:
Amplitude :
The maximum value of displacement from
equilibrium position is called as amplitude.
Phase :
The state of particle with respect to its position
and direction of motion is called as phase.
Initial phase :
The initial state of particle is said to be initial
phase.
Free oscillation :
When a system is displaced from its
equilibrium position and released, it oscillates
with the natural frequency and the oscillations
are called as free oscillation.
Forced or driven oscillation :
If an external agency maintain the oscillations
then it is called as forced or driven oscillation.
Resonance :
The phenomenon of increase in amplitude
when driving force is close to natural
frequenccy of oscillator is called resonance.
 :
K.E. :
P.E. :
T.E. :
R :
s
:
Fd :
b
:
F(t) :
F0 :
Initial phase or epech
8)
Kinetic energy [ J ]
Potential energy [ J ]
9)
d :
Forced frequency
0 :
Natural frequency

k
m
I
C
Moment of inertia
Torsinal constant
T
2
m
 2

k
:
:
Total energy [ J ]
Resultant amplitude [m]
Resultant initial phase angle
Damped force [N]
Damping constant
External force [N]
Amplitude of external force [N]
DEFINITIONS
DEFINITIONS
1)
Periodic motion :
A motion that repeat itself at regular interval
of time is called periodic motion.
2)
3)
Time period :
The smallest time interval after which the
motion repeats itslef is called the time period.
Simple harmonic motion :
Incase of motion of particle moves back and
forth about fixed point throught a force which
is directly proportional to displacement but
opposite in direction the motion is called as
simple harmonic motion.
FORMULAE
FORMULAE
1)
Linear simple harmonic motion
F = –kx
a
F  kx

 2 x
m
m
F = –kx
ma = –kx
a
k
x
m
d2 x
 2 x
2
dt
d2 x
 2 x  0
dt 2
V   A2  x2
x  A sin(t  )
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45
Simple Harmonic Motion
2)
Equation of SHM
i) Particle starting from extreme right
6)
x  A cos(t )
T  2
Angle measured from positive x axis.
ii) Particle starting from mean position
3)
T  2
1
1
m2 ( A2  x 2 )  K( A2  x 2 )
2
2
T  2
T 
1 1
t   0
T 
v) Pendulum accelerating horizontally
1
1
1
P.E.  m2 x 2  kx2  KA2 sin2 t
2
2
2
1
KA 2
2
1
KA2
4
Comparison of two SHM
x1  A1 sin( t  1 )
7)
8)
Second pendulum
T=2S
g
l 2

Pendulum of large length
T  2
x  x1  x2
x  R sin( t  )
5)
Simple pendulum
T  2
l
g
1
1 1
g 
l R
if l  
x  A1 sin( t  1 )  x  A2 sin( t   2 )
A sin 1  A2 sin  2
tan   1
A1 cos 1  A2 cos  2
1
( g2  a2 ) 2
x2  A2 sin( t   2 )
R  A12  A22  2 A1 A2 cos( 1   2 )
l
T  2
P.E.average 
4)
l
ga
iv) If lift is under free fall
1
 KA2 cos2 t
2
1
K.E.average  KA 2
4
l
ga
iii) If lift is moving downward with an
acceleration [a]
1
K.E.  K( A 2  A2 sin 2 t )
2
T .E.  K .E.  P.E. 
l
g
ii) If lift is moving upward with an acceleration
[a]
x  A sin t
Angle measure from negative y axis.
Energy in SHM
If x  A sin t
K.E. 
Pendulum in lift
i) If lift is at rest
T  2
9)
R
 84.6 min
g
Compound pendulum
T  2
I
mgl
10) Torsional pendulum
T  2
I
C
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Simple Harmonic Motion
11) Motin of a ball in a tunnel through earth
16) Damped simple harmonic motion
F  Fd  Fs
T  2
R
 84.6 min
g
12) Oscillating of a floating body in a liquid
T  2
h
g
s
M
d2 x
dx
 b  kx
2
dt
dt
M
d2 x
dx
 b  kx  0
2
dt
dt
l
13) Oscillating of a liquid column in a V-tube.
T  2
M a  bV  kx
h
g

' 
14) Simple pendulum in a liquid
T  2
l
s
g( s
T
l
)
15) Spring System
F   kx

k
m
bt
x  Ae 2 m cos(  ' t  )
k  b 


m  2m 
2
2
k  b 

m  2 m 
2
17) Forced oscillation and resonance
F (t )  F0 cos( t )
m
k
Spring in sereis
T  2
1
1
1


K eff K1 K 2
m
d2 x
dx
b
 kx  F0 cos t
2
dt
dt
x(t )  A cos( d t  )
F0
A
1
[m2 ( 2  2d )2  2d b2 ]2
tan 
Spring in a parallel
Keff  K1  K 2
If mass of spring [Ms] considered then
Ms
3
T  2
K
Reduced mass
 v0
d x0
V0 = Initial velocity
i) Small damping driving frequency far from
natural frequency
m
Mr
K
M1 M 2
Mr 
M1  M 2
T  2
A
F0
M(2  2d )
ii) Driving frequency closed to natural
frequency
A
F0
d b
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Simple Harmonic Motion
GRAPHS
GRAPHS
1)
F Vs x
3)
x  A cos t
V  A sin t
a  2 A cos t
2)
x  A sin t
V  A cos t
a  2 A sin t
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Waves
WAVES
Quick Revision
TERMINOLOGY
VALUES
A
v
K
T
Y
B
P
R
T
M
I


:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
Amplitude
Wavelength
Wave velocity
Wave number
Tension in string
Mass per unit length
Density of medium
Young’s modulus of elasticity
Bulk modulus of elasticity
Pressure
Universal gas constant
Temperature
Molar mass
Intensity
Angular
Modulus of rigidity
TYPES OF WAVES
DEFINITIONS
1)
2)
3)
4)
5)
6)
7)
FORMULAE
FORMULAE
1)
Wave velocity (v) :
2)

T k
Intensity of wave (I) :
v f 

I  2 2 f 2 A 2 v
3)
Energy density :
2 2 f 2 a 2 v
V
Velocity of transverse wave :
Energy density 
4)
5)
On The Basis Of Medium:
i) Mechanical Waves : Required medium for
their propagation ex.: Waves on string, and
spring etc.
ii) Non-mechanical Waves : Do not require
medium for their propagation ex.: Light,
radio waves, X-rays etc.
On The Basis Of Vibration Of Particle:
i) Transverse waves : Particle of medium
vibrates in a direction perpendicular to the
direction of propagation of waves ex.:
movement of string of sitar
ii) Longitudinal waves : Particles of medium
vibrate in the direction of wave motion. ex.: 6)
sound wave travel through air.
On The Basis Of Energy Propagation:
i) Progressive wave : These waves propagates
7)
energy in medium. Ex. : Sound wave
ii) Stationary wave : Energy is not propagated
by these waves. ex.: waves in a string, waves
in organ pipes.
Amplitude : Maximum displacement from
mean position.
Wavelength : It is equal to the distance travelled
by the wave during the time in which any 8)
particle of medium completes one vibration.
Angular wave number : Number of wavelengths
in the distance 2 .
Wave velocity (v) : It is the distance travelled
by the disturbance in one time period.
v
v
T
(in stretched string)

(in solid)
Velocity of sound wave :
v
Elasticity of medium
Density of medium
v
E
v
Y
v
B
(in solids)
(in liquid and gaslong medium)
Newton’s formula :
v
P
Laplace correction :
v
YP
Y = 1.41 for air
YRT
M
Equation of plane progressive wave :
v
y( x1t )  A sin( t kx
where
) (general equation)
= Initial phase
 = Wave number
A = amplitude
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Waves
Various Form
Destructive interference:
i) y  A sin( t  kx) ,
Let
Phase difference,
0
 180 or (2n  1)
 = 1, 2, 3, .........

2 
x
ii) y  A sin  t 


Path difference  (2 n  1)
 2
2 
x
iii) y  A sin  t 
 T

I min  I1  I 2  2 I1 I 2 
2
(odd multiple of
2
)
Amin  A1  A2
I1  I 2
2
11) Stationary wave :
equation, y  2 A sin kx cos t
 t x
y  A sin 2   
T

9)
Amplitude of wave
Node : The points where amplitude is
minimum
Particle velocity
VP = –v × slope of wave at that point.
Distance between two successive nodes is
10) Interference of sound waves : When two waves
of same frequency, same velocity moves in same
direction.
2
Antinode : The points of maximum amplitude.
Distance between two successive antinodes
is
y1  A1 sin(t )
2
.
y2  A2 sin( t  )
after superposition.
12) Stenting wave can string
frequency of vibration = frequency of wave
Anet  A12  A22  2 A1 A2 cos

Intensity ( I )  I 1  I 2  2 I1 I 2 cos
0

1 T
13) i) Fundamental frequency or first harmonic
Constructive interference:
Phase difference
v
n1 
or 2n
1 T

1 T
2l
ii) Second harmonic or first overtone :
Path difference  
(even multiple of
2
)
h2 
1 T
h2 
1 T
 2n2
l
Resultant amplitude,
Amax  A1  A2
iii) Third harmonic or second over tone.
Resultant intensity
I max  I 1  I 2  2 I 1 I 2

I1  I 2
2
h3 
1 T
h3 
3 T
 3n1
2l
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Waves
14) Standing wave in organ pipe :
Closed organ pipe {Vs = Velocity of sound}
Resonance frequency
f1 
f2 
Open organ pipe
f1 
Vs
[fundamental frequency or 1st harmonic]
2l
f2 
2V s
[2nd harmonic or 1st overtone]
2l
Vs
[fundamental frequency 1st harmonic]
4l
3Vs
[3rd harmonic or 1st over tone]
4l
15) Beat frequency
Beat frequency = No. of beats per second
= Difference in frequency of
two source
5V
f3  s [5th harmonic or 2nd overtone]
4l
= n1  n2
Doppler effect :
apparent frequency
 v v0 
fapp  
 factual
 v vsource 
Note : choose plus / minus signs based on
situation
v = velocity of sound.
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