Physics - Properties of Matter - Basics

Properties of Matter
Ms Dhivya R
Assistant Professor
Department of Physics
Sri Ramakrishna College of Arts and Science
Coimbatore - 641 006
Tamil Nadu, India
1

Overview
1. Elasticity
2. Stress
3. Strain
4. Stress – Strain Diagram
5. Different Moduli of Elasticity
1. Young’s
2. Bulk
3. Rigidity
6. Poisson’s Ratio
2

Elasticity
What do you mean by Elasticity?
External force applied - Deformation
- Change in shape or size
External force removed - Regains original shape
and size
- Perfectly Elastic
Property - Elasticity
3

Elasticity
External force applied - Deformation
- Change in shape or size
External force removed – does not regain
original shape and size
- Perfectly Plastic
Perfectly elastic/Perfectly Plastic – Ideal
Quartz Fibre – Nearly Perfect Elastic
4

Stress
Restoring Force per unit area
Stress = F/A (N/m2)
Dimension - ML-1T-2
Tensile Stress - Increase in length
Compressive Stress - Decrease in length
Tangential Stress - Slide the layer of the body
5

Strain
External force – Change in length, Volume or
shape
Types:
1. Longitudinal strain
2. Shearing strain
3. Volume strain
6
ension
original
ension
in
change
strain
dim
dim


Hooke’s Law
Within Elastic limits,
Where E = Constant – Modulus of Elasticity
7
E
strain
stress
strain
stress



Stress – Strain Diagram
8

Different Moduli of Elasticity
Young’s Modulus:
Rigidity Modulus:
Bulk Modulus:
9
Strain
al
Longitudin
Stress
al
Longitudin
Strain
Shearing
Stress
Shearing
Strain
Volume
Stress
Volume

Young’s Modulus (E)
Young’s Modulus:
10
Al
FL
L
l
A
F
E 


Rigidity Modulus (G)
Rigidity Modulus:
11

A
F
Strain
Shearing
Stress
Tangential
G 


Bulk Modulus (K)
Bulk Modulus:
12
V
v
A
F
Strain
Volume
Stress
Bulk
G




Poisson’s Ratio (ℽ)
Wire Stretched – Becomes thin
Length increases – Diameter decreases
13


 

Elongation
al
Longitudin
n
Contractio
Lateral

Summary
14

Reference Books
Elements of Properties of Matter
- DS Mathur
Properties of matter
- N. Subramanyam &
Brij Lal
15

Ms Dhivya R
Assistant Professor
Tamil Nadu, India
16

Recall
1. Elasticity
2. Stress
3. Strain
4. Stress – Strain Diagram
5. Different Moduli of Elasticity
1. Young’s
2. Bulk
3. Rigidity
6. Poisson’s Ratio
17

Overview
1. Relation between different Strains
1. Relation between angle of shear and linear strain
2. Relation between volume strain and linear strain
2. Work done in a strain
1. Linear strain
2. Shearing strain
3. Volume strain
18

Relation between angle of shear
and linear strain
19

Relation between angle of shear
and linear strain
20
DB
KB
DB
DK
DB
DB
DB
DB
e
DB
diagonal
the
along
strain
Tensilet
'
'
'






2
2
2
'
2
2
'
'
2
/
'
45
cos
'
'
45
cos
'
/
'
,
'











AC
diagonal
along
strain
e
compressiv
L
BB
L
BB
DB
KB
e
BB
BB
KB
or
BB
KB
BKB
in
but



Relation between Volume strain
and linear strain
Cube – Subjected to three equal stresses –
tending to expand.
Side becomes (1+e)
New volume = (1+e)3 which is approximately
equal to 1+3e
So increase in volume = 3e.
Volume strain =
21
e
e
volume
original
volume
in
increase
3
1
3



Work done in a strain
Body strained – work has to be done to deform
Work done is stored as potential energy
Can be shown that
22
)]
(
)
(
2
1
[ strain
x
stress
x
volume
unit
per
done
work 

Work done in a Linear strain
23







l
dl
F
W
l
to
form
wire
the
of
stretching
a
produce
to
done
work
total
dl
L
EAl
dl
F
dl
stretch
a
producing
in
done
work
L
EAl
F
or
Al
FL
E
ulus
s
Young
0
.
0
.
.
mod
'

Work done in a Linear strain
24
)
(
)
(
2
1
.
2
1
.
2
1
.
,
2
1
.
2
1
2
1
2
.
0
2
strain
stress
L
l
A
F
AL
l
F
volume
unit
per
done
work
L
A
wire
the
of
volume
now
produced
Elongation
force
stretching
l
F
l
L
EAl
l
L
EA
dl
L
EAl
l


















 

Work done in a Shearing strain
25

Work done in a Volume strain
26

Summary
27

Reference Books
- DS Mathur
- N. Subramanyam &
Brij Lal
28

Ms Dhivya R
Assistant Professor
Tamil Nadu, India
29

Recall
1. Relation between different Strains
1. Relation between angle of shear and linear strain
2. Relation between volume strain and linear strain
2. Work done in a strain
1. Linear strain
2. Shearing strain
3. Volume strain
30

Overview
1. Torsion
2. Torsion of a cylinder – Expression for torque
per unit twist
3. Work done in twisting a wire
31

Torsion
The action of twisting or the state of being twisted,
especially of one end of an object relative to the
other.
When a body is fixed at one end and twisted about
its axis by means of a torque at the other end, the
body is said to be under torsion.
Torsion involves shearing strain and the modulus
involved is the rigidity modulus.
32

Torsion of a cylinder
33

34
L
x
or
L
x
BB
now
BAB
shear
of
angle
The





.
.
'
'






acts
force
the
which
on
area
stress
shearing
force
shearing
acts
force
the
which
on
area
force
shearing
stress
shearing
but
Gx
G
stress
shearing
shear
of
angle
stress
shearing









,
L
.
)
(
G
modulus
rigidity
have,
We




35
xdx
L
Gx
F
force
shearing
the
hence
xdx
acts
force
shearing
the
which
over
area
the



2
,
2








36

 L
Ga
radian
when
torque
the
e
i
twist
unit
per
torque
the
L
Ga
C
or
a
dx
x
L
G
C
cylinnder
whole
the
on
torque
twisting
dx
x
L
G
x
xdx
L
Gx
cylinder
the
of
OO
axis
the
about
force
this
of
moment
The
2
4
1
.,
.
(
2
4
0
3
2
3
2
.
2
'

























Work done in twisting a wire
37

2
.
2
1
0
.








c
W
d
c
W
angle
an
through
wire
the
twisting
in
done
work
total
The
c
C
twist
a
produce
to
required
torque
c
twist
angular
per
torque
angle
an
through
twisted
a
radius
L
length
wire
l
Cylindrica











Reference Books
- DS Mathur
- N. Subramanyam &
Brij Lal
38

Ms Dhivya R
Assistant Professor
Tamil Nadu, India
39

Recall
1. Torsion
2. Torsion of a cylinder – expression for torque
per unit twist
3. Work done in twisting a wire
40

Overview
1. Determination of rigidity modulus
1. Static Torsion method
2. Dynamic Torsion method
41

Determination of rigidity modulus
– Static Torsion method
42

43






4
2
4
4
360
180
.
2
,
180
.
2
Re
a
MgRLG
G
or
L
Ga
MgR
m
equilibriu
for
L
Ga
que
storingTor




44

Static Torsion method (Scale and
Telescope)
45
s
a
mgRLD
G 4
4



Static Torsion method (Scale and
Telescope)
46

Dynamic Torsion method
• Torsional Oscillations
of a body
• Moment of inertia = I
• Length = l
• Radius = a
• Rigidity modulus = G
• Torsional oscillation,
torsional pendulum
• Angle of twist = theta
• Angular velocity=omega
47

Torsional Oscillations of a body
48
t
cons
c
dt
d
I
system
the
of
energy
total
dt
d
I
I
energy
Kinetic
c
energy
potential
tan
.
2
1
2
1
2
1
2
1
.
2
1
2
2
2
2
2























0
0
.
.
0
2
.
2
1
.
2
.
2
1
,
2
2
2
2
2
2














I
c
dt
d
or
c
dt
d
I
or
dt
d
c
dt
d
dt
d
I
t
to
respect
with
ating
Differenti
c
I
T
by
given
is
period
its
motion
harmonic
simple
has
body
the

2


Torsional Pendulum
49
1
2
2
1
1
1
4
2
I
c
T
c
I
T





Torsional Pendulum
50
 
 
)
(
)
(
16
2
)
(
2
4
2
/
)
(
2
4
2
2
4
min
det
tan
2
2
4
2
2
,
2
1
2
2
4
2
1
2
2
2
4
2
1
2
2
2
2
1
2
2
4
2
1
2
2
2
2
1
2
2
2
2
0
2
2
2
2
2
2
1
0
2
2
1
2
1
0
1
T
T
a
d
d
Lm
G
or
Ga
L
d
d
m
T
T
hence
L
Ga
c
but
d
d
m
c
T
T
md
i
I
c
T
ed
er
is
T
period
time
ing
correspond
the
and
disc
the
of
center
the
from
d
ces
dis
equal
at
kept
are
masses
two
if
md
i
I
c
T
md
i
I
I
theorem
axes
parallel
By





























Torsional Pendulum
51
)
(
)
(
2
)
(
4
)
(
2
4
)
(
2
4
,
4
4
2
1
2
2
2
0
2
1
2
2
0
2
1
2
2
2
2
0
2
1
2
2
2
0
2
1
2
2
2
1
2
2
2
2
2
0
0
0
2
2
0
T
T
T
d
d
m
I
T
T
T
d
d
m
I
Hence
T
T
d
d
m
c
steps
previous
the
from
cT
I
or
I
c
T

















Summary
52

Reference Books
- DS Mathur
- N. Subramanyam &
Brij Lal
53

Ms Dhivya R
Assistant Professor
Tamil Nadu, India
54

Recall
1. Determination of rigidity modulus
1. Static Torsion method
2. Dynamic Torsion method
55

Overview
1. Bending of beams
2. Expression for bending moment
56

Bending of beams
Beam: A beam is a defined as a rod or a bar of
uniform cross-section(circular or rectangular)
whose length is very much greater than its
thickness.
Bending Couple:
57
moment
bending
ernal
moment
bending
External int


Bending of beams
Plane of bending:
The plane in which bending takes place and the
bending couple acts in this plane
Neutral Axis:
Filament ab – Elongated
Filament cd – Compressed
Filament ef – neither
elongated nor compressed
58

Expression for bending moment
• ef – neutral axis
• R – radius of curvature
of the neutral axis
• Ɵ – angle subtended at
its centre of curvature C
• Filaments above ef –
elongated
• Filaments below ef –
compressed
59

• a’b’ – filament at a
distance z from ef
• Length of a’b’ before
bending is equal to the
corresponding filament
ab on the neutral axis
60





z
R
z
R
ab
b
a
length
in
increase
z
R
b
a
length
elongated
R
ab
length
Original











)
(
'
'
)
(
'
'

61
)
(
mod
'
R
z
E
strain
Linear
E
stress
strain
linear
stress
E
ulus
s
young




R
z
R
z
length
original
length
in
increase
strain
Linear 




2
.
.
.
z
A
R
E
z
A
R
z
E
ef
axis
neutral
the
about
force
this
of
Moment

 






A
R
z
E
area
stress
A
area
the
on
force
Tensile 

.



,
sec filament
the
of
tion
cross
of
area
the
is
A
if 

62







2
2
.
.
z
A
R
E
z
A
R
E
filaments
the
all
on
acting
forces
of
moments
the
of
sum
The















2
.
sec
z
A
bending
of
plane
the
to
lar
perpendicu
centre
its
through
axis
an
about
beam
the
of
tion
cross
the
of
inertia
of
moment
l
geometrica


63







2
2
.
.
z
A
R
E
z
A
R
E
filaments
the
all
on
acting
forces
of
moments
the
of
sum
The


R
EAk
moment
bending
gyration
of
radius
K
and
tion
cross
of
Area
A
Ak
z
A
2
2
2
sec
.






• Notes:
– For rectangular beam of breadth b, and depth
(thickness)d, A=bd and ,
– For a beam of circular cross section of radius r,
A=∏r2 and ,
– EAk2 is called the flexural rigidity of the beam
64
12
2
2 d
k 
12
3
2 bd
Ak 
4
2
2 r
k 
4
4
2 r
Ak



Summary
65

Reference Books
- DS Mathur
- N. Subramanyam &
Brij Lal
66

Ms Dhivya R
Assistant Professor
Tamil Nadu, India
67

Recall
1. Bending of beams
68

Overview
1. Cantilever
3. Measurement of Young’s modulus E
4. Uniform Bending
5. Non Uniform Bending
69

Cantilever
Cantilever:
A beam fixed horizontally at one and left free at
the other end is called a cantilever.
70

Cantilever
Depression for the loaded
end of a Cantilever:
• External BM = Wx
• Internal BM = EAK2/R
• At equi Wx= EAK2/R
• R=EAK2/Wx ----- (1)
• Angle PCQ = dѲ
• PQ = dx = RdѲ
• dѲ = dx/R= Wx dx/EAk2
(From 1)
71

Cantilever
Depression for the
loaded end of a
Cantilever:
• we have
• dy = xdѲ
• dy = x .Wx dx /EAk2 =
Wx2dx/EAk2 ------- (2)
72

Cantilever
Depression for the
loaded end of a
Cantilever:
73
 

l
EAk
Wl
dx
EAk
Wx
y
0
2
3
2
2
3

Cantilever
Measurement of E in a
Cantilever:
74
y
bd
Mgl
E
Ebd
Mgl
y
bd
Ak
Mg
W
EAk
Wl
y
3
3
3
3
3
2
2
3
4
4
12
;
;
3








Bending
• Uniform Bending
75
• Non - Uniform Bending

Bending
76
Depression at D below A
2
3
2
3
48
3
)
2
)(
2
(
EAk
Wl
EAk
l
W
y 


Bending
• Uniform Bending
77
• Uniform Bending
• External bending
moment with respect to
P
Wa
AC
W
AP
CP
W
AP
W
CP
W






.
)
(
.
.

Bending
• Uniform Bending
78
• Uniform Bending
• Int BM = Ext BM
R
EAk
Wa
2

R
l
y
le
isnegligib
y
l
R
y
l
y
R
y
AF
EF
R
EF
8
)
(
4
2
.
)
2
(
)
2
(
)
2
(
2
2
2
2
2







Bending
• Uniform Bending
79
• Uniform Bending
• WKT
y
Ak
Wal
E
EAk
Wal
y
EAk
Wa
R
2
2
2
2
2
8
&
8
1





Konig’s Method
80
2
2
3
2
16
2
)
2
(
3
EAk
Wl
s
bd
L
D
Mgl
E





Konig’s Method
81
2
2
2
2
16
2
)
2
(
16
2
)
2
(
4
2
EAk
Wl
D
L
s
EAk
Wl
D
L
D
L
KH
EG
KH
CK
s



















Konig’s Method
82
s
bd
D
L
Mgl
E
s
bd
D
L
Mgl
E
Mg
W
bd
Ak
s
Ak
D
L
Wl
E
3
2
3
2
3
2
2
2
2
)
2
(
3
)
12
(
8
)
2
(
;
12
8
)
2
(









Summary
83

Reference Books
- DS Mathur
- N. Subramanyam &
Brij Lal
84

Physics - Properties of Matter - Basics

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