JEE Main Numericals: Mechanical Properties of Solids | Physics for JEE Main & Advanced PDF Download

 Page 1


1 . Two wires A and B are of the same material. Their lengths
are in the ratio 1 : 2 and the diameter are in the ratio 2 : 1. If
they are pulled by the same force, then increase in length
will be in the ratio
(a) 2 : 1 (b) 1 : 4 (c) 1 : 8 (d) 8 : 1
2 . For a constant hydraulic stress on an object, the fractional
change in the object volume 
V
V
Dæö
ç÷
èø
 and its bulk modulus
(B) are related as
( a)
V
B
V
D
µ (b)
1 V
VB
D
µ
(c)
2
V
B
V
D
µ (d)
2
V
B
V
-
D
µ
3 . The load versus elongation graphs for four wires of same
length and made of the same material are shown in the figure.
The thinnest wire is represented by the line
( a) OA
A
B
C
D
Load
Elongation
O
(b) O C
(c) OD
(d) OB
4 . A metal wire of length L
1
 and area of cross-section A is
attached to a rigid support. Another metal wire of length L
2
and of the same cross-sectional area is attached to the free
end of the first wire. A body of mass M is then suspended
from the free end of the second wire. If Y
1
 and Y
2
 are the
Young’s moduli of the wires respectively , the effective force
constant of the system of two wires is
( a)
12
1221
()
2()+
YYA
YLYL
(b)
12
1/2
12
()
()
YYA
LL
(c)
12
1221
()
+
YYA
YLYL
(d)
1/2
12
1/2
21
()
()
YYA
LL
l a c i n a h c e M 	 s e i t r e p o r P 	 f o 	 s d i l o S
J E E 	 n i a M 	 s l a c i r e m u N
Page 2


1 . Two wires A and B are of the same material. Their lengths
are in the ratio 1 : 2 and the diameter are in the ratio 2 : 1. If
they are pulled by the same force, then increase in length
will be in the ratio
(a) 2 : 1 (b) 1 : 4 (c) 1 : 8 (d) 8 : 1
2 . For a constant hydraulic stress on an object, the fractional
change in the object volume 
V
V
Dæö
ç÷
èø
 and its bulk modulus
(B) are related as
( a)
V
B
V
D
µ (b)
1 V
VB
D
µ
(c)
2
V
B
V
D
µ (d)
2
V
B
V
-
D
µ
3 . The load versus elongation graphs for four wires of same
length and made of the same material are shown in the figure.
The thinnest wire is represented by the line
( a) OA
A
B
C
D
Load
Elongation
O
(b) O C
(c) OD
(d) OB
4 . A metal wire of length L
1
 and area of cross-section A is
attached to a rigid support. Another metal wire of length L
2
and of the same cross-sectional area is attached to the free
end of the first wire. A body of mass M is then suspended
from the free end of the second wire. If Y
1
 and Y
2
 are the
Young’s moduli of the wires respectively , the effective force
constant of the system of two wires is
( a)
12
1221
()
2()+
YYA
YLYL
(b)
12
1/2
12
()
()
YYA
LL
(c)
12
1221
()
+
YYA
YLYL
(d)
1/2
12
1/2
21
()
()
YYA
LL
l a c i n a h c e M 	 s e i t r e p o r P 	 f o 	 s d i l o S
J E E 	 n i a M 	 s l a c i r e m u N
5 . Choose the wrong statement.
(a) The bulk modulus for solids is much larger than for
liquids.
(b) Gases are least compressible.
(c) The incompressibility of the solids is due to the tight
coupling between neighbouring atoms.
(d) The reciprocal of the bulk modulus is called
compressibility.
6 . K is the force constant of a spring. The work done in
increasing its extension from l
1
 t o l
2
 will be
( a) K(l
2
 – l
1
) (b)
21
()
2
K
ll +
(c)
22
21
()Kll - (d)
22
21
()
2
K
ll -
7 . A steel wire of length l and cross sectional area A is stretched
by 1 cm under a given load. When the same load is applied
to another steel wire of double its length and half of its
cross section area, the amount of stretching (extension) is
(a) 0.5 cm (b) 2 cm (c) 4 cm (d) 1.5 cm
8 . A cube at temperature 0ºC is compressed equally from all
sides by an external pressure P. By what amount should its
temperature be raised to bring it back to the size it had before
the external pressure was applied. The bulk modulus of the
material of the cube is B and the coefficient of linear
expansion is a.
(a) P/B a (b) P/3 B a (c) 3 p a/B (d) 3 B/P
9 . Stress Vs strain for the elastic tissue of the aorta, the large
tube (vessel) carrying blood from the heart, will be : [stress
is proportional to square of the strain for the elastic tissue
of the aorta]
( a) (b)
(c) (d)
10. In materials like aluminium and copper, the correct order of
magnitude of various elastic modului is:
(a) Young’s modulus < shear modulus < bulk modulus.
(b) Bulk modulus < shear modulus < Young’s modulus
(c) Shear modulus < Young’s modulus < bulk modulus.
(d) Bulk modulus < Young’s modulus < shear modulus.
11. An elastic string of unstretched length L and force constant
k is stretched by a small length x. It is further stretched by
another small length y. The work done in the second
stretching is :
( a)
2
1
2
ky (b)
22
1
()
2
kxy +
(c)
2
1
()
2
kxy + (d)
1
(2)
2
kyxy +
12. Steel ruptures when a shear of 3.5 × 10
8
 N m
–2
 is applied.
The force needed to punch a 1 cm diameter hole in a steel
sheet 0.3 cm thick is nearly:
(a) 1.4 × 10
4
 N (b) 2.7 × 10
4
 N
(c) 3.3 × 10
4
 N (d) 1.1 × 10
4
 N
13. Copper of fixed volume ‘V; is drawn into wire of length ‘l ’ .
When this wire is subjected to a constant force ‘F’, the
extension produced in the wire is ‘Dl’. Which of the following
graphs is a straight line?
( a) Dl versus 
1
l
(b) Dl versus l
2
(c) Dl versus 
2
1
l
(d) Dl versus l
Page 3


1 . Two wires A and B are of the same material. Their lengths
are in the ratio 1 : 2 and the diameter are in the ratio 2 : 1. If
they are pulled by the same force, then increase in length
will be in the ratio
(a) 2 : 1 (b) 1 : 4 (c) 1 : 8 (d) 8 : 1
2 . For a constant hydraulic stress on an object, the fractional
change in the object volume 
V
V
Dæö
ç÷
èø
 and its bulk modulus
(B) are related as
( a)
V
B
V
D
µ (b)
1 V
VB
D
µ
(c)
2
V
B
V
D
µ (d)
2
V
B
V
-
D
µ
3 . The load versus elongation graphs for four wires of same
length and made of the same material are shown in the figure.
The thinnest wire is represented by the line
( a) OA
A
B
C
D
Load
Elongation
O
(b) O C
(c) OD
(d) OB
4 . A metal wire of length L
1
 and area of cross-section A is
attached to a rigid support. Another metal wire of length L
2
and of the same cross-sectional area is attached to the free
end of the first wire. A body of mass M is then suspended
from the free end of the second wire. If Y
1
 and Y
2
 are the
Young’s moduli of the wires respectively , the effective force
constant of the system of two wires is
( a)
12
1221
()
2()+
YYA
YLYL
(b)
12
1/2
12
()
()
YYA
LL
(c)
12
1221
()
+
YYA
YLYL
(d)
1/2
12
1/2
21
()
()
YYA
LL
l a c i n a h c e M 	 s e i t r e p o r P 	 f o 	 s d i l o S
J E E 	 n i a M 	 s l a c i r e m u N
5 . Choose the wrong statement.
(a) The bulk modulus for solids is much larger than for
liquids.
(b) Gases are least compressible.
(c) The incompressibility of the solids is due to the tight
coupling between neighbouring atoms.
(d) The reciprocal of the bulk modulus is called
compressibility.
6 . K is the force constant of a spring. The work done in
increasing its extension from l
1
 t o l
2
 will be
( a) K(l
2
 – l
1
) (b)
21
()
2
K
ll +
(c)
22
21
()Kll - (d)
22
21
()
2
K
ll -
7 . A steel wire of length l and cross sectional area A is stretched
by 1 cm under a given load. When the same load is applied
to another steel wire of double its length and half of its
cross section area, the amount of stretching (extension) is
(a) 0.5 cm (b) 2 cm (c) 4 cm (d) 1.5 cm
8 . A cube at temperature 0ºC is compressed equally from all
sides by an external pressure P. By what amount should its
temperature be raised to bring it back to the size it had before
the external pressure was applied. The bulk modulus of the
material of the cube is B and the coefficient of linear
expansion is a.
(a) P/B a (b) P/3 B a (c) 3 p a/B (d) 3 B/P
9 . Stress Vs strain for the elastic tissue of the aorta, the large
tube (vessel) carrying blood from the heart, will be : [stress
is proportional to square of the strain for the elastic tissue
of the aorta]
( a) (b)
(c) (d)
10. In materials like aluminium and copper, the correct order of
magnitude of various elastic modului is:
(a) Young’s modulus < shear modulus < bulk modulus.
(b) Bulk modulus < shear modulus < Young’s modulus
(c) Shear modulus < Young’s modulus < bulk modulus.
(d) Bulk modulus < Young’s modulus < shear modulus.
11. An elastic string of unstretched length L and force constant
k is stretched by a small length x. It is further stretched by
another small length y. The work done in the second
stretching is :
( a)
2
1
2
ky (b)
22
1
()
2
kxy +
(c)
2
1
()
2
kxy + (d)
1
(2)
2
kyxy +
12. Steel ruptures when a shear of 3.5 × 10
8
 N m
–2
 is applied.
The force needed to punch a 1 cm diameter hole in a steel
sheet 0.3 cm thick is nearly:
(a) 1.4 × 10
4
 N (b) 2.7 × 10
4
 N
(c) 3.3 × 10
4
 N (d) 1.1 × 10
4
 N
13. Copper of fixed volume ‘V; is drawn into wire of length ‘l ’ .
When this wire is subjected to a constant force ‘F’, the
extension produced in the wire is ‘Dl’. Which of the following
graphs is a straight line?
( a) Dl versus 
1
l
(b) Dl versus l
2
(c) Dl versus 
2
1
l
(d) Dl versus l
14. If the ratio of lengths, radii and Young's moduli of steel and
brass wires in the figure are a, b and c respectively, then the
corresponding ratio of increase in their lengths is :
Steel
Brass
M
2M
( a)
2
3
2
c
ab
(b)
2
2ac
b
(c)
2
3
2
a
bc
(d)
2
2ac
b
15. The Poisson’s ratio of a material is 0.5. If a force is applied to a
wire of this material, there is a decrease in the cross-sectional
area by 4%. The percentage increase in the length is:
(a) 1% (b) 2% (c) 2.5% (d) 4%
16. The end of a uniform wire of length L and of weight W is
attached rigidly to a point in the roof and a weight W
1
 is
suspended from its lower end. If s is the area of cross-section
of the wire, the stress in the wire at a height 
3
4
L
 from its
lower end is :
( a)
1
W
s
(b)
1
4
W
W
s
+
(c)
1
3
4
W
W
s
+
(d)
1
WW
s
+
17. The diagram shows a force-
extension graph for a rubber
band. Consider the following
statements :
Extension
Force
I. It will be easier to compress
this rubber than expand it
II. Rubber does not return to its original length after it is
stretched
III. The rubber band will get heated if it is stretched and
released
Which of these can be deduced from the graph:
(a) III only (b) II and III (c) I and III (d) I only
18. When a 4 kg mass is hung vertically on a light spring that
obeys Hooke’s law , the spring stretches by 2 cms. The work
required to be done by an external agent in stretching this
spring by 5 cm will be (g = 9.8 m/sec
2
)
(a) 4.900 joule (b) 2.450 joule
(c) 0.495 joule (d) 0.245 joule
19. A circular tube of mean radius 8 cm and thickness 0.04 cm is
melted up and recast into a solid rod of the same length. The
ratio of the torsional rigidities of the circular tube and the
solid rod is
( a)
4
4 4
) 8 . 0 (
)98 . 7 ( )02 . 8 ( -
(b)
2
2 2
) 8 . 0 (
)98 . 7 ( )02 . 8 ( -
(c)
4 4
2
)98 . 7 ( )02 . 8 (
) 8 . 0 (
-
(d)
2 3
2
)98 . 7 ( )02 . 8 (
) 8 . 0 (
-
20. A mild steel wire of length 2L and cross-sectional area A is
stretched, well within elastic limit, horizontally between two
pillars. A mass m is suspended from the mid point of the
wire. Strain in the wire is
2L
x
m
( a)
2
x
2L
(b)
x
L
(c)
2
x
L
(d)
2
x
2L
21. Two wires A and B of same material and of equal length with
the radii in the ratio 1 : 2 are subjected to identical loads. If
the length of A increases by 8 mm, then the increase in
length of B is
(a) 2  mm (b) 4  mm (c) 8  mm (d) 16  mm
Page 4


1 . Two wires A and B are of the same material. Their lengths
are in the ratio 1 : 2 and the diameter are in the ratio 2 : 1. If
they are pulled by the same force, then increase in length
will be in the ratio
(a) 2 : 1 (b) 1 : 4 (c) 1 : 8 (d) 8 : 1
2 . For a constant hydraulic stress on an object, the fractional
change in the object volume 
V
V
Dæö
ç÷
èø
 and its bulk modulus
(B) are related as
( a)
V
B
V
D
µ (b)
1 V
VB
D
µ
(c)
2
V
B
V
D
µ (d)
2
V
B
V
-
D
µ
3 . The load versus elongation graphs for four wires of same
length and made of the same material are shown in the figure.
The thinnest wire is represented by the line
( a) OA
A
B
C
D
Load
Elongation
O
(b) O C
(c) OD
(d) OB
4 . A metal wire of length L
1
 and area of cross-section A is
attached to a rigid support. Another metal wire of length L
2
and of the same cross-sectional area is attached to the free
end of the first wire. A body of mass M is then suspended
from the free end of the second wire. If Y
1
 and Y
2
 are the
Young’s moduli of the wires respectively , the effective force
constant of the system of two wires is
( a)
12
1221
()
2()+
YYA
YLYL
(b)
12
1/2
12
()
()
YYA
LL
(c)
12
1221
()
+
YYA
YLYL
(d)
1/2
12
1/2
21
()
()
YYA
LL
l a c i n a h c e M 	 s e i t r e p o r P 	 f o 	 s d i l o S
J E E 	 n i a M 	 s l a c i r e m u N
5 . Choose the wrong statement.
(a) The bulk modulus for solids is much larger than for
liquids.
(b) Gases are least compressible.
(c) The incompressibility of the solids is due to the tight
coupling between neighbouring atoms.
(d) The reciprocal of the bulk modulus is called
compressibility.
6 . K is the force constant of a spring. The work done in
increasing its extension from l
1
 t o l
2
 will be
( a) K(l
2
 – l
1
) (b)
21
()
2
K
ll +
(c)
22
21
()Kll - (d)
22
21
()
2
K
ll -
7 . A steel wire of length l and cross sectional area A is stretched
by 1 cm under a given load. When the same load is applied
to another steel wire of double its length and half of its
cross section area, the amount of stretching (extension) is
(a) 0.5 cm (b) 2 cm (c) 4 cm (d) 1.5 cm
8 . A cube at temperature 0ºC is compressed equally from all
sides by an external pressure P. By what amount should its
temperature be raised to bring it back to the size it had before
the external pressure was applied. The bulk modulus of the
material of the cube is B and the coefficient of linear
expansion is a.
(a) P/B a (b) P/3 B a (c) 3 p a/B (d) 3 B/P
9 . Stress Vs strain for the elastic tissue of the aorta, the large
tube (vessel) carrying blood from the heart, will be : [stress
is proportional to square of the strain for the elastic tissue
of the aorta]
( a) (b)
(c) (d)
10. In materials like aluminium and copper, the correct order of
magnitude of various elastic modului is:
(a) Young’s modulus < shear modulus < bulk modulus.
(b) Bulk modulus < shear modulus < Young’s modulus
(c) Shear modulus < Young’s modulus < bulk modulus.
(d) Bulk modulus < Young’s modulus < shear modulus.
11. An elastic string of unstretched length L and force constant
k is stretched by a small length x. It is further stretched by
another small length y. The work done in the second
stretching is :
( a)
2
1
2
ky (b)
22
1
()
2
kxy +
(c)
2
1
()
2
kxy + (d)
1
(2)
2
kyxy +
12. Steel ruptures when a shear of 3.5 × 10
8
 N m
–2
 is applied.
The force needed to punch a 1 cm diameter hole in a steel
sheet 0.3 cm thick is nearly:
(a) 1.4 × 10
4
 N (b) 2.7 × 10
4
 N
(c) 3.3 × 10
4
 N (d) 1.1 × 10
4
 N
13. Copper of fixed volume ‘V; is drawn into wire of length ‘l ’ .
When this wire is subjected to a constant force ‘F’, the
extension produced in the wire is ‘Dl’. Which of the following
graphs is a straight line?
( a) Dl versus 
1
l
(b) Dl versus l
2
(c) Dl versus 
2
1
l
(d) Dl versus l
14. If the ratio of lengths, radii and Young's moduli of steel and
brass wires in the figure are a, b and c respectively, then the
corresponding ratio of increase in their lengths is :
Steel
Brass
M
2M
( a)
2
3
2
c
ab
(b)
2
2ac
b
(c)
2
3
2
a
bc
(d)
2
2ac
b
15. The Poisson’s ratio of a material is 0.5. If a force is applied to a
wire of this material, there is a decrease in the cross-sectional
area by 4%. The percentage increase in the length is:
(a) 1% (b) 2% (c) 2.5% (d) 4%
16. The end of a uniform wire of length L and of weight W is
attached rigidly to a point in the roof and a weight W
1
 is
suspended from its lower end. If s is the area of cross-section
of the wire, the stress in the wire at a height 
3
4
L
 from its
lower end is :
( a)
1
W
s
(b)
1
4
W
W
s
+
(c)
1
3
4
W
W
s
+
(d)
1
WW
s
+
17. The diagram shows a force-
extension graph for a rubber
band. Consider the following
statements :
Extension
Force
I. It will be easier to compress
this rubber than expand it
II. Rubber does not return to its original length after it is
stretched
III. The rubber band will get heated if it is stretched and
released
Which of these can be deduced from the graph:
(a) III only (b) II and III (c) I and III (d) I only
18. When a 4 kg mass is hung vertically on a light spring that
obeys Hooke’s law , the spring stretches by 2 cms. The work
required to be done by an external agent in stretching this
spring by 5 cm will be (g = 9.8 m/sec
2
)
(a) 4.900 joule (b) 2.450 joule
(c) 0.495 joule (d) 0.245 joule
19. A circular tube of mean radius 8 cm and thickness 0.04 cm is
melted up and recast into a solid rod of the same length. The
ratio of the torsional rigidities of the circular tube and the
solid rod is
( a)
4
4 4
) 8 . 0 (
)98 . 7 ( )02 . 8 ( -
(b)
2
2 2
) 8 . 0 (
)98 . 7 ( )02 . 8 ( -
(c)
4 4
2
)98 . 7 ( )02 . 8 (
) 8 . 0 (
-
(d)
2 3
2
)98 . 7 ( )02 . 8 (
) 8 . 0 (
-
20. A mild steel wire of length 2L and cross-sectional area A is
stretched, well within elastic limit, horizontally between two
pillars. A mass m is suspended from the mid point of the
wire. Strain in the wire is
2L
x
m
( a)
2
x
2L
(b)
x
L
(c)
2
x
L
(d)
2
x
2L
21. Two wires A and B of same material and of equal length with
the radii in the ratio 1 : 2 are subjected to identical loads. If
the length of A increases by 8 mm, then the increase in
length of B is
(a) 2  mm (b) 4  mm (c) 8  mm (d) 16  mm
22. Select the correct statement(s) from the following.
I. Modulus of rigidity for a liquid is zero
II. Young's modulus of a material decreases with rise in
temperature
III. Poisson's ratio is unitless
(a) I only (b) II only (c) I and II (d) I, II and III
23. The upper end of a wire of diameter 12mm and length 1m is
clamped and its other end is twisted through an angle of
30°. The angle of shear is
(a) 18° (b) 0.18° (c) 36° (d) 0.36°
24. A structural steel rod has a radius of 10 mm and length of
1.0 m. A 100 kN force stretches it along its length. Young’s
modulus of structural steel is 2 × 10
11
 Nm
–2
. The percentage
strain is about
(a) 0.16% (b) 0.32% (c) 0.08% (d) 0.24%
25. The graph given is a stress-
strain curve for
1.0
0.5
0
0. 5 1.0
Strain
Stress (N/ m )
2
(a) elastic objects
(b) plastics
(c) elastomers
(d) None of these
26. A beam of metal supported at the two edges is loaded at the
centre. The depression at the centre is proportional to
d
( a) Y 
2
(b) Y (c) 1/Y (d) 1/Y
2
27. The length of an elastic string is a metre  when the
longitudinal tension is 4 N and b metre when the longitudinal
tension is 5 N. The length of the string in metre when the
longitudinal tension is 9 N is
( a) a – b (b) 5b – 4a (c)
1
2–
4
ba (d) 4a – 3b
28. A thick rope of density r and length L is hung from a rigid
support. The Young’s modulus of the material of rope is Y .
The increase in length of the rope due to its own weight is
(a) (1/4) r g L
2
/Y (b) (1/2) r g L
2
/Y
(c) r g L
2
/Y (d) r g L/Y
29. A metal rod of Young's modulus 2 × 10
10
 N m
–2
 undergoes
an elastic strain of 0.06%. The energy per unit volume stored
in J m
–3
 is
(a) 3600 (b) 7200 (c) 10800 (d) 14400
30. For the same cross-sectional area and for a given  load, the
ratio of depressions for the beam of a square cross-section
and circular cross-section is
(a) 3 : p (b) p : 3 (c) 1 : p (d) p : 1
Page 5


1 . Two wires A and B are of the same material. Their lengths
are in the ratio 1 : 2 and the diameter are in the ratio 2 : 1. If
they are pulled by the same force, then increase in length
will be in the ratio
(a) 2 : 1 (b) 1 : 4 (c) 1 : 8 (d) 8 : 1
2 . For a constant hydraulic stress on an object, the fractional
change in the object volume 
V
V
Dæö
ç÷
èø
 and its bulk modulus
(B) are related as
( a)
V
B
V
D
µ (b)
1 V
VB
D
µ
(c)
2
V
B
V
D
µ (d)
2
V
B
V
-
D
µ
3 . The load versus elongation graphs for four wires of same
length and made of the same material are shown in the figure.
The thinnest wire is represented by the line
( a) OA
A
B
C
D
Load
Elongation
O
(b) O C
(c) OD
(d) OB
4 . A metal wire of length L
1
 and area of cross-section A is
attached to a rigid support. Another metal wire of length L
2
and of the same cross-sectional area is attached to the free
end of the first wire. A body of mass M is then suspended
from the free end of the second wire. If Y
1
 and Y
2
 are the
Young’s moduli of the wires respectively , the effective force
constant of the system of two wires is
( a)
12
1221
()
2()+
YYA
YLYL
(b)
12
1/2
12
()
()
YYA
LL
(c)
12
1221
()
+
YYA
YLYL
(d)
1/2
12
1/2
21
()
()
YYA
LL
l a c i n a h c e M 	 s e i t r e p o r P 	 f o 	 s d i l o S
J E E 	 n i a M 	 s l a c i r e m u N
5 . Choose the wrong statement.
(a) The bulk modulus for solids is much larger than for
liquids.
(b) Gases are least compressible.
(c) The incompressibility of the solids is due to the tight
coupling between neighbouring atoms.
(d) The reciprocal of the bulk modulus is called
compressibility.
6 . K is the force constant of a spring. The work done in
increasing its extension from l
1
 t o l
2
 will be
( a) K(l
2
 – l
1
) (b)
21
()
2
K
ll +
(c)
22
21
()Kll - (d)
22
21
()
2
K
ll -
7 . A steel wire of length l and cross sectional area A is stretched
by 1 cm under a given load. When the same load is applied
to another steel wire of double its length and half of its
cross section area, the amount of stretching (extension) is
(a) 0.5 cm (b) 2 cm (c) 4 cm (d) 1.5 cm
8 . A cube at temperature 0ºC is compressed equally from all
sides by an external pressure P. By what amount should its
temperature be raised to bring it back to the size it had before
the external pressure was applied. The bulk modulus of the
material of the cube is B and the coefficient of linear
expansion is a.
(a) P/B a (b) P/3 B a (c) 3 p a/B (d) 3 B/P
9 . Stress Vs strain for the elastic tissue of the aorta, the large
tube (vessel) carrying blood from the heart, will be : [stress
is proportional to square of the strain for the elastic tissue
of the aorta]
( a) (b)
(c) (d)
10. In materials like aluminium and copper, the correct order of
magnitude of various elastic modului is:
(a) Young’s modulus < shear modulus < bulk modulus.
(b) Bulk modulus < shear modulus < Young’s modulus
(c) Shear modulus < Young’s modulus < bulk modulus.
(d) Bulk modulus < Young’s modulus < shear modulus.
11. An elastic string of unstretched length L and force constant
k is stretched by a small length x. It is further stretched by
another small length y. The work done in the second
stretching is :
( a)
2
1
2
ky (b)
22
1
()
2
kxy +
(c)
2
1
()
2
kxy + (d)
1
(2)
2
kyxy +
12. Steel ruptures when a shear of 3.5 × 10
8
 N m
–2
 is applied.
The force needed to punch a 1 cm diameter hole in a steel
sheet 0.3 cm thick is nearly:
(a) 1.4 × 10
4
 N (b) 2.7 × 10
4
 N
(c) 3.3 × 10
4
 N (d) 1.1 × 10
4
 N
13. Copper of fixed volume ‘V; is drawn into wire of length ‘l ’ .
When this wire is subjected to a constant force ‘F’, the
extension produced in the wire is ‘Dl’. Which of the following
graphs is a straight line?
( a) Dl versus 
1
l
(b) Dl versus l
2
(c) Dl versus 
2
1
l
(d) Dl versus l
14. If the ratio of lengths, radii and Young's moduli of steel and
brass wires in the figure are a, b and c respectively, then the
corresponding ratio of increase in their lengths is :
Steel
Brass
M
2M
( a)
2
3
2
c
ab
(b)
2
2ac
b
(c)
2
3
2
a
bc
(d)
2
2ac
b
15. The Poisson’s ratio of a material is 0.5. If a force is applied to a
wire of this material, there is a decrease in the cross-sectional
area by 4%. The percentage increase in the length is:
(a) 1% (b) 2% (c) 2.5% (d) 4%
16. The end of a uniform wire of length L and of weight W is
attached rigidly to a point in the roof and a weight W
1
 is
suspended from its lower end. If s is the area of cross-section
of the wire, the stress in the wire at a height 
3
4
L
 from its
lower end is :
( a)
1
W
s
(b)
1
4
W
W
s
+
(c)
1
3
4
W
W
s
+
(d)
1
WW
s
+
17. The diagram shows a force-
extension graph for a rubber
band. Consider the following
statements :
Extension
Force
I. It will be easier to compress
this rubber than expand it
II. Rubber does not return to its original length after it is
stretched
III. The rubber band will get heated if it is stretched and
released
Which of these can be deduced from the graph:
(a) III only (b) II and III (c) I and III (d) I only
18. When a 4 kg mass is hung vertically on a light spring that
obeys Hooke’s law , the spring stretches by 2 cms. The work
required to be done by an external agent in stretching this
spring by 5 cm will be (g = 9.8 m/sec
2
)
(a) 4.900 joule (b) 2.450 joule
(c) 0.495 joule (d) 0.245 joule
19. A circular tube of mean radius 8 cm and thickness 0.04 cm is
melted up and recast into a solid rod of the same length. The
ratio of the torsional rigidities of the circular tube and the
solid rod is
( a)
4
4 4
) 8 . 0 (
)98 . 7 ( )02 . 8 ( -
(b)
2
2 2
) 8 . 0 (
)98 . 7 ( )02 . 8 ( -
(c)
4 4
2
)98 . 7 ( )02 . 8 (
) 8 . 0 (
-
(d)
2 3
2
)98 . 7 ( )02 . 8 (
) 8 . 0 (
-
20. A mild steel wire of length 2L and cross-sectional area A is
stretched, well within elastic limit, horizontally between two
pillars. A mass m is suspended from the mid point of the
wire. Strain in the wire is
2L
x
m
( a)
2
x
2L
(b)
x
L
(c)
2
x
L
(d)
2
x
2L
21. Two wires A and B of same material and of equal length with
the radii in the ratio 1 : 2 are subjected to identical loads. If
the length of A increases by 8 mm, then the increase in
length of B is
(a) 2  mm (b) 4  mm (c) 8  mm (d) 16  mm
22. Select the correct statement(s) from the following.
I. Modulus of rigidity for a liquid is zero
II. Young's modulus of a material decreases with rise in
temperature
III. Poisson's ratio is unitless
(a) I only (b) II only (c) I and II (d) I, II and III
23. The upper end of a wire of diameter 12mm and length 1m is
clamped and its other end is twisted through an angle of
30°. The angle of shear is
(a) 18° (b) 0.18° (c) 36° (d) 0.36°
24. A structural steel rod has a radius of 10 mm and length of
1.0 m. A 100 kN force stretches it along its length. Young’s
modulus of structural steel is 2 × 10
11
 Nm
–2
. The percentage
strain is about
(a) 0.16% (b) 0.32% (c) 0.08% (d) 0.24%
25. The graph given is a stress-
strain curve for
1.0
0.5
0
0. 5 1.0
Strain
Stress (N/ m )
2
(a) elastic objects
(b) plastics
(c) elastomers
(d) None of these
26. A beam of metal supported at the two edges is loaded at the
centre. The depression at the centre is proportional to
d
( a) Y 
2
(b) Y (c) 1/Y (d) 1/Y
2
27. The length of an elastic string is a metre  when the
longitudinal tension is 4 N and b metre when the longitudinal
tension is 5 N. The length of the string in metre when the
longitudinal tension is 9 N is
( a) a – b (b) 5b – 4a (c)
1
2–
4
ba (d) 4a – 3b
28. A thick rope of density r and length L is hung from a rigid
support. The Young’s modulus of the material of rope is Y .
The increase in length of the rope due to its own weight is
(a) (1/4) r g L
2
/Y (b) (1/2) r g L
2
/Y
(c) r g L
2
/Y (d) r g L/Y
29. A metal rod of Young's modulus 2 × 10
10
 N m
–2
 undergoes
an elastic strain of 0.06%. The energy per unit volume stored
in J m
–3
 is
(a) 3600 (b) 7200 (c) 10800 (d) 14400
30. For the same cross-sectional area and for a given  load, the
ratio of depressions for the beam of a square cross-section
and circular cross-section is
(a) 3 : p (b) p : 3 (c) 1 : p (d) p : 1
1. (c) We know that Young's modulus
l
L
r
F
Y
2
´
p
=
Since Y , F are same for both the wires, we have,
2
2
2
2
1
1
2
1
L
r
1 L
r
1
l l
=
 or,
2
2
1
1
2
2
2
1
L r
L r
´
´
=
l
l
 = 
2
2
1
1
2
2
L ) 2 / D (
L ) 2 / D (
´
´
or,
8
1
L 2
L
) D 2 (
D
L D
L D
2
2
2
2
2
2
2
2
1
1
2
2
2
1
= ´ =
´
´
=
l
l
So,   l
1
 : l
2
 = 1 : 8
2. (b)
1
/
pV
B
VVBV
DD
=Þµ
D
[Dp = constant]
3. (a) From the graph, it is clear that for the same value of
load, elongation is maximum for wire OA. Hence OA is
the thinnest wire among the four wires.
4. (c) Using the usual expression for the Young’s modulus,
the force constant for the wire can be written as
k = =
D
F YA
lL
 where the symbols have their usual
meanings. Now the two wires together will have an
effective force constant 
12
12
éù
êú
+
ëû
kk
kk
. Substituting the
corresponding lengths and the Young’s moduli we get
the answer.
5. (b) Solids are least compressible whereas gases are highly
compressible.
6. (d) At extension l
1
, the stored energy = 
2
1
1
2
Kl
At extension l
2
, the stored energy = 
2
2
1
2
Kl
Work done in increasing its extension from l
1
 t o l
2
22
21
1
(–)
2
Kll =
7. (c) Young’s modulus of elasticity is
F/A
Y
L/L
=
D
FL
L
AY
\D=
So, 
L
L
A
Dµ
2 2 1
112
LLA22
4
LLA11
D
\=´=´=
D
DL
2
 = 4 × DL
1
 = 4  × 1 = 4 cm
8. (b) Bulk modulus B =
and    T . V . 3 T V V a = D g = D o r
. T . 3
1
V
V
a
=
D
-
    ...(2)
From eqs. (1) and (2), ) T . 3 /( P B a = or 
B 3
P
T
a
=
9. (a)
10. (c) Poisson’s ratio, 
( )
()
lateral strain
longitudinal strain
b
s=
a
For material like copper, s = 0.33
And, y = 3k (1 – 2 s)
Also, 
913
ykn
=+
y = 2n (1+ s )
Hence, n < y < k
11. ( d) W
1
=
2
1
2
kx
and W
2
=
2
1
()
2
kxy +
\ W=
22
21
11
()
22
W W k x y kx -=+-
=
1
(2)
2
kyxy+
12. (c)
D
F
h
Shearing strain is created along the side surface of the
punched disk. Note that the forces exerted on the disk are
exerted along the circumference of the disk, and the total
force exerted on its center only.
Let us assume that the shearing stress along the side surface
of the disk is uniform, then
max max max
surface surface surface
F dF dA dA > = s =s
òòò
= max max
D
.A.2h
2
æö
s=sp
ç÷
èø
ò
= 
822
1
3.5 10 10 0.3 10 2
2
--
æö
´´´´´´p
ç÷
èø
= 
44
3.297 10 3.3 10 N´´;
13. (b) As Y = 
F
F
A
A Y
ÞD=
D
l
l
l
l
V
HINTS & SOLUTIONS