Elasticity

ELASTICITY

CHAPTER - 11

ELASTICITY

Rigid Body :A body whose size and shape cannot bechanged, however large the applied force maybe is called rigid body.There is no perfectly rigid body in nature.

Deformation force :The force which changes the size or shape orboth of a body without moving it as a whole iscalled deformation force.

Restoring force :The force which restores the size and shape ofthe body when deformation force is removed iscalled restoring force.Magnitude of restoring force is equal to the defor-mation force. But they are in opposite direction.They do not form action, reaction pair. This forceis responsible for the elastic nature of thebody.

ELASTIC BEHAVIOUR OF SOLIDSIn a solid, each atom or molecules are bondedtogether by interatomic or intermolecular forcesand stay in a stable equilibrium position.When a solid is deformed, the atoms or moleculesare displaced from their equilibrium positions caus-ing a change in the interatomic distances.When the deforming force is removed, the inter-atomic forces tend to drive them back to theiroriginal positions. Thus the body regains its origi-nal shape and size.

Elasticity :The property of a material by virtue of which itregains its original size and shape when defor-mation force is removed is called elasticity.Ex : Steel, Rubber.No body is perfectly elastic, but quartz is thenearest example.Elasticity is molecular property of matter.

Plasticity :The property of a material by virtue of which itdoes not regain the size and shape when the deformation force is removed is called Plasticity.Ex : Putty dough, Chewing gum, Soldering leadNo body is perfectly plastic but putty is nearestexample.

Stress :The restoring force per unit area is called stress.

Stress = sectioncrossofareaforcerestoring = A

F

Unit : 2N/m or Pascal. If the stress is normal to the surface, it is called

normal stress. If the stress is tangential to thesurface, it is called tangential stress.

Normal restoring forceNormal stress Area of cross section

ELASTICITY

ELASTICITYTangential restoring forceShearing stress Area of cross section

When normal stress changes the volume of thebody then it is called Volume Stress or BulkStress. It is denoted by P.

Longitudinal stress is called tensile stress whenthere is an increase in length and compressivestress when there is a decrease in length.

Longitudinal stress and bulk stress are normalstresses which produce change in size, shear-ing stress is a tangential stress which producechange in shape.

Strain :The change produced per unit dimension is calledstrain.

dimensionoriginaldimensioninchangeStrain

3 3/ 4W l b d Y Shearing strain:-

relative displacement of layersdistance between the layersShearing strain

Lx

This strain is due to the change in shape of thebody.Volumetric strain (or) Bulk strain

VV

volumeoriginalin volumechange

Negative sign indicates decrease in volumeFor isotrophic substances:Shearing strain = 2 x e(e is longitudinal strain).Bulk strain (or) Volumetric strain = 3 x eLongitudinal strain : Shearing strain :Bulk strain = 1 : 2 : 3.

. Elastic limit :The maximum value of the stress with in whichthe body regains its original size and shape is

called elastic limit. . Hooke's law :

With in the elastic limit, stress is directly proportional to strain.StressStrain EWith in the elastic limit, stress-strain graph is astraight line passing through the origin.

Strain

Stress

Slope of the graph is E.Spring balance works on the principle of Hooke'slaw.Modulus of elasticity depends on the nature ofthe material, but it is independent of dimensions.

Young's modulus (Y) :

AeF

strainallongitudinstressallongitudinY

If load attached to the wire is M, then F = Mg,and 2rA

2MgY r e

If the load attached to the wire and Y are constant

2re

If length is also constant 2r1e

If volume of the wire is constant

422

r1e;A

1e;e Elongation of a wire under its own weight

2

2dge Y

( d is density).

If 21, are the lengths of a wire under tensions21 T&T then the length of the unstretched wire.

ELASTICITY

121221

TTTT

When a load is suspended from a wire its elon-gation is e. If the load is immersed in a liquid ofdensity '' then the new elongation.

)d/1(ee1 d is density of load

Youngs moduli, elastic limit and tensile strengths of some materials

Stress required to double the length of the wireis numerically equal to the Young's modulus.If a load 'M' produced elongation e in a wire,the raise in temperature required to produce sameelongation.

AYMg ; AY

MgTwo wires of same length and radius are joinedend to end and loaded. If Young's moduli of thematerials are 21 Y&Y , the Young's modulus of

the combination is21 Y1

Y1

Y2 ;

2121YYYY2Y

For a perfectly elastic material e = 0. So Young'smodulus is infinity.For a plastic material the Young's modulus is zero.From Searle's experiment the Young's modulusof the material of wire.

erMgY 2

The graph between load and elongation is astraight line passing through the origin.

Rigidity modulus :With in the elastic limit, the ratio of tangentialstress to the shearing strain is called rigiditymodulus.

Rigidity modulus = strainshearingstresstangential

FA

a) If is small for a wire, it can be twisted easily.

b) As the rigidity modulus of phsphor bronze islow, it is used as supspension wire in movingcoil galvanometer.

c) A rod of length 'l' and radius r is fixed at oneend.If the other free end is twisted through anangle . Then the angle of shear '' is givenby

r .

One end of the rod is fixed the other free end istwisted through an angle by applying a torque . The work done on the rod is

21W

When a spring is stretched, the strain involved isshear.When a helical spring (thickness is large) isstretched, the strain involved is longitudinal andshearing strain.

Bulk modulus (K)With in the elastic limit the ratio between volumestress and bulk strain is called bulk modulus.

Bulk modulus strainbulkstressvolume

VPV

AF

VV

VV

AF

K

(- sign shows decrease in volume)

SubstanceYoungsmodule

109 N/m2Elasticlimit

107 N/m2%Tensilestrength1 0

7 N/m2

AluminiumCopperIron (wrought)SteelBone(Tensile)(Compressive)

70120190200169

18201730

204033501212

ELASTICITYi) If a block of coefficient of cubical expansin is heated through a rise in temperature of ,the pressure to be applied on it to prevent itsexpansion = K , where K is its bulk modulus.ii) When a rubber ball of volume V, bulk modulus K is taken to a depth 'h' in water decreases

in its volume KhdgVV .

(d = density of material)For an incompressible material, 0V , sobulk modulus is infinity.Solid possess y,n and kLiquids and gases possess only K.Isothermal bulk modulus of the gas =P(pressure)Adiabatic bulk modulusof the gas = p

(where Vp

CC

)The reciprocal of bulk modulus is called compressibility

Poisson's ratio ( )The ratio of lateral contraction strain to the longitudinal elongation strain is called Poisson'sratio.

strainelongationallongitudinstrainncontractiolateral

/r/r

strainallongitudinstraintransverse

= rx

r

i) As it is a ratio, it has no units and dimension.ii) Theoretical limits of 5.0to1Practical limits of 5.0to0iii) For an incompressible substance 5.0

Relation among elastic constants y,n,k.

i)9 1 3Y K (or)

93

kY K

ii) 2 (1 )Y iii) 3 (1 2 )Y k iv) 3 26 2

KK

Behaviour of a wire under increasing load.

Proportional limit (A) :Upto the point A stress is directly proportionalto the strain and the wire obeys Hooke's law.Elastic limit (B) :In the region AB, the wire does not obey Hooke'slaw but it regains, the original length when loadis removed.Yield point C:The point where elasticity ends and plasticitybegins is called yield point.Here the wire behaves like fluid and it elongateswith time.Permanent set (OP) :The permanent increase in length of the wire after removing the load is called permanent set.Permanent set = xOP

Breaking Point : The point in the graph wherethe wire breaks.

Breaking Stress :The stress required to break the wire is calledbreaking stress.The maximum stress that the material can withstandwithout breaking is called breaking stress.Breaking stress

= sectioncrossofareainitialforcebreaking

dgAdgA

AMg

The maximum length of the wire that can hangwithout breaking under its own weight

dgstressbreaking

Breaking stress depends on the nature of thematerial, but it is independent of dimensions.

ELASTICITYBreaking stress per unit area is called tensilestrength.Breaking force = Breaking stress x areaBreaking force is independent of length of thewire, but it depends on the nature of materialand area of cross section.

2rF,AF If we cut a wire that can support a maximumload W into two equal parts, then each part ofthe wire can support a maximum load W.

ElastomersThe tress -strain behaviour varies from materialto material.The figure show stress-strain curve for the elas-tic tissue of aorta, present in the heart.

z

1.0

0.5

00 0.5 1.0

Stress (

106 N

m-2 )

Strain

1.0

0.5

00 0.5 1.0

Stress (

106 N

m-2 )

Stress (

106 N

m-2 )

StrainStrain

Although elastic region is very large, the materialdoes not obey Hookes law over most of the re-gion.There is no well defined plastic region.Substances like tissue of aorta, rubber etc.,whichcan be stretched to cause large strains are calledelastomers.

Elastic fatigue :The state of temporary loss of elastic propertydue to continuous strain is called elastic fatigue.If some rest is given to the wire, it regains theelastic property.Due to elastic fatiguea) A wire can be broken with in the elastic limit.b) A wire can be cut into two pieces withoutusing instruments.c) Railway tracks and bridges declared unsafeafter long use.d) Spring balance show wrong reading on longuse.e) If the material doesn't break after the elasticlimit, it is used to prepare thin wires. It is calledductile metal.Ex : Copper, Silver, Gold etc.f) If the material braeaking after elastic limit, it isnot used to prepare thin wires . It is called brittlemetal. Ex : Glass.

Elastic after effect :The delay in regaining the original state on removal of the deforming force on a body is calledelastic after effect.For a perfectly elastic body, the elastic aftereffect is zero.For a perfectly plastic body, the elastic aftereffect is infinity.For a given load the elongation of steel wire isless than rubber. So steel is more elastic thanrubber.Springs are made of steel, because it is moreelastic.Elasticity of a material will decrease with increaseof temperature.For rubber elasticity will increase withtemperature.For invar steel elasticity is independent oftemperature.Annealing means slow cooling after heating. Itdecreases the elastic property.Hammering and rolling increases with elasticproperty.Elastic property of a material increase withaddition of impurity.

Thermal force :When a metal bar is fixed between two supportsand heated, it tries to expand and exerts forceon the walls. This is called thermal force.

AYFThermal force is independent of length of the bar.Thermal stress (linear compressive stress)

YAAY

areaforce .

Punching a holePunching a hole of radius 'r' in a metal plate ofthickness 't'. The force required is calaculatedby the formula.F = breaking stress x rt2

A beam loaded at the centerA bridge has to be designed such that it can with-stand the load of the following traffic, the force ofwinds and its own weight.A bar of length l, breadth b and depth d whenloaded at the centre by a load W sags by anamount given by 3 3/ 4Wl bd Y

W

d

ELASTICITYTo reduce the bending for given load one shoulduse a material with a large Youngs modulus Y.For a given material increasing depth d ratherthan the bredth b is more effective in reducingthe bending.

SpringFor a given spring xF ; kxF ; x

Fk k is called spring constant (or) force constant(or) stiffness constant.Spring constant interms of Young's modulus area

of cross section and length yAk .

P.E.energy of a streched spring2

21 12 2 2

FE Kx Fx K Two springs having force constants 21 K&K( )KK 21 are stretched by same amount thenmore work is done on the first spring KW .Two springs having force constants

)KK(K,K 2121 are stretched by same forcethen more work is done on the second spring.

K/1W .If energy is same for both the springs the relationbetween force and spring constant KF .

The work done in stretching a wire12W Stress x Strain x Volume

21 12 2

e AYW Fe Strain energy per unit volume (energy density)

strainxstressx21E 21 ( )2Y Strain

Also,2( )

2Stress StressE YY Strain

CONCEPTUAL QUESTIONS1. Assertion (A): Stress is restoring force per unit

area.Reason (R) : Interatomic forces in solids areresponsible for the property of elasticity1) Both Assertion and Reason are true and thereason is correct explanation of the assertion2) Both Assertion and Reason are true, butreason is not correct explanation of the assertion3) Assertion is true, but the Reason is false4) Assertion is false, but the reason is true

2. A spiral spring is stretched by a force, the resultantstrain produced in the spring is1) Volume strain 2) shearing strain3) longitudinal strain 4) all the above

3. Reason for the deformation of a regular body is1) bulk strain 2) shearing strain3) linear strain 4) lateral strain

4. Hooke's law states that1) Stress is inversely proportional to strain2) Stress is directly proportional to strain3) Stress is independent of strain4) Stress is proportional to elastic modulus

5. According to Hook's law of elasticity, the ratio ofstress to strain (1996 M)1) does not remain constant 2) remains constant3) increases 4) decreases

6. Steel is preferred for making springs over copperbecause1) Y of steel is more than that of copper2) Steel is cheaper3) Y of copper is more than steel4) Steel is less likely to be oxidised

7. Elongation of a wire under its own weight isindependent of1) Length 2) Area of cross section3) Density 4) Young's modulus

8. The young's modulus of a wire of length 'L' andradius 'r' is 'Y'. If length is reduced to L/2 and radius r/2, Youngs modulus will be1) Y/2 2) Y 3) 2Y 4) 4Y

9. A copper wire and a steel wire of the samediameter and length are connected end to endand a force is applied so that their combinedlength stretched by 1cm. The two wires have1) same stress, same strain2) same stress, different strains3) same strain different stress4) different stress, different strains

10. Three wires A, B, C made of different materialselongated by 1.5, 2.5, 3.5 mm, under a load of5kg. If the diameters of the wires are the same,the most elastic material is that of1) A 2) B 3) C 4) All

11. The modulus of elasticity is dimensionallyequivalent to1) Stress 2) Surface tension3) Strain 4) Coefficient of viscosity

ELASTICITY12. Modulus of elasticity for a perfectly elastic body

is1) Zero 2) Infinity 3) 1 4) 2

13. A wire of length 'L' and radius 'r' is fixed at one end.A force 'F' applied to the other end produces anextension 'l'. The extension produced in anotherwire of the same material of length '2L', radius '2 r'by a force F is

1) l 2) 2l 3) 8 l 4) 2 l

14. As temperature increases the Young's modulusof the material of a wire1) increases 2) decreases3) remains the same 4) becomes infinite

15. If stress is numerically equal to young's modulus,the elongation will be1) 1/4 the original length2) 1/2 the original length3) equal to the original length4) Twice the original length

16. A wire elongates by 1 mm when a load W ishung from it. If the wire goes over a pulley andtwo weights W each are hung at the two ends,the elongation of the wire will be1) 0.5mm 2)1 mm 3) 2 mm 4) 4mm

17. Dimensional formula of youngs modulus is1) M1 L-1 T-2 2) M1L1 T13) M-1L2T-1 4) M-1L3T-2

18. An iron bar of length L, cross-section A andYoung's modulus Y is pulled by a force F fromends so as to produce an elongation .Whichof the following statements is correct ?

1) L1 2) A

3) A1 4) Y

19. A wire of length L and radius r fixed at one endand a force F applied to the other end producesand extension . The extension produced inanother wire of the same material of length 2Land radius 2r by a force 2 F is

1) 2) 2 3) 2 4) 4

20. Consider the following two statements A and Band identify the correct answer .A. A metal wire held vertically is longer than whenit placed on a horizontal table.B) Due to its own weight, some elongation isproduced when it is held vertically.1) Both A & B are true 2) A is false but B is true3) A is true but B is false 4) Both A & B are false

21. Assertion (A): Steel is more elastic than rubber.Reason (R) : Under a given deforming force, steelis deformed less than rubber.A) Both Assertion and Reason are true and thereason is correct explanation of the assertionB)Both Assertion and Reason are true, but reasonis not correct explanation of the assertionC) Assertion is true, but the reason is falseD) Assertion is false, but the reason is true1) A 2) B 3) C 4) D

22. Assertion (A) : Lead is more elastic than rubber.Reason (R) : If the same load is attached to leadand rubber wires of the same cross-sectionalarea, the strain of lead is very much less thanthat of rubber.1) Both assertion and reason are true and the

reason is correct explanation of the assertion2) Both assertion and reason are true, but reason

is not correct explanation of the assertion3) Assertion is true, but the reason is false4) Assertion is false, but the reason is true

23. Assertion (A) : Ductile metals are used toprepare thin wires.

Reason (R) : In the stress-strain curve ofductile metals, the length between the pointsrepresenting elastic limit and breaking point isvery small. (2006 E)

1.Both (A) and ( R) are true and ( R) is thecorrect explanation of (A)

2.Both (A) and (R) are true and (R) is not correctexplanation of (A)

3.(A) is true but (R ) is false. 4.(A) is false but (R ) is true.24. Consider the statements A and B, identify the

correct answer given below :(A) : If the volume of a body remains unchangedwhen subjected to tensile strain, the value ofpoisson's ratio is 1/2.(B) : Phosper bronze has low Young's modulusand high rigidity modulus. (2003 M)1) A and B are correct 2) A and B are wrong3) A is correct and B is wrong4) A is wrong and B is right

25. Assertion (A) : With increase of temperature,elastic property of a substance decreases.Reason (R) : Elasticity is due to intermolecularforces which decreases with the increase ofintermolecular distance.1) Both assertion and reason are true and thereason is correct explanation of the assertion2) Both assertion and reason are true, but reasonis not correct explanation of the assertion3) Assertion is true, but the reason is false4) Assertion is false, but the reason is true

ELASTICITY26. Assertion (A) : Stress is the internal force per unit

area of the body.Reason (R) : Rubber is more elastic than steel1) Both assertion and reason are true and the



27. Assertion (A) : Young's modulus for a perfectlyplastic body is zero.Reason (R) : For a perfectly plastic body, restoringforce is zero.1) Both assertion and reason are true and the



28. Consider the following two statements A and Band identify the correct answer.A) The bulk modulus for an incompressible liquid isinfinite.B) Young's modulus increases with raise oftemperature.1) Both A & B are true 2) A is true but B is false3) Both A & B are false 4) A is false but B is true

29. Assertion (A) : Bulk modulus of elasticity (K)represents incompressibility of the material.

Reason (R) : V/VPK

, where symbols havetheir standard meaning.1) Both assertion and reason are true and thereason is correct explanation of the assertion2) Both assertion and reason are true, but reasonis not correct explanation of the assertion3) Assertion is true, but the reason is false4) Assertion is false, but the reason is true

30. The following substances which possess rigiditymodulus1) Only Solids 2) Only liquids3) Liquids and Gases4) Solids, Liquids and Gases

31. Consider the following two statements A and Band identify the correct answer.A) The product of bulk modulus of elasticity andcompressibility is one.B) Tangential stress applied on the body onlyprodues change in shape but not in size.1) Both A & B are true 2) A is true but B is false3) A is false but B is true 4) Both A & B are false

32. The poisson's ratio cannot have the value1) 0.7 2) 0.2 3) 0.1 4) 0.3

33. When a rubber cord is stretched, the change involume is negligible compared to the change inits linear dimension. Then poisson's ratio forrubber is1) infinite 2) zero 3) 0.5 4) -1

34. A material has poisson's ratio of 0.5. If a uniformrod suffers a longitudinal strain of 2 x 10-3 thepercentage increase in its volume is1) 2% 2) 0.5% 3) 4% 4) 0%

35. The Poisson's ratio should satisfy the relation (1998 M)

1) -1

ELASTICITY42. The breaking stress of a wire depends upon

1) material of the wire 2) length of the wire3) radius of the wire 4) shape of the cross section

43. A wire can sustain the weight of 40kg beforebreaking. If the wire is cut into 4 equal parts,each part can sustain a weight of ...kg1) 40 2) 160 3)10 4) 20

44. Three wires A,B, C made of the same material andradius have different lenghts. The graphs in the figureshow the elongation-load variation. The longest wireis

1) A 2) B 3) C 4) All 45. A body is subjected to strain several times will

not obey Hookes law due to1) Yeild point 2) Permanent state3) Elastic fatigue 4) Breaking stress

46. Force vs Elongation graph of a wire is shown inthe figure. Then, at two different temperatures T1& T2

1) T1 = T2 2) T1 < T2 3) T1 > T24) cannot be predicted.47. The elastic after effect shows that the

1) strain in a material is lagging behind stress2) strain produced is quick3) elasticity of the material vanishes4) stress is developed very slowly.

48. The graph shows the behaviour of a steel wire inthe region for which the wire obeys Hooke's law.The graph is a part of a parabola. The variables xand y might represent.

1) x = stress ; y = strain2) x = strain ; y = stress3) x = strain ; y = elastic energy4) x = elastic energy ; y = strain

49. Assertion (A): Rigidity modulus of a liquid is infinity.Reason (R): For a ductile material yield point andbreaking point are separated by larger distancethan for brittle materials on the stress-starincurve.1) Both assertion and reason are true and the



50. Assertion (A) : Silver is a ductile material,Reason (R): For a ductile material yield point andbreaking point are separated by larger distancethan for brittle materials on the stress-straincurve.1) Both assertion and reason are true and the



51. When an elastic material with young's modulus'y' is subjected to a stretching stress 's', theelastic energy stored per unit volume is

1) y2s2 2) 2

ys2 3) y2s

4) 2ys

52. A uniform heavy rod of length L and area of crosssection 'A' is hanging from a fixed support. Ifyoung's modulus of the material of the rod is Y,the increase the length of rod is ( is density ofthe mateiral of the rod)

1)2

2L Yg 2)

2

2L gY

3)2

2L gY 4)

2

3L gY

53. A wire extends by 'I' on the application of load'mg'. Then, the energy stored in it is1) mgl 2) mgl/2 3) mg/1 4) mgl2

54. The stress required to double the length of a wireof Youngs modulus E is (1990 M)1) 2E 2) E 3) E/2 4) 3E

ELASTICITY55. A metallic rod of length 'L' and cross-section 'A'

has Young's modulus 'Y' and coefficient of linearexpansion 'a'. If the rod is heated to a temperature'T' the energy stored per unit volume is :

1) 2TY21 2) 2TYA2

1

3) YA21 4) 22 TYA2

1 56. A copper and steel wire of same diameter and

length are connected end to end and a force isapplied which stretches their combined lengthby 1 cm, the two wires will have (1988 M)1) The same stress and strain2) The same stress but different strains3) the same strain but different stresses4) different stress and strains.

57. An iron bar of length l, cross section A andYoung's modulus Y is heated from 0 to 1000C. Ifthis bar is held so that it is not permitted to bendto expand, the force F that is develped, isproportional to1) l 2) l 3) l0 4) l-1

58. A student plotted a graph from his readings onthe determination of Young's modulus of a metalwire but forgotten to label. The quantities on xand y cannot represent.

1) weight hung and extension2) stress applied and extension3) stress applied and strain produced4) stress applied and energy stored

59. Assertion (A) : The elastic potential energy of aspring increases when it is elongated anddecreases when it is compressed.Reason (R) : Work done on spring is stored in itas elastic potential energy.

1) Both assertion and reason are true and the reasonis correct explanation of the assertion

2) Both assertion and reason are true, but reasonis not correct explanation of the assertion

3) Assertion is true, but the reason is false4) Assertion is false, but the reason is true

LEVEL - ISTRESS AND TYPES OF STRESS

MODEL QUESTIONS60. A 20 Kg load is suspended by a wire of cross

section 0.4 mm2. The stress produced in N/m2is1) 4.9 x 10-6 2) 4.9 x 1083) 49 x 108 4) 2.45 x 10-6

PRACTICE QUESTIONS61. A force of 30N acts on a rod of area of cross

section 5 x 10-6 m2. The stress produced in dyne/cm2 is1) 6 x 106 2) 6 x 07 3) 6 x 105 4) 0.16 x 10-6

62. A steel wire of 2mm in diameter is stretched byapplying a force of 72N. stress in the wire is1) 7 22.29 10 /N m 2) 7 21.7 10 /N m3) 7 23.6 10 /N m 4) 7 20.8 10 /N mSTRAIN AND TYPES OF STRAIN

MODEL QUESTIONS63. The length of a wire is 4m. Its length is increased

by 2mm when a force acts on it. The strain is1) 0.5 x 10-3 2) 5 x 10-3 3) 2 x 10-3 4) 0.05

PRACTICE QUESTIONS64. The length of a wire under stress changes by

0.01%. The strain produced is1) 1 x 10-4 2) 0.01 3) 1 4) 104

65. The length of a cube increases by 0.1%, what isthe bulk strain ? (1995 E)1) 0.003 2) 0.006 3) 0.002 4) 0.001YOUNG'S MODULUS OF ELASTICITY

MODEL QUESTIONS66. A wire of length 'l' and radius 'r' is clamped rigidly

at one end. When the other end of the wire ispulled by a force 'F', its length increases by 'x'.Another wire of same material of length '2l' andradius '2r' is pulled by a force '2F', the increasein its length will be1) x 2) 2x 3) x/2 4) 4x

67. Force required to increase the length of a wire ofarea of cross-section 'A' by one percent, (Y isyoung's modulus) is1) AY 2) AY/10 3) AY/100 4) AY/1000

68. Four wires made of same materials are stretchedby the same load. Their dimensions are givenbelow. The one which elongates more is ?1) Wire of length 1 m and diameter 1 mm2) Length 2m, diameter 2 mm3) Length 3m, diameter 3 mm4) Length 0.5m, diameter 0.5mm

ELASTICITY69. An elongation of 0.1% in a wire of cross section

610 m2 causes a tension of 100N. Y for the wireis1) 1012N/m22) 1011N/m23) 1010N/m2 4) 100 N/m2

70. The force required to double the length of a steelwire of area of cross section 5 x 10-5m2 (y=20 x1010pa.) (in N) is1) 107 2) 106 3) 10-7 4) 105

71. The lengths of two wires of the same material anddiameter are 100cm and 125cm. If same force isapplied on them, the elongation in the first wire is4mm, then the elongation in the second wire (inmm) is1) 4 2) 5 3) 0.8 4) 1.25

72. The length of two wires are in the ratio 3 : 4.Ratio of the diameters is 1:2; young's modulusof the wires are in the ratio 3:2; If they aresubjected to same tensile force, the ratio of theelongation produced is1) 1 :1 2) 1 :2 3) 2 : 3 4) 2 : 1

73. A solid sphere hung at the lower end of a wire issuspended from a fixed point so as to give anelongation of 0.4mm. When the first solid sphereis replaced by another one made of samematerial but twice the radius, the new elongationis1) 0.8mm 2)1.6mm 3) 3.2mm 4)1.2mm

74. The elongation produced in a copper wire of length2m and diameter 3mm, when a force of 30N isapplied is [Y=1x1011N.m-2]1) 8.5mm 2) 0.85mm 3) 0.085mm 4) 85mm

75. The strain is 0.01%. Young's modulus for the materialof the wire is 20 x 1010 N/m2. The stress on thewire is1) 5 x 104 N/m2 2) 20 x 104 N/m23) 20 x 106 N/m2 4) 20 x 108 N/m2

76. A copper wire and a steel wire of radii in theratio 1:2 lengths in the ratio 2:1 are stretchedby the same foces. If young's modulus ofcopper 11 21.1 10 /N m , young's modulusof steel = 11 22 10 /N m . Ratio of theirextensions is1) 160 : 11 2) 16 : 110 3) 1 : 61 4) 6 : 11

PRACTICE QUESTIONS77. Two steel wires have equal volumes. Their

diameters are in the ratio 2 : 1. When same forceis applied on them, the elongation produced willbe in the ratio of1) 1:8 2) 8:1 3) 1:16 4) 16:1

78. The diameters of two steel wires are in the ratio 2: 3. Their lengths are equal. When same force isapplied on them, the ratio of the elongationproduced is

1) 4 : 9 2) 9 : 4 3) 3 : 2 4) 2 : 379. The force that must be applied to a steel wire 6m

long and diameter 1.6mm to produce anextension of 1mm [y=2.0 x 1011 N.m -2] isapproximately.1) 100N 2) 50N 3) 67N 4) 33.5N

80. Two wires A and B of same material have lengthsin the ratio 1:2 and diameter in the ratio 2:1. Ifthey are stretched by same force, the ratio ofthe increase in their lengths will be1) 2:1 2) 1:4 3) 8:1 4) 1: 8

81. The area of cross section of a wire is 10-6m2.When its length is increased by 0.1%, a tensionof 1000N is produced. The young's modulus ofthe wire will be1) 1012N/m2 2) 1011N/m23) 1010N/m2 4) 109N/m2

82. A steel wire of length 1m has cross sectional area1cm2. If young's modulus of steel is 11 210 /N m ,then force required to increase the length of wireby 1mm will be1) 1011N 2) 107N 3) 104N 4) 102N

83. Young's modulus of material of a wire is 200G.Pa.The stress required to produce a strain of 4 x 10-3 in pa is1) 8 x 1012 2) 0.2 x 10-113) 5 x 107 4) 8 x 108

84. A wire of 10m long and 1mm2 area of cross sectionis stretched by a force of 20N. If the elongation is2mm, the young's modulus of the material of thewire (in pa) is1) 1 x 109 2) 2 x 10-9 3) 1 x 1011 4) 1 x 1012

85. Ratio of lengths of two brass wires is 3 : 4; theirareas of cross section are in the ratio 2:3. Whensame force is applied on them, the elongationsproduced will be in the ratio:1) 9 : 8 2) 8 : 9 3 ) 22 :3 4) 1 : 1

86. The ratio of lengths of two wires made of samematerial is 2 :3. The ratio of their respectivelongitudinal stress to produce same elongationis1) 4 : 9 2) 9 : 4 3) 2 : 3 4) 3: 2

87. The length of a wire of cross sectional area1 x 10-6m2 is 10m. The young's modulus of thematerial of the wire is 25 G.pa. When the wire issubjected to a tensile force of 100N, the elongationproduced in mm is1) 0.04 2) 0.4 3) 4 4) 40

ELASTICITYBULK MODULUS OF ELASTICITY

MODEL QUESTIONS88. The fractional change in the volume of oil is

1 percent when a presure of 2 x 107 N/m2 isapplied. The bulk modulus and its compressibilityis1) 3 x 108 N/m2, 0.33 x 10-9 m2/N2) 5 x 109 N/m2, 1010x2 m2/N3) 2x 109 N/m2, 1010x5 m2/N4) 3 x 10+9 N/m2, 910x5 m2/N

89. A volume of 3 310 m is subjected to a pressureof 10 atmospheres. The change in volume is

6 310 m . Bulk modulus of water is (Atmo-sphere pressure = 5 21 10 /N m )1) 9 21 10 /N m 2) 10 21 10 /N m3) 12 21 10 /N m 4) 7 21 10 /N m

PRACTICE QUESTIONS90. The compressibility of water is 4 x 10-5 per unit

atmosphere pressure. The decrease in volumeof 100cm3 water under a pressure of 100atm willbe1) 0.4 cm3 2) 4 x 10-5cm33)0.025 cm3 4) 0.04cm3

SHEAR MODULUS OR THE MODULUS OF RIGIDITYMODEL QUESTIONS

91. A metal cube of side length 8.0 cm has its upersurface displaced with respect to the bottom by0.10 mm when a tangential force of 4 x 109 N isapplied at the top with bottom surface fixed. Therigidity modulus of the material of the cube is

(2001 M)1) 4 x 109 N/m2 2) 5 x 109 N/m23) 8 x 109 N/m2 4) 1x 108 N/m2

92. A wire of 1m length and 4mm radius is clampedat upper end. The lower end is twisted by anangle of 300. The angle of shear is1) 0.120 2) 1.20 3) 120 4) 0.0120

PRACTICE QUESTIONS93. A tangential force of 2100 N is applied on a

surface of area 6 23 10 m which is 0.1m froma fixed face. The force produces a shift of 7mmof upper surface with respect to bottom. Therigidity modulus of material is

1) 10 210 /N m 2) 8 210 /N m3) 6 210 /N m 4) 12 210 /N m

94. A cube of gelatin of 7cm side on an edge isresting in a dish with bottom face fixed. Pushinghorizontally on the top face with a force of 0.21Ncauses the top to undergo a displacement of3mm. Then, the rigidity modulus of gelatin is1) 100N/m2 2) 10N/m2 3) 1000N/m2 4) 104N/m2

POISSONS RATIOMODEL QUESTIONS

95. A rod has poisson's ratio 0.2. If a rod suffers alongitudinal strain of 2 x 10-3, then thepercentage change in volume is1) +0.12 2) -0.123) 0.28 4) -0.28

96. The fractional increase in volume of a wire ofcircular cross section, if its lingitudinal strainis 1% 0.3 1) 0.4 2) 0.04 3) 0.0044) 4

PRACTICE QUESTIONS97. A 3 cm long copper wire is stretched to

increase its length by 0.3cm. If poisson's ratiofor copper is 0.26, the lateral strain in the wireis1) 0.26 2) 2.6 3) 0.026 4) 0.0026

98. A wire is subjected to a longitudinal strain of 0.05.If its material has a Poisson's ratio 0.25, thelateral strain experienced by it is (1986 E)1) 0.00625 2) 0.125 3) 0.0125 4) 0.0625

IMPORTANT RELATIONS BETWEEN Y,K AND MODEL QUESTIONS

99. Young's modulus of a metal is 15 x 1011 pa. If itspoisson's ratio is 0.4. The bulk modulus of the metalin pa is1) 25 x 1011 2) 2.5 x 10113) 250 x 1011 4) 0.25 x 1011

100. Y, k, n represent respectively the young'smodulus,bulk modulus and rigidity modulus of abody. If rigidity modulus is twice the bulk modulus,then (1999 M)1) Y = 5k/18 2) Y = 5n/93) Y = 9k/5 4) Y = 18k/5

101. For a given material the Young's modulus is2.4 times that of its rigidity modulus. Its Poisson'sraio is (1990 M)1) 2.4 2) 1.2 3) 0.4 4) 0.2

ELASTICITYPRACTICE QUESTIONS

102. For a material Y = 6.6x1010 N/m2 and bulk modulusK = 11x1010N/m2, then its Poissons's ratio is1) 0.8 2) 0.35 3) 0.7 4) 0.4

103. If the poisson's ratio of a solid is 2/5, then theratio of its young's modulus to the rigiditymodulus is1) 5/4 2) 7/15 3) 14/9 4) 14/5

ENERGY STORED IN A WIRE , ELASTIC FATIGUE,STRAIN ENERGY,THERMAL STRESS

MODEL QUESTIONS104. When the load on a wire is slowly increased from

3 to 5kg wt, the elongation increases from 0.61to 1.02 mm. The work done during the extensionof wire is (g=10ms-2)1) 0.16J 2) 0.016J 3) 1.6J 4) 16J

105. The 'Y' of a material of a rod is 200G pa. Itscoefficient of linear expansion is 15 x 10-6/0C. Itis fixed at two ends and is cooled by 1000C. Thethermal stress produced in the rod (in 2. N m )is1) 3 x 10-7 2) 3 x 108 3) 3 x 106 4) 0.33 x 107

106. Two rods of different materials havingcoefficients of linear expansion 21 : andYoung's modulii Y1 : Y2 are fixed between tworigid massive walls. The rods are heated suchthat they undergo the same increase intemperature. There is no bending of the rods. If

21 : =2 :3, the thermal stress developed inthe rods are equal, provided Y1 : Y2 is equla to1) 2 : 3 2) 1 : 1 3) 3 : 2 4) 4 : 9

PRACTICE QUESTIONS107. A uniform wire of length 4m and area of cross

section 2mm2 is subjected to longitudinal forceproduced an elongation of 1mm.If Y=0.2 x 1011 NM-2, elastic potential energystored in the body is1) 0.5J 2) 0.05J 3) 0.005J 4) 5.0J

108. A metallic rod undergoes a strain of 0.05%. Theenergy stored per unit volume is(Y=2 x 1011

2Nm )1) 0.5 x 104 Jm-3 2) 0.5 x 105J m -33) 2.5 x 105 J m-3 4) 2.5 x 104 Jm-3

109. A steel wire of length 4m is stretched by a forceof 100N. The work done to increase the length ofthe wire by 2mm is1) 0.4J 2) 0.2J 3) 0.1J 4) 1J

LEVEL - IIYOUNG'S MODULUS OF ELASTICITY

MODEL QUESTIONS110. A load of 4.0 kg is suspended from a ceiling

through a steel wire of length 20m and radius2.0mm. It is found that the length of the wireincreases by0.031 mm. as equilibrium is achieved.If g=3.1 xms-2 , the value of young's modulusin N.m-2 is1) 2.0 x 1012 2) 4.0 x 10113) 2.0 x 1011 4) 0.02 x 109

111. Two wires of equal cross section, but one madeup of steel and the other copper are joined endto end. When the combination is kept undertension, the elongations in the two wires arefound to be equal. If Ysteel = 2.0 x 1011 N.m-2 andYcopper = 1.1 x 1011 Nm-2, the ratio of the lengths ofthe two wires is1) 20 : 11 2) 11:20 3) 5 : 4 4) 4 : 5

112. If Young's modulus of iron be 2 x 1011 N/m-2andinteratomic distance be 3 x 10 -10m, theinteratomic force constant will be1) 60 N/m 2) 120 N/m3) 30 N/m 4) 180 N/m

113. Two exactly similar wires of steel (y=20 x 1011dyne/cm2) and copper (y = 12 x 1011 dyne/cm2)are stretched by equal forces. If the totalelongation is 1cm, elongation of copper wire is1) 3/5 cm 2) 5/3 cm 3) 3/8 cm 4) 5/8 cm

114. Two wires A and B of the same dimensions areunder loads of 4 and5.5 kg respectively. The ratioof Young's moduli of the materials of the wiresfor the same elongation is1) 64 : 121 2) 8:11 3) 11:8 4) 8 : 111

115. A load of 1kg weight is attached to one end of asteel wire of cross sectional area 3 2mm andYoungs modulus 1110 N/ 2m . The other end issuspended vertically from a hook on a wall, thenthe load is pulled horizontally and released.When the load passes through its lowest positionthe fractional change in length is (g=10m/ 2s )

(2008 E)1) 410 2) 310 3) 310 4) 410

116. The radii and youngs modulus of two uniform wiresA & B are in the ratio 2:1 and 1:2 respectively.Both the wires are subjected to the samelongitudinal force. If increase in the length of wireA is 1% . Then the percentage increase in lengthof wire B is (2005 E)1) 1 2) 1.5 3) 2 4) 3

ELASTICITY117. A metallic ring of radius 'r' cross sectional area

'A' is fitted in to a wooden circular disk of radius'R' (R>r). If the Young's modulus of the materialof the ring is 'Y', the force with which the metalring expands is : (2004 M)

1) rAYR 2) r

)rR(AY

3) Ar)rR(Y 4) Ar

YR

118. When a tension F is applied, the elongationproduced in uniform wire of length 'L', radius 'r' is'e'. When tension 2 F is applied, the elongationproduced in another uniform wire of length '2L'and radius '2r' made of same material is

(2000 M)1) 0.5e 2) 1.0e 3) 1.5e 4) 2.0e

119. When a uniform wire of radius r is stretched by a2 kg weight, the increase in its length is2.00 mm. If the radius of the wire is r/2 and otherconditions remain in the same, increase in itslength is (2000 E)1) 2.00mm 2) 4.00 mm 3) 6.00 mm 4) 8.00mm

120. The elongation of a steel wire stretched by a forceis 'e'. If a wire of the same material of double thelength and half the diameter is subjected todouble the force, its elongatin will be

(1999 E)1) 16e 2) 4e 3) (1/4)e 4)(1/16)e

121. When a certain force is applied on a string itextends by 0.01cm. When the same force isapplied on another string of same material, twicethe length and double the diameter, then theextension in second string is (1995 M)1) 0.005 cm 2) 0.02 cm 3) 0.08 cm 4) 0.04 cm

122. An aluminium wire and steel wire of the samelength and cross section are joined end to end.The composite wire is hung from a rigid supportand a load is suspended from the free end. Theyoung's modulus of steel is 20/7 times thealuminium. The ratio of increase of length of steeland aluminium is (1993 M)1) 20/7 2) 400/49 3) 7/20 4) 49/400

123. A tensile force of 2 x 105 dynes doubles the lengthof an elastic cord whose area of cross-section is 2 sq. cm. Young's modulus of the material of thecord is (1985 M)

1) 1 x 105 dynes /cm2 2) 2 x 105 dynes /cm23) 0.5 x 105 dynes /cm2 4) 4 x 105 dynes /cm2

124. Four wires P,Q,R and S of same materials havediameters and stretching forces as shown below.Arrange their strains in the decreasing order.Wire Diameter Stretching force P 2 mm 10 N Q 1 mm 20 N R 4 mm 30 N S 3 mm 40 N

1) Q,S,P,R 2) R,P,S,Q 3) P,Q,R,S 4) P,R,Q,S125. What percent of length of a wire will increase by

applying a stress of 1 kg. wt/mm2 on it. [Y=1x1011Nm-2 and 1 kg wt = 9.8N]1) 0.0078% 2) 0.0088%3) 0.0098% 4) 0.0067%

126. The greatest length of the wire made of materialof breaking stress 8 28 10 /N m and density

3 38 10 /kg m that can be suspended from arigid support without breaking is 210 /g m s1) 1km 2) 0.1km 3) 10km 4) 100km

127. A lift is tied with thick iron wire and its mass is1000kg. If the maximum acceleration of the liftis 1.2 ms-2 and the maximum stress of the wireis1.4 x 108 Nm-2 what should be the minimumdiameter of the wire ?1) 10-2m 2) 10-4m 3) m10 6 4) m10x5.0 2

PRACTICE QUESTIONS128. The stress produced in a wire of young's modulus

1.2 x 1011 N.m-2 is 7 x 106 pa. The strain itundergoes is1) 5.8 x 10-5 2) 3.5 x 10-3 3) 3.5 4) 0.35

129. A steel wire is 1 m long and 1mm2 in area ofcross-section. If it takes 200N to stretch thiswire by 1mm, the force that will be required tostretch the wire of the same material and cross-sectional area from a legnth of 10m to 1002 cm1) 100N 2) 200 N 3) 400 N 4) 2000N

130. When load is applied to a wire, the extension is3mm. The extension in the wire of same materialand length but half the radius extended by thesame load is :1) 0.75mm 2) 6mm 3) 1.5mm 4) 12.0mm

131. An iron wire of length 4m and diameter 2 mm isloaded with a weight of 8kg. If the young's modulus'Y' for iron is 2 x 1011 N/m2, then the increase inthe length of the wire is (1998 M)1) 0.2 mm 2) 0.5mm3) 2mm 4) 1 mm

132. A wire whose cross sectional area is 2 mm2 isstretched by 0.1 mm by a certain load. If a similarwire of triple the area of cross section is stretchedby the same load, the elongation of the secondwire would be (1996 E)1) 0.33m 2) 0.033mm3) 0.3mm 4) 0.0033 mm

133. Two wires made of same material have theirlengths L and 2 L and the radii 2R and R. If theyare stretched by the same force, their extensionsare e1 and e2, then e1 : e2 (1994 M)1) 1:8 2) 8:1 3) 1:4 4) 2:1

134. The extension of a wire by the application of aload is 0.3 cm. The extension in the wire of thesame material but of double the length and halfthe radius of cross section in cm is (1993 E)1) 3 2) 0.3 3) 1.2 4) 2.4

135. The Y of a material having a cross sectional areaof 1 cm2 is 2 x 1012 dynes/cm2. The forcerequired to double the length of the wire is

(1989 M)1) 1 x 1012dynes 2) 2 x 1012dynes3) 0.5 x 1012dynes 4) 4 x 1012dynes

136. A wire extends by 1 mm when a force is applied.Double the force is applied to another wire of samematerial and length but of half the radius of cross-section. The elongation of wire in mm will be

(1986 M)1) 8 2) 4 3) 2 4) 1

137. A stress of 106 N/m2 is required for breaking amaterial. If the density of the material is3 x 103 kg/m3, then what should be the minimumlength of the wire made of the same material sothat it breaks by its own weight ? (g=10m/s2)1) 33.4 m 2) 3.4 m3) 34 cm 4) 3.4 cm

138. The breaking force for a wire of radius 'r' is 'F'.The breaking force for the wire of same materialbut double the length and double the radius is1) F 2) 2F 3) 4F 4) 8F

139. One end of a wire 2m long and 0.2cm2 in crosssection is fixed to a ceiling and a load of 4.8kgis attached to its free end. The elongation in thewire in mm is (if y = 2.0 x 1011 N.m-2, g =10ms-2)1) 2.4 x 10-5 2) 2.4 3) 0.024 4) 0.0024

140. A rope of 1 cm in diameter breaks if tension in itexceeds 500N. The maximum tension that may begiven to a smaller rope of diameter 2cm (in N) is1) 500 2) 250 3) 1000 4) 2000

141. A wire can be broken by 400kg.wt. The loadrequired to break the wire of double the thicknessof the same material will be1) 800 kg.wt 2) 1600 kg.wt.3) 3200 kg. wt 4) 6400 kg. wt

142. A wire elongates by l mm when a load W ishanged from it. If the wire goes over a pulley andtwo weights W each are hung at the two ends,the elongation of the wire will be (in mm)

(AIEEE-2006)1) zero 2) / 2l 3) l 4) 2l

BULK MODULUS OF ELASTICITYMODEL QUESTIONS

143. A sphere contracts in volume by 0.01% whentaken to the bottom of lake 1km deep. If thedensity of water is 1gm/cc, the bulk modulus ofwater is1) 9.8 x 105N/m2 2) 9.8 x 108N/m23)9.8 x 1010N/m2 4) 9.8 x 106N/m2

144. A sphere contracts in volume by 0.01%. Whensubjected to a pressure of 100 atmosphere.The bulk modulus of the material of the ball is(1 atm = 105pa)1) 1010N/m2 2) 109 N/m23) 5 x 1010N/m2 4) 1011N/m2

145. The increase in pressure required to decreasethe 200 litres volume of a liquid by 0.004% inkPa is : (bulk modulus of the liquid = 2100 MPa)

(2002 M)1) 8.4 2) 84 3) 92.4 4) 168

146. A copper solid cube of 60mm side is subjectedto a compressible pressure of 2.5 x 107 Pa. Ifthe bulk modulus of copper is 1.25 x 1011 pascals,the change in the volume of cube is (1997 M)1) -43.2 mm3 2) -43.2m33) -43.2 cm3 4) -432 m m3

PRACTICE QUESTIONS

ELASTICITY147. On taking a solid rubber ball from the surface to

the bottom of a lake 200m deep, the reductionin volume is found to be 0.5%. If the density ofwater is 103 kgm-3 and g=10 ms-2, find the bulkmodulus of rubber.1) Pa10x2 8 2) 4 x 108 Pa3) Pa10x6 8 4) Pa10x8 8

148. Bulk modulus of water is 2 x 109 N/m2. Thepressure required to increase the density of waterby 0.1% in N/m2 is : (2003 M)1) 2 x 109 2) 2 x 1083) 2 x106 4) 2 x 104

149. A ball falling in a lake to a depth 200m showsa decrease of 0.1% in its volume at the bottom.the bulk modulus of the ball is1) 19.6 x 108 N/m2 2) 19.6 x 10-10 N/m23) 19.6 x 1010 N/m2 4) 19.6 x 10-8 N/m2

POISSONS RATIOMODEL QUESTIONS

150. When a wire is subjected to a force along itslength, it length increases by 0.4% and its radiusdecreases by 0.2% . Then the Poisons ratio ofthe material of the wire is ( 2008 E)

1) 0.8 2) 0.5 3) 0.2 4) 0.1151. When a wire of length 10m is subjected to a force

of 100 N along its length , the lateral strainproduced is 0.01X 10-3. The Poissons ratio wasfound to be 0.4. If the area of cross-section ofwire is 0.025 2m , its Youngs modulus is

( 2007 E)1) 1.6X 8 210 /N m 2) 10 22.5 10 /X N m3) 11 212.5 10 /X N m 4) 9 216 10 /X N m

PRACTICE QUESTIONS152. The increase in length of a wire on stretching is

0.025%. If its Poisson's ratio is 0.4, then thepercentage decrease in the diameter is

(2004 M)1) 0.01 2) 0.02 3) 0.03 4) 0.04

153. The Poisson's ratio of a material is 0.4. If a forceis applied to a wire of this material, there is adecrease of cross-sectional area by 2%. Thepercentage increase in its length is : (2002 M)1) 3% 2) 2.5% 3) 1% 4) 0.5%

ENERGY STORED IN A WIRE , ELASTIC FATIGUE,

STRAIN ENERGY,THERMAL STRESSMODEL QUESTIONS

154. A rubber cord of length 40cm and area of crosssection 4 x 10-6 m2 is extended by 10cm. If theenergy gained is 20 joule, young's modulus ofrubber is1) 108 Nm-2 2) 2 x 108 Nm-23) 4 x 108 Nm-2 4) 1.5 x 106Nm-2

155. If the work done in stretching a wire by 1mm is2 J, the work necessary for stretching anotherwire of same material but withdouble radius ofcross section and half the length by 1mm is injoules (1991 M)1) 16 2) 8 3) 4 4) 1/4

156. A steel wire of mass 3.16 Kg is stretched to atensile strain of 1 x 10-3. What is the elasticdeformation energy if density =7.9 g/cc andY=2x1011 N/m21) 4 KJ 2) 0.4 KJ3) 0.04KJ 4) 4J

PRACTICE QUESTIONS157. A stress 107 pa produces a strain of 4 x 10-3. The

energy stored per unit volume of the body (in J.m-3) is1) 2 x 103 2) 2 x 104 3) 2.5 x 1010 4) 0.8 x 104

158. Two wires of same material and same diameterhave lengths in the ratio 2:5. They are strechedby same force. The ratio of workdone in strechingthem is (2005 E)1) 5 : 2 2) 2 : 5 3) 1:3 4) 3 : 1

159. A steel wire of length 5m is pulled to have anextension of 1mm. Its Y is 1.9 x 104 N/m2. Theenergy per unit volume stored in it is (1994 E)1) 3.8 x 10-4 J/m3 2) 7.6 x 10-4 J/m33) 1.9x10-4 J/m3 4) 0.95 x 10-4 J/m3

160. A wrie suspended bertically from one of its endsis stretched by attaching a weight of 200 N tothe lower end. The weight stretches the wrie by1 mm. Then the elastic energy stored in the wireis (AIEEE-2003)1) 0.1 J 2) 0.2 J 3) 10 J 4) 20 J

161. A spring of spring constant 3 15 10 Nm isstretched initially by 5 cm from the unstretchedposition. Then the work required to stretch itfurther by anotehr 5 cm is AIEEE-2003)1) 6.25 Nm 2) 12.50 Nm3) 18.75 Nm 4) 25.00 Nm

162. A wire fixed at the upper end stretches by length

ELASTICITYl by applying a force F. The work done instretching is (AIEEE-2004)1) / 2F l 2) Fl 3) 2Fl 4) / 2Fl

163. If S is stress and Y is Youngs modulus ofmaterial of a wire, the energy stored in the wireper unit volume is (AIEEE-2005)

1) 2SY 2) 2

2YS 3) 3

2YS 4) 22S Y

LEVEL - IIIMODEL QUESTIONS

164. A uniform metal rod of 2mm2 cross section isheated from 00C to 200C. The coefficient of linearexpansion of the rod is 12 x 10-6/0C. Its young'smodulus of elasticity is 1011 N/m2. The energystored per unit volume of the rod is1) 2880 J/m3 2) 1500 J/m33) 5760 J/m3 4) 1440 J/m3

165. A brass rod has a length of 0.2m, area of crosssection 1.0cm2 and young's modulus 1011 Nm-2.If it is compressed by 5kg.wt. along its length,then the change in its energy will be1) an increase of 2.4 x 10-5 J2) a decrease of 2.4 x 10-5J3) an increase of 2.4 x 107 J4) a decrease of 2.4 x 107 J

166. The rubber cord of a catapult has a cross-sectionalarea 1mm2 and total unstretched length 10cm.It is stretched to 12cm and then released toproject a missile of mass 5gm. Taking 'Y' forrubber 5 x 108 N/m2, the velocity of projection is1) 10m/s 2) 15 m/s 3) 20 m/s 4) 25 m/s

167. On loading a metal wire of cross section610 sq.m. and length 2m by a mass of 210 kg,

it extends by 16mm and suddenly broke fromthe point of support. If density of that metal is8000K 3gm and its specif ic heat is420 1 1JKg K , the rise in temperature of wireis1) 02.5 C 2) 05 C 3) 06 C 4) 010 C

168. A wire of length 1 m fixed at one end has a sphereattached to it at the other end. The sphere isprojected horizontally with a velocity of g9 .When it describes a vertical circle, the ratio ofelongations of the wire when the sphere is at thetop and bottom of the circle is

1) 2 : 5 2) 5 : 2 3) 3 : 5 4) 5 : 3169. A piece of copper wire has twice the radius of

steel wire. One end of the copper wire is joinedto one end of steel wire so that both of them canbe subjected to the same longitudinal force. Yfor steel is twice that of copper. When the lengthof copper wire is increased by 1%, the steel wirewill be stretched by1) 2% of its original length2) 1% of its original length3) 4% of its original length4) 0.5% of its original length

170. A 20 kg load is suspended from the lower end of awire 10cm long and 1mm2 in cross sectional area.The upper half of the wire is made of iron and thelower half with aluminium. The total elongation inthe wire is(Yiron = 20 x 1010 N/m2, YAl = 7 x 1010 N/m2)1) 18.9 x 10-3m 2) 17.8 x 10-3m3) 1.78 x 10-3 m 4) 1.89 x 10-4 m

171. A body of mass 10kg is attached to a wire 0.3mlong. Its breaking stress is 4.8 x 107 N/m2. Thearea of cross section of the wire is 10-6m2. Themaximum angular velocity with which it can berotated in a horizontal circle without breaking is1) 2 rad/s 2) 4 rad/s 3) 6 rad/s 4) 8 rad/s

172. A mass 'm' kg is whirled in a vertical plane bytieing it at the end of a flexible wire of length 'I'and area of cross-section 'A' such that it justcompletes the vertical circle. When the mass isat its lowest position, the strain produced inthe wire is (Young's modulus of the wire is 'Y')1) AY/ 6 mg 2) 6mg/AY 4) 5mg/AY 4) AY/5mg

173. The length of an elastic spring is 'a' when tensionis 4 N and 'b' when the tension is 5N. The lengthof the spring when tensionis 9N is1) 4a - 5b 2) 5b - 4a 3) 5b + 4a 4) 5(b-a)

174. A steel wire is suspended vertically to a rigidsupport. When loaded with a weight in air, itextends by la and when the weight is immersedcompletely in water, the extension is reduced tolw Then, the relative density of the material ofweight is

1. aw

ll 2)

aa w

ll l 3)

wa w

ll l 4)

wa

ll

175. The length of a metal wire is 1l when the tension

ELASTICITYin it is 1F and 2l when the tension is 2F . Thennatural length of the wire is

1) 1 1 2 21 2

l F l FF F 2)

2 12 1

l lF F

3) 1 2 2 12 1

l F l FF F

4)

1 1 2 22 1

l F l FF F

176. One end of uniform wire of length L and weight Wis attached to rigid point in the roof and a weightw1 is suspended from its lower end. If S is thearea of cross-section of the wire the stress inthe wire at a height (3L/4) from its lower end is

1) Sw1 2)

S4ww1

3)

S4w3w1

4) S

ww1

177. A uniform pressure 'P' is exerted on all sides of asolid cube at temperature 00C. In order to bringthe volume of the cube to the original volume,the temperature of the cube must be increasedby t0c. If is the linear coefficient and K thebulk modulus of the material of the cube, then tis equal to

1) KP3 2) K2

P 3) K3

P 4) K

P

178. A solid sphere of radius R made of material ofbulk modulus K is surrounded by a liquid in acylindrical container. A massless piston of areaA floats on the surface of liquid. When a mass'm' is placed on the piston to compress the liquid,the fractional change in the radius of the sphere

RR is

1) AKmg 2) AK3

mg 3) Amg 4) AK

mg3

PRACTICE QUESTIONS179. A steel wire of length 20cm and cross sectional

area 1.0 mm2 is clamped at both the ends. Thecoefficient of linear expansion for steel = 1.1 x10-5/0C and Y=2 x 1011 N/m2. If the temperature ofwire is reduced from 400C to 200C, the tensiondeveloped in the wire will be1) 2.2 x 106N 2) 8 N 3) 16N 4) 44N

180. A 8m long string of rubber, having density

1.5 x 103 kg/m3 and young's modulus 5 x 106N/m2 is suspended from the ceiling of a room. Theincrease in its length due to its own weight willbe (g=10m/s2)1) 9.6 x 10-2m 2) 19.2 x 10-5m3) 9.6 x 10-3m 4) 9.6 m

181. A force of 15N increases the length of a wire by1mm. The additional force required to increasethe length by 2.5mm in N is1) 22.5 2) 37.5 3) 52.5 4) 75

182. When a sphere of radius 2 cm is suspended atthe end of a wire, elongation is 'e'. When thesame wire is loaded with a sphere of radius 3cm and made of the same material, theelongation would be :

1) e278 2) e8

27 3) e94 4) e4

9

183. A wire suspended from one end carries a sphereat its other end. The elongation in the wirereduces from 2 mm to 1.6mm on completelyimmersing the sphere in water. The density ofthe material of the sphere is1) 3200 kg/m3 2) 800 kg/m33) 1250 kg/m3 4) 5000 kg/m3

184. Two bars A and B of circular section and of thesame volume and made of the same materialare subjected to tension. If the diameter of A ishalf that of B and if the force supplied to boththe rods is the same and is within the elasticlimit, the ratio of the extension of A to that of Bwill be1) 16 2) 8 3) 4 4) 2

185. When a mass is suspended from the end of awire the top end of which is attached to the roofof the lift, the extension is e when the lift isstationary. If the lift moves up with a constantacceleration g/2, the extension of the wire wouldbe

1) 23e 2) 32

e 3) 2e 4) 3e186. Two wires of different materials each of length l

and cross sectional area A are joined in seriesto form a composite wire. If their Youngs moduliiare Y and 2Y, the total elongation produced byapplying a force F to stretch the composite wire.1) 3FA/2Yl 2) 2FA/3Yl3) 2FA/3AY 4) 3Fl/2AY

187. The volume of oil contained in a certain hydraulic

ELASTICITYpress is 0.2m3. The compressibility of oil is 20 x 10-6per atmosphere. The decrease in volumeof the oil when subjected to 200 atmospheres is(1 atmosphere = 1.02 x 105 N/m2)1) 4 x 10-4m3 2) 8 x 10-4m33) 16 x 10-4m3 4) 2 x 10-4m3

LEVEL-IV188. If the force constant of a wire is K, the work done

in increasing the length l of the wire by l is1) 3K l 2) 2K l3)

2

2Kl 4)

2

3Kl

189. The density of water at the surface of the oceanis . If the bulk modulus of water is B, what isthe density of ocean water at a depth where thepressure is onP , where oP is the atmosphreicpressure

1) 01B

B n P

2) 01B

B n P

3)0

BB nP 4) 0

BB nP

190. A light rod of length L is suspended from a supporthorizontally by means of two vertical wires A andB of equal length as shown in Fig. The cross-sectional area of A is half that of B and theYoungs modulus of A is twice that of B. A weightW is hung as shown. The value of x so that Wproduces equal stress in wires A and B is

1) 3L 2) 2

L 3) 23L 4) 34

L

191. The density of a metal at a normal pressure .Its density when it is subjected to an excesspressure p is ' . If B is the bulk modulus ofthe metal, the ratio '/ is

1)1

1 /p B 2) 1PB

3)1

1 /B p 4) 1 /B p192. A thin uniform metallic rod of mass M and length

L is rotated with a angular velocity in ahorizontal plane about a vertical axis passingthrough one its ends. The tension in the middleof the rod is

1) 212ML 2)21

4ML

3) 218ML 4)23

8ML193. Two wires of same material but radii 1r and 2r

support a mass M in between. If a force of13F Mg is applied then

1) for 1 2r r , 2 should break before 12) for 1r less than 22r , 2 should break before 13) data is insufficient4) for 1 22r r , any of the two may break

194. Three blocks, each of same mass m, areconnected with wire 1W and 2W of same cross-sectional area a Youngs modulus Y. Neglectingfriction the strain developed in wire 2W is

1) 23mgaY 2)

32mgaY

3) 13mgaY 4)

3mgaY

ELASTICITY195. A steel wire of diametr d, area of cross-section A

and length 2l is clapmed firmly at two points Aand B which are 2l m apart and in the sameplane. A body of mass m is hung from themiddle point of wire such that the middle pointsags by x lower from original position. If Youngsmodulus is Y then m is given by

1)2

212YAxgl 2)

22

12YAlgx

3)3

3YAxgl 4)

22

YAlgx

196. A wire of radius r, Youngs modulus Y and lengthl is hung from a fixed point and supports a heavymetal cylinder of volume V at its lower end. Thechange in length of wire when cylinder isimmersed in a liquid of density is in fact1) decreases by 2

Vl gY r

2) increases by 2Vr gY l

3) decreases by V gY r

4) increases by V grl

197. Two blocks of masses m and 2m are connectedthrough a wire of breaking stress S passing overa frictionless pulley. The maximum radius of wireused so that the wire may not break is

1) 34mgS 2)

43mgS

3) 43mgS 4)

12mgS

198. A stone of 0.5 kg mass is attached to one end ofa 0.8 m long aluminium wire of 0.7 mm diameterand suspended vertically. The stone is nowrotated in a horizontal plane at a rate such thatwire makes an angle of 085 with the vertical. If

10 27 10Y Nm , 0sin 85 0.9962 and0cos85 0.0872 , the increase in length of

wire is

1) 31.67 10 m 2) 36.17 10 m3) 31.76 10 m 4) 37.16 10 m

199. What is the density of lead under a pressureof 8 22.0 10 /N m , if the bulk modulus of leadis 9 28.0 10 /N m and initially the density oflead is 311.4 /g cm1) 312.89 /gm cm 2) 314 /gm cm3) 311.69 /gm cm 4) zero

200. A light rod of length 2.00 m is suspended fromthe ceiling horizontally by means of two verticalwires of equal length tied to its ends. One of thewires is made of steel and is of cross section

210m and the other is of brass of cross-section3 22 10 m . Find out the position along the rod

at which a weight may be hung to produce.(Youngs modulus for steel is 11 22 10 /N mand for brass is 11 210 /N m )a) equal stress in both wiresb) equal strains on both wires1) 1.33 m, 1m 2) 1m, 1.33 m3) 1.5 m, 1.33 m 4) 1.33m, 1.5 m

ELASTICITY201. A steel rod of length 6.0 m and diameter 20 mm

is fixed between two rigid supports. The stressin the rod ( in kg/cm2) is........., when thetemperature increases by 80oC (take

6 22.0 10 /Y kg cm and612 10 oper C )

a) the ends do not yieldb) the ends yield by 1 mm1) 1587,1920 2) 1920, 15873) 1000,2000 3) 2000, 1000

202. A steel rod of cross-sectional area 216cm andtwo brass rods each of cross-sectional area

210cm together support a load of 5000 kg asshown in figure. Find the stress in the rods(inkg/cm2). Take Y for steel = 6 22.0 10 /kg cmand for brass = 6 21.0 10 /kg cm

1) 120, 161 2) 161, 1203) 120, 140 4) 141, 120

LEVEL - V203. Figure shows the stress-strain graphs for

materials A and B

From the graph it follows that1) material A has a higher Youngs modulus2) material B is more ductile3) material A is more brittle4) material A can withstand a greater stress

204. Two wires A and B have the same cross-sectionand are made of the same material, but the lengthof wire A is twice that of B. Then, for a given load1) the extension of A will be twice that of B2) the extensions of A and B will be equal3) the strain in A will be half that is B4) the strains in A and B will be equal

205. Two wires A and B have equal lengths and aremade of the same material, but the diameter ofwire A is twice that of wire B. Then, for a givenload1) the extension of B will be four times that of A2) the extensions of A and B will be equal3) the strain in B is four times that in A4) the strains in A and B will be equal

206. Two wires A and B have equal lengths and aremade of the same material, but the diameter ofwire A is twice that of wire B. The, for a givenload1) Youngs modulus of A is twice that of B2) Youngs modulus is same for both A and B3) A can withstand a greater load before breakingthan B4) A and B will withstand the same load beforebreaking

207. Choose the correct statements from the following:1) steel is more elastic than rubber2) the stretching of a coil spring is determined bythe Youngs modulus of the wire of the spring.3) the frequency of a tuning fork is determined bythe shear modulus of the material of the fork4) when a material is subjected to a tensile(stretching) stress, the restoring forces arecaused by interatomic attraction

208. A uniform wire of Youngs modulus Y is stretchedby a force within the elastic limit. If S is the stressproduced in the wire and is the strain in it, thepotential energy stored per unit volume is givenby

1) 12 S 2)21

2Y

3)2

2SY 4)

12Y S

ELASTICITY209. A wire AB of length 2l and cross-sectional area

a is stretched without tension between fixedpoints A and B . The wire is pulled at the centreinto shape ACB such that d l . The tensionin the string is T and the strain produced in it is . If the Youngs modulus of the material of wireis Y. Then

1) 2dl 2)

222

dl

3) 2dT aY l

4)222

dT aY l

Passage IQuestions 210 to 213 are based on thefollowing passageElasticity : The atoms in solids are held togetherby interatomic forces.The average locations ofthe atoms in a lattice do not change with time.Since the atoms are almost lacking in mobility,their kinetic energy is negligibly small. It is thislack of mobility which makes a solid rigid. Thisrigidity is the cause of elasticity in solids. In somesolids such as steel, the atoms are boundtogether by larger interatomic forces than insolids such as aluminium. Thus, the elasticbehaviour varies from solid to solid. Even fluidsexhibit elasticity. All material bodies get deformedwhen subjected to a suitable force. The ability ofa body to regain its original shape and size iscalled elasticity. The deforming force per unitarea is called stress. The change in thedimension(length, shape or volume) divided bythe original dimension is called strain. The threekinds of stress are tensile stress, shearing stressand volumetric stress. The corresponding strainsare called tensile strain, shearing strain andvolume strain. According to Hookes law, withinthe elastic limit stress is proportional to strain.The ratio stress/strain is called the modulus ofelasticity.

210. Figure is the load-extension curve for a metallicwire. Over whcih region is Hookes la obeyed ?

1) OA 2) AB 3) BC 4) CD211. In the above,question.the metal shows plastic

behaviour beyound point1) A 2) B 3) C 4) D

212. Figure shows the strain-stress graphs formaterials A and B. From the graph it follows that

1) material A has a higher Youngs modulus than B 2) material B has a higher Youngs modulus than A 3) for a given strain, material B can withstand greater

stress than A 4) for a given stress, the strain produced in material

B is more than that in material A213. Choose the correct statements from the following

1) Steel is more elestic than rubber2) Fluids have Youngs modulus as well as shearmodulus3) Solids have Youngs modulus, bulk modulusas well as shear modulus4) Bulk modulus of water is greater than that ofcopper.Passage IIQuestions 214 to 216 are based on thefollowing passageTwo rods P and Q of different metals having thesame area A and the same length L are placedbetween two rigid walls as shown in fig. Thecoefficients of linear expansion of P and Q are

1 and 2 respectively and their Youngssmodulii are 1Y and 2Y . The temperature of bothrods is now raised by T degrees.

ELASTICITY214. The force exerted by one rod on the other is

1 )

1 21 2

1 1TAF

Y Y

2) 1 2 1 2F TAYY 3) 1 2 1 2F TA Y Y 4) None of these

215. The new length of the rod P is

1) 1 11

1 FL L T AY

2) 1 11

1 FL L T AY

3) 1 11

1 FL L T AY

4) 1 11

1 FL L T AY

216. The new length of rod Q is

1) 2 22

1 FL T AY

2) 2 22

1 FL L T AY

3) 2 22

1 FL L T AY

4) 2 22

1 FL L T AY

Passage IIIQuestions 217 to 220 are based on thefollowing passageOne end of a string of length L and cross-sectionalarea A is fixed to a support and the other end isfixed to a bob of mass m. The bob is revolved ina horizontal circle of radius r with an angularvelocity such that the string makes an angle with the vertical.

217. The angular velocity is equal to

1) sing r 2) cosg r

3) tang r 4) cotg r

218. The tension T in the string is

1) cosmg 2) sin

mg

3) tanmg 4) 2 2 4 1/ 2( )m g r

219. The increase L in length of the string is1) TLAY 2) cos

MgLAY

3) sinMgL

AY 4)MgLAY

220. The stress in the string is

1) mgA 2) 1mg rA L

3) 1mg rA L 4)

mg rA L

Passage IVQuestions 221 to 223 are based on thefollowing passageA thin rod of negligible mass and cross-sectionalarea 6 24 10 m , suspended vertically from oneend, has a length of 0.5 m at 100oC . The rod iscooled to 0oC .Youngs modulus = 11 210 Nm ,coefficient of linear expansion = 5 110 K and

210g ms221. The decrease in the length of the rod on cooling

is1) 42 10 m 2) 43 10 m3) 44 10 m 4) 45 10 m

222. What mass must be attached at the lower end ofthe rod so that the rod is rpevented fromcontracting on cooling1) 40 kg 2) 30 kg3) 20 kg 4) 10 kg

223. The total energy stored in the rod is1) 0.1 J 2) 0.2 J3) 0.3 J 4) 0.4 J

ELASTICITYPassage VQuestions 224 to 226 are based on thefollwoing passageTwo blocks of masses m and M = 2m areconnected by means of a metal wire of cross-sectional area A passing over a frictionless fixedpulley as shown in figure. The system is thenreleased.

224. The common acceleration of the blocks is

1) g 2) 3g 3) 23

g 4) 32g

225. The stress produced in the wire is

1) mgA 2)23mgA 3)

34mgA 4 )

43mgA

226. If 9 21 , 8 10m kg A m , the breaking stress= 9 22 10 Nm and 210g ms , themaximum value of M for which the wire will notbreak is1) 4 kg 2) 6 kg3) 8 kg 4) 10 kg

227. Figure shows the applied force versus extensiongraph for a metal wire.

Column I Column II(a) Point A on curve (p) Yield point(b) Point B on curve (q) Breaking point(c) Portion ABC on curve (r) Elastic limit(d) Point C on curve (s) Plastic rangea b c d

1) r s q p2) s q p r3) r p q s4) q p r s

KEY01.1 02.3 03.2 04.2 05.206.1 07.2 08.2 09.2 10.111.1 12.2 13.2 14.2 15.316.2 17.1 18.3 19.1 20.121.1 22.1 23.3 24.3 25.126.3 27.1 28.2 29.1 30.131.1 32.1 33.3 34.4 35.136.4 37.1 38.4 39.3 40.341.1 42.1 43.1 44.3 45.346.2 47.1 48.4 49.4 50.151.1 52.2 53.2 54.2 55.156. 2 57.3 58.4 59.4 60.261.2 62.1 63.1 64.1 65.166.1 67.3 68.4 69.2 70.171.2 72.4 73.3 74.3 75.376.1 77.3 78.2 79.3 80.481.1 82.3 83.4 84.3 85.186.4 87.4 88.3 89.1 90.191.2 92.1 93.1 94.3 95.196.3 97.3 98.3 99.1 100.4101.4 102.4 103.4 104.2 105.2106.3 107.3 108.4 109.3 110.1111.1 112.1 113.4 114.4 115.1116.3 117.2 118.2 119.4 120.1121.1 122.3 123.1 124.1 125.3126.3 127.1 128.1 129.3 130.4131.2 132.2 133.1 134.4 135.2136.1 137.1 138.3 139.3 140.4141.2 142.3 143.3 144.4 145.2146.1 147.2 148.3 149.1 150.2151.1 152.1 153.2 154.3 155.1156.3 157.2 158.2 159.1 160.1161.3 162.4 163.3 164.1 165.1166.3 167.1 168.1 169.1 170.4171.2 172.2 173.2 174.2 175.3176.3 177.3 178.2 179.4 180.1181.1 182.2 183.4 184.1 185.2186.4 187.2 188.3 189.1 190.3191.1 192.4 193.4 194.1 195.3196.1 197.3 198.1 199.3 200.1201.2 202.1 203.1,4 204.1,4205.1,3 206.2,3 207.1, 4208. 1,2,3209.2,4 210.1 211.2 212. 2,3213.1,3 214.1 215.3 216.3217.3 218.1,4 219.1,2 220.3221.4 222.1 223.1 224.2225.4 226.1 227.1

ELASTICITYHINTS AND SOLUTIONS LEVEL-I

STRESS AND TYPES OF STRESS(MODEL QUESTIONS)

60. F mgS A A (PRACTICE QUESTIONS)

61. F mgS A A

62. 2F mgS A r

STRAIN AND TYPES OF STRAIN(MODEL QUESTIONS)

63. lstrain l

(PRACTICE QUESTIONS)

64. 100 0.01ll

40.01 10100ll

65. 100 0.1ll

3 13 0.0031000V lV l YOUNGS MODULUS OF ELASTICITY(MODEL QUESTIONS)

66. Fle A

67. YAeF l1

100el

68. 2le r

69. FlY Ae70. 2 1e l l

YAeF l

71. 21e r

72. 2le Yr

73. 1 3e n e , 2n 74. 2

Fle r Y75. stress Y strain 76. 2

le r Y(PRACTICE QUESTIONS)

77. 41e r

78. 21e r

79. YAeF l

80. 2le r

81. FlY Ae

82. 2;YAeF A rl

83. FAYestress l

84. FlY Ae

85. le A

86. 1S l

87. Fle AYBULK MODULUS OF ELASTICITY(MODEL QUESTIONS)

88.1;PK compressibilityV K

V

89.PK VV


90. PVV K

ELASTICITYSHEAR MODULUS OR THE MODULUS OF RIGIDITY(MODEL QUESTIONS)

91.FxA l

92. rl


93.3; 0.1 ; 7 10F l m x mxA l

94.2 3; 7 10 ; 3 10F l m x mxA l

POISSONS RATIO

(MODEL QUESTIONS)

95. 100 (1 2 ) 100V lV l

96. 100 (1 2 ) 100V lV l


97.r lr l

98.r lr l

IMPORTANT RELATIONS BETWEEN Y, K, (MODEL QUESTIONS)

99. 3 (1 2 )Y K 100.

9 1 3Y K

101. 2 (1 )Y (PRACTICE QUESTIONS)

102. 3 (1 2 )Y K 103. 2 (1 )Y ENERGY STORED IN A WIRE, ELASTIC FATIGUE,

STRAIN ENERGY, THERMAL STRESS(MODEL QUESTIONS)104. 2 2 1 1

1 ( )2W F e Fe 105. Thermal stress = Y t106. 1 1 2 2Y Y (PRACTICE QUESTIONS)107.

2

2e AYE l

108. 2( )2E Y strainV

109. 12W Fe

LEVEL-IIYOUNG'S MODULUS OF ELASTICITY

(MODEL QUESTIONS)

110. mglY Ae

111. s sc c

l Yl Y

112. oK YrK = interatomic force constantro = interatomic distance

113. scs c

Y ee Y Y e = Total elongation

114. 1 12 2

Y WY W ; W mg

115. 3e mgl AY

116.2

2t B A At A BB

S r YS Yr

117. 2 11

( )YA l lF l

118. 2Fle r

119. 21e r

120. 2Fle r

121. 2le r

122. s s Al AlY e Y e123. FlY Ae

124. 2Fstrain r

ELASTICITY125. 100 100e Fl AY 126. Breaking stress = ldg

127. Breaking stress = 2( )m g ar


128. stressstrain Y

129. YAeF l

130. 21e r

131. mgle AY

132. 21e A

133. 2le r

134. 2le r

135. 2 1 2 11

( ) ; 2YA l lF l ll

136. 2Fle r

137. Breaking stress = ldg138. 2F r139. mgle AY140. 2F r141. 2F r142. In both the cases, the tension in the wire remains

the same. So, elongation will be the same.

BULK MODULUS OF ELASTICITY(MODEL QUESTIONS)

143.h gK VV

144.PK VV

145.VP K V

146. PVV K


147.h gK VV

148. 1 ;1000

d K d Pd d

149.PK VV

POISSONS RATIO(MODEL QUESTIONS)

150.

rrll

151. ;r

Flr Yl Ael


152.r lr l

153. ( / ) ; 2l r r A rl A r

ENERGY STORED IN A WIRE , ELASTIC FATIGUE,STRAIN ENERGY,THERMALSTRESS(MODEL QUESTIONS)

154.2

2e AYE l

155.2

; 2A eAYE El l

156. 2( )2Y mE strain d (PRACTICE QUESTIONS)

157. 12E stress strainV

ELASTICITY158.

21; 2F lE l E AY

159. 2( )2E Y strainV

160. Energy stored 31 1 200 1 102 2F l J

161.2 2

31 10 55 102 100 100W

162. work done = 12 stress strain volume

163. 2 2

2 2stress SE y y

LEVEL-III(MODEL QUESTIONS)

164. 21 ( )2E Y tV

165.2

2F lE AY

166. YAv e ml

167. 2Fet AL S

168. rg4xV;xrgV bottomtop g5Vtop

mg10mgrmvT

2b

bottom

mg4mgrmvT

2t

top

mg10mg4

eebt

169.

1eAYF = constant

ccc

sss 1

eYA1eYA

170. 1 2e e e ,FLe AY

171. 2mr = Breaking stress x AA172. 6 , FF mg strain AY 173. Let 'l' be the original length, then

xA9

bA5

aA4y

a4b5x

174. Fed11mg

mgll

relwa

175. FLY Ae , 2 1e l l

176. Force acting at 43 length from bottom

134F W W

177. t3tVV;

VVPk

178. AKmg

AKF

VV

VV

31

RR

(PRACTICE QUESTIONS)179. F YA t

180.2

2l dge Y

181. F = Kx1 1 21 2

F F FK x x

215 151 2.5

F

2 22.5F N

ELASTICITY182. 1 3e n e , 2 1r nr

183.s

wsaw

ddd

ee

3s

sws m/kg5000dd

dd26.1

184. 41e r

185. 2 1 1 ae e g

186.1 2

1 1Fle A Y Y

187. V P VC 1C compressibilityK

LEVEL-IV188. Total work done= 12 stress strain volume

Force constant

K=force stressarealength l

Kl stressArea

Toral work done= 1 ( )2Kl l A lengthArea l

21 12 2Kl l Kl

189. increase in pressure ( 1)o o oP nP P n P Let V be the volume of a certain mass M ofwater at the surface, so that M V . Thedecrease in volume under pressure P is

V PV B

' V PV V V V B ( )V B PB

'' '

M VV V

190.1 ( )2

A AB B

T a givenT a since the system is in equilibrium

( )A BT x T L x 23Lx

191. pVV B New volume of metal is

' 1pV pV V VB B

'' 11

m VppV V BB

192. Let m be the mass per unit length of the

rod. Then M= mLConsider a small element of length dxlocated at C at a distance x from AThe mass of element of length dx = mdx.The centripetal force at C is

2( )dF mdx x2 2 2 21( ) ( )2

x L

x xF mdx x m L x

Now , Mm L , therefore,2

22

1 12xF ML L

2

2 2 22

1 12xF r L L

Tension in the middle put /2x L

193. stress for wire 1, 1 21

43MgS r and

stress for wire 2, 2 223

MgS r

ELASTICITYfor 1 2 1 2, 4r r S S any of the two wires may breakbecause stress is same

194. '1T ma , where acceleration of system istaken as 'a

'2 1T T ma

'2mg T ma

on solving 1 3mgT

223T mg

22

23

stress T mgW Y aY aY

195.

For equilibrium,2 sinmg T

here sin tan xl

2 2 1/ 2, ( )YA YAT l l x ll l 2

22YAxl

from (i)2

22 2YAx xmg l l or

33

YAxm gl

196.2

..........(1)Y rMg l Wl

2' '..........(2)Y rW ll

2' 'Y rW W l ll

'2

V gll l Y r

197. stress, 2TS r

here T - mg = ma............(i)2mg - T = 2ma..............(ii)

on solving 43T mg

then 243mgS r

198. cosT mg T lY A l

2 cosmgll r Y

199. The changed density'

1 dpB

substituting the value, we have'

89

11.42.0 101 8.0 10

' 311.69 /g cm

200. Given,stress in steel = stress in brass

S BS B

T TA A

S SB B

T AT A

13

10 1 ....................( )2 10 2 i

As the system is in equilibrium, takingmoments about D, we have

(2 )S BT x T x 2 ............( )S

B

T x iiT x

from equation (i) and (ii), we get x = 1.33 m

ELASTICITYstressstrain Y

given strain in steel = strain in brass/ /S S B Bs B

T A T AY Y

3 113 11

(1 10 )(2 10 ) 1(2 10 )(10 )S S SB S s

T A YT A Y

......(iii)from equations (ii) and (iii) we haveX= 1.0 m

201. Given length of the rod 1=6m = 600 cmdiameter of the rod d = 20 cm = 2cmincrease in temperature t = 80oCYoungs modulus 6 22.0 10 /Y kg cm and thermal coefficient of linear expansion

612 10 oper C When the ends do not yieldLet,

6 6 21 (12 10 )(80)(2 10 ) 1920 /kg cm When the ends do not yield by 1mm,increase in length due to increase intemperature l l t

of this 1 mm or 0.1 cm is allowed to expandTherefore, net compression in the rod

( 0.1)netl l t or compressive strain in the rod

0.1netl tl l

2 0.1stress Y t l

substituting the values6 6 2

20.12 10 12 10 80 1587 /600 kg cm

202. given area of steel rod 216sA cm

area of two brass rods22 10 20BA cm

Load, 5000F kgY for steel 6 22.0 10 /AY kg cm Y for brass 6 21.0 10 /BY kg cm

length of steel rod 30Sl cm andlength of brass rod 20Bl cmLet s stress in steel and s stressin brassdecrease in length of steel rod = decreasein length of brass rodor

S BS B

S Bl lY Y

or66

2.0 10 201.0 10 30

S BS B B

B S

Y lY l

4 ........( )3S B i

now using the relation,S S B BF A A or

5000 16 20.......( )S B ii solving equ. (i) and (ii) we get

2120.9 /B kg cm and2161.2 /S kg cm

203 Concepual204. l L

i.e extension is proportional to the length ofthe wire. Hence choice(a) is correct. The

strain in a wire is given by l FL AYsince F, A and Y are the same, the strain inwires A and B will be equal. Thus thecorrect choices are (1) and (2)

205. Area of cross-section 2 / 4A d , where dis the diameter of the wire.

Therefore 24FLl d Y

since F, L and Y are the same for wires Aand B, 21/l d , i.e the extension isinversely proportional to the square of thediameter. Hence choice (a) is correct. Thestrain is

24l F

L d Y

ELASTICITYThus, strain = 21/ d . Hence the correctchoices are (1) and (3)

206. Since wires A and B are made of the samematerial they have the same Youngsmodulus. Now breaking load - breakingstress x cross-section can withstand agreater load. Hence the correct choices are(2) and (3)

207. Concepual208. Concepual209. L AC CB AB

2 2 1/ 22( ) 22

L l d lstrain L l

2 22 2 1/ 2

22( ) 2 2 1 22d dl d l l ll l

thus222

L dstrain L l

Tstress a210. Hookes law is obeyed in the linear portion

of the graph. hence the correct choice is (1)211. Concepual

212. Youngs modulus = stressstrain , which is,therefore, given by the reciprocal of thestrain versus stress graph. The correctchoices are (2) and (3)

213. Concepual214. 1 2 1 2( )L L LT

since the walls are rigid, this increase inlength cannot occur. Hence a force Fappears between them. The decrease inlength of the rods due to this force is

1' '

21 2

1 1FLL L A Y Y

since the total length cannot change, theincrease in total length due to heating =decrease in total length due to thecompressive force F. Equating (1) and (2),we find that the correct choice is (1)

215. New length of rod P =original length +increase in length due to heating - decrease

in length due to F. Hence,

11

FLL L L T AY 216. Concepual

217.2

tan rg , cosT mg --(1)

2sinT m ...(2)

or tang r

hence the correct choice is (3)218. from equation (1) and (2) we find that the

correct choices are (1) and (4)

219. TStress A alsoL stressstrain L Y

cosTL mg LL AY AY

hence the correct choices are (1) and (2)220. Concepual221. The decrease in the length of the rod on

cooling is L L t 222. If the rod is to be prevented from

contracting, the mass m attached at thelower end must increase its length by an

amount 45 10L m , now mgLY A L 223. Energy stored in the rod is

1 12 2U F L mg L

224. Mg - T = Ma ............(1)T - mg = ma ..........(2)adding eq.(1) and eq.(2), we get

( )( )M m ga M m

225. From eqs. (1) and (2),

T = m(g+a) = m(g + g/3)= 4 mg/343

T mgstress A A , which is choice (4)226. Breaking stress is the maximum stress the

wire can withstand. From eqs(1) and (2)2( )mMgT M m

ELASTICITY

Breaking strest=21

max

T mgA mA M

and max 4M kg227. Concepual

24l F

L d YThus, strain = 21/ d . Hence the correctchoices are (1) and (3)

206. Since wires A and B are made of the samematerial they have the same Youngsmodulus. Now breaking load - breakingstress x cross-section can withstand agreater load. Hence the correct choices are(2) and (3)

207. Concepual208. Concepual209. L AC CB AB

2 2 1/ 22( ) 22

L l d lstrain L l

2 22 2 1/ 2

22( ) 2 2 1 22d dl d l l ll l

thus222

L dstrain L l

Tstress a210. Hookes law is obeyed in the linear portion

of the graph. hence the correct choice is (1)211. Concepual

212. Youngs modulus = stressstrain , which is,therefore, given by the reciprocal of thestrain versus stress graph. The correctchoices are (2) and (3)

213. Concepual214. 1 2 1 2( )L L LT

since the walls are rigid, this increase inlength cannot occur. Hence a force Fappears between them. The decrease inlength of the rods due to this force is

1' '

21 2

1 1FLL L A Y Y

since the total length cannot change, theincrease in total length due to heating =decrease in total length due to thecompressive force F. Equating (1) and (2),we find that the correct choice is (1)

215. New length of rod P =original length +increase in length due to heating - decreasein length due to F. Hence,

11

FLL L L T AY 216. Concepual

217.2

tan rg , cosT mg --(1)

2sinT m ...(2)

or tang r

hence the correct choice is (3)218. from equation (1) and (2) we find that the

correct choices are (1) and (4)

219. TStress A alsoL stressstrain L Y

cosTL mg LL AY AY

hence the correct choices are (1) and (2)220. Concepual221. The decrease in the length of the rod on

cooling is L L t 222. If the rod is to be prevented from

contracting, the mass m attached at thelower end must increase its length by an

amount 45 10L m , now mgLY A L 223. Energy stored in the rod is

1 12 2U F L mg L

224. Mg - T = Ma ............(1)T - mg = ma ..........(2)adding eq.(1) and eq.(2), we get

ELASTICITY( )( )M m ga M m

225. From eqs. (1) and (2),

T = m(g+a) = m(g + g/3)= 4 mg/343

T mgstress A A , which is choice (4)226. Breaking stress is the maximum stress the

wire can withstand. From eqs(1) and (2)2( )mMgT M m

Breaking strest=21

max

T mgA mA M

and max 4M kg227. Concepual

Elasticity

Documents

Transcript of Elasticity