Elasticity 2

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PHYSICS - I C ELASTICITY

i) Longitudinal stress : If the restoring forcesare perpendicular to the area of cross sectionand are along the length of the wire, the stressis called longitudinal stress.

During longitudinal stress, the bodyundergoes change in length but no change inshape and volume.

ii) Tangential stress (or shear ing stress) : Ifthe restoring forces are parallel to the surface,the stress is called shearing stress.

During tangential stress the shape of thebody changes, size and volume remain same.

iii) Bulk stress (or volume stress) : If a bodyis subjected to equal forces normally on all thefaces the stress involved is called bulk stress.

13. Str ain : The deformation produced per unitmagnitude is called strain

a) longitudinal strain = lengthoriginallengthinchange

= ll

b) Bulk strain = volumeoriginalvolumeinchange

= vv

c) Shearing strain = =lateral displacement between two layers

perpendicular distance between the two layers

d) Transverse strain = radiusoriginalradiusinchange

= rre) Shearing strain = 2 x longitudinal strainf) Bulk strain = 3 x longitudinal straing) Longitudinal strain : shearing strain : Bulkstrain = 1 : 2 : 3

14. The maximum value of the stress within whichthe body regain its original size and shapecompletely is called elastic limit.

15. Hookes law : Within the elastic limit, stress isdirectly proportional to strain.

strainstress

= E = constant

E is modulus of elasticity

4. Elasticity : The property of a body by virtue ofwhich it regains its original size and shape whendeformation forces are removed is calledelasticity.Elasticity is a molecular phenomenon.

5. Elast ic body : A body which shows elasticbehaviour is called elastic bodyEg. Steel, rubberNearest approach to perfectly elastic body isquartz.

6. Plast ic body : A body which does not showelastic behaviour is called plastic body.Eg : Putty, chewing gum, lead solder, wax.

7. A body which is more difficult to strain is moreelastic.

Eg: i) Steel is more elastic than rubberii) Springs are made of steel but not of copper

because steel is more elastic than copper.8. Hammering, rolling and addition of impurity

increases elasticity.9. Annealing and rise in temperature decreases

elasticity.10. For invar steel, the elastic property does not

change with temperature.11. Stress : The restoring force developed per unit

area of the deformed body is called stress

Stress = AF

areationalseccrossforcestoringRe

F = External force applied

A = r2 = area of cross section

r = radius of the wire

Unit :N

m2 or pascal ,

dynecm2

Dimensional Formula : M1L-1T-2

Stress is a tensor12. Stress is of three types :

i) Longitudinal stress

ii) Tangential stress

iii) Bulk stress

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PHYSICS-1C

Unit of E : N

m2 or pascal ,

dynecm2

i) within the elastic limit, stress - strain graphis a straight line passing through the origin

Slope of the straight line is Eii) A spring balance works on the principle ofHookes law.iii) Modulus of elasticity does not depend uponthe dimensions of the body but it is a propertyof the material of the body.iv) Within the elastic l imit, the load-extensiongraph is a straight l ine passing through theorigin.

16. Youngs modulus :Y= longitudinalstresslongitudinalstrain

Y = FIAe

If load is M, then F = Mg and 2Mgly=

r ep where r

is radius of the wire.i) The stress required to double the length ofa wire or to produce 100%longitudinal strain is equal to Youngs modulusof the wire.ii) Y of a perfectly elastic material is infiniteand that of a perfectly inelastic material is zero.iii) If a load M produces an elongation e in awire, the rise in temperature required to producethe same elongation is t . Then

a M g Y A twhere A is area of cross section of the wire

and is coefficient of linear expansion of thematerial of the wire.iv) when a metal bar is fixed between two wallshas a rise in temperature, the bar tries to expandand exerts force on the walls. This force is calledthermal force.Thermal force F = Y A a qwhere is co-efficient of linear expansion ofthe bar is rise in temperature.Y is youngs modulus,A is area of cross section.Thermal force is independent of length of the bar.v) Thermal stress : Thermal force per unit areais called thermal stress

Thermal stress = Thermalforcearea = YA

A = YY

Thermal stress is independent of area ofcross section and length of the bar.vi) A long wire suspended vertically canelongate due to its own weight. Elongation of a wire due to its own weight

2

2d geY

l =length of the wire, d=density of the wireY = youngs modulus of the material of

the wireg = acceleration due to gravity

vii)Two wires made of same material havinglengths l1 & l2 and radii r1 and r2 are subjectedto the same force. Then the ratio of theirelongations is

21

22

2

1

2

1

rr

ll

ee

(since e 2rl

)

viii) Two wires made of same mater ialhaving lengths l1 and l2 and masses m1 and m2are subjected to the same force. Then the ratioof their elongations is

1

222

21

2

1

mm

ll

ee

(since e ml 2

)

ix) Two wires of same mater ial and samevolume having areas of cross section A1 & A2are subjected to the same force. Then the ratioof their elogations is

21

22

2

1

AA

ee

(since e 21A

)

x) Two wires of same mater ial and samevolume having radii r1 and r2 are subjected tothe same force. Then the ratio of theirelongations is

41

42

2

1

rr

ee

(since e 41r

)

viii) When a body of mass 'm' and density'dB' is suspended from a wire its elongation is'e' when it is in air. If it is completely immersedin a non-viscous liquid of density dL then itsnew elongation is

e1 = e(1B

L

dd

)

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ix) Two wires of same length and radius haveyoungs modulii Y1 & Y2 are subjected to thesame force. Then the ratio of their elongationsis

1

2

2

1YY

ee

(since e Yl

)

x) Two wires of same length and radius arejoined end to end and loaded. If the youngsmodulii of the materials of the wires are Y1 &Y2 and the combination behaves as a single wirethen its young's modulus is

1 2

1 2

2Y YYY Y

xi) Two wires of same length and radius arejoined in parallel and loaded. If the youngsmoduli of the materials of the wires are Y1 & Y2and the combination behaves as a single wirethen its young's modulus is

1 2

2Y YY

17. Rigidity modulus, n =strain shearingstress Tangential

= F

Awhere is shearing straini) If n is low for a wire then it can be twistedvery easily.ii) Since phosphorbronze has very lowrigidity modulus, it is used as suspension wirein moving coil galvanometer.iii) For a fluid (liquid or gas) the rigiditymodulus is zero.

18. 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 given by

rf q

19. One end of the rod is fixed. The other free endis twisted through an angle '' by applying atorque '' then the work done on the rod (or)energy stored in the rod is

21W

where is in radians.

20. Bulk modulus K = volumestressvolumestrain

= VPV

i) If a block of coefficient of cubical expansion is heated through a rise in temperature of ,the pressure to be applied on it to prevent itsexpansion, P Kg q where K is its bulkmodulus.ii) When a rubber ball of volume V, bulkmodulus K is taken to a depth h in water, then

decrease in its volume h d gVV

K

(d = density of material)21. Solids posses Y, n and K,but liquids and gases

posses only K.22. Isothermal bulk modulus of a gas = pressure of

the gas (P)Adiabatic bulk modulus of a gas = Pwhere is the ratio of two specific heats.

23. Compr essibi l i ty : The reciprocal of bulkmodulus is called compressibility, C = 1/KFor incompressible substances , C = 0; K = .

24. Possions ratio () =

lateral contraction strainlongitudinal elongation strain =

r/r/

l l

i) Poissons ratio has no units and no dimensionsii) Theoretical limits of : -1 to 0.5iii) Practical limit of : 0 to 0.5iv) If = 0.5 then the substance is perfectlyincompressible.v) The Poissons ratio of a given material is ' '.When the load is applied on it, the longitudinal

elongation strain is ll

. Then the fractional

change in its volume is

1 2V lV l

s

25. i) If large deformation occurs between theelastic limit and the breaking point, the materialis ductile.

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PHYSICS-1C

Ex : copper, silver, gold etc.,ii) If the deformation between the elastic limitand the breaking point is very small or if thewire breaks immediately after crossing theelastic limit, the material is brittle.Ex : glass, ceramic etc.

26. Breaking stress :i) The breaking stress of a wire is themaximum stress the material can withstand.

ii) Breaking stress = BreakingForceinitialareaofcrosssection

iii) Breaking force = Breaking stress x area ofcross section.iv) Breaking stress a) depends only on thenature of material of the wire b) independentof the length and area of cross section of the wire.v) Breaking force a) is independent of lengthof the wire b) depends on the area of crosssection and nature of material of the wire.vi) Breaking force is proportional to area of cross-section.vii) If we cut a cable that can support a maximumload of W into two equal parts, then each part cansupport a maximum load of W.viii) A very long wire suspended verticallycan break due to its own weight

The maximum length of a wire that can hungwithout breaking under its own weight is

Breaking Stressdg

l

27. Elastic fatigue :a) The state of temporary loss of elastic nature

due to continuous strain is called elasticfatigue.

b) Due to elastic fatiguei) a wire can be broken within the elastic limit.ii) a wire can be cut into pieces without

using instruments.iii) Railway tracks and bridges are declared

unsafe after long use.iv) spring balances show wrong readings

after long use.28. Elastic after affect:Within the elastic limit , the

delay in regaining the original length after theremoval of deforming force by a material iscalled its elastic after effect.

i) For a perfectly elastic body, elastic aftereffect is zero.ii) For a perfectly inelastic body, elastic aftereffect is infinite.

29. Work done in stretching a wire,

w = 12 x stretching force x extension.

i) w = 12 F e = lYAe

21 2

= 12

F2 lAY =

12

F2 lr 2y

ii) w =12 x stress x strain x volume of the wire

iii) workdonevolume

(a strain energy per unit volume

or elastic potential energy per unit volume) =12

x stress x strain

= 12 x Y x (strain)2 =

2Stress12 Y

iv) The total work done per unit volume instretching an object upto the limits of perfectelasticity is known as its resiliance.v) Area under F-e graph gives the work doneor the strain energy stored in the wire. Area =12 F.e = WW

30. Spr ings :i) A spring that obeys Hookes law has forceconstant or spring constant

Y = FlAx ; F = lYA x and F = Kx

x is elongationWhere K is called the force constant or

spring constant. It is the force needed to stretch

a wire or spring over unit distance. i.e., K = lYA

ii) K Y, K A, K l1

iii) If a spring (or a wire) of force constant K iscut into n equal parts, the force constant ofeach part of the wire is nk.iv) Potential energy of a stretched spring =

12

Fx = 12

Kx2 = KF

21 2

v) Two springs have force constant K1 and K2a) When they are stretched by the same

force and if their elastic energies are

E1 and E2 then E

1

E2

= K

2

K1

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b) When they are extended by the samelength,

E1

E2

= K

1

K2

c) When they are extended till their

energies are same then F

1

F2

= K1K2

d) The potential energy of a springincreases, whether it is stretched or compressed.

e) Springs in series, Keff = K

1K

2K

1+K

2

f) Springs in parallel , Keff = K1 + K2g) The reciprocal of spring constant is

called compliance.

31. i) When a spring is stretched strain involvedis shearing strainii) When a wire is stretched modulus ofelasticity involved is Youngs modulusiii) When a wire is twisted modulus of elasticityinvolved is rigidity modulus.

32. Torsion of a cylinder :When one end of a cylinder of length 'l' andradius 'r' is clamped and the other end is givena twist ' ', then the restoring torque is givenby

2nr

l

Where 'n' is the modulus of rigidity of thematerial of the cylinder.

EXERCISE - I

THEORITICAL QUESTIONS1. A spiral spr ing is stretched by a force the

resultant strain produced in the spr ing is1) Volume strain2) Shearing strain3) Tensile strain 4) None of the above

2. Reason for the deformation of a regular body1) Bulk strain2) Shearing strain3) Linear strain4) Lateral strain

3. When a spr ing is stretched by a force, theresultant strain in known as1) Longitudinal strain2) Sheariing strain3) Volume strain 4) Bulk strain

4. For a gas elastic limit1) Exists 2) Does not exist3) Exists only at absolute zero4) Exists for a perfect gas

5. 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

6. Within elastic limit, if stress is doubled thestrain becomes1) Half 2) Twice3) Thrice 4) None

7. The law which governs the working of a spr ingbalance is1) Kepler's law2) Robert Hooke's law3) Newton's law 4) Young's law

8. According to Hooke's law of elast icity theratio of stress to strain1) Does not remain constant2) Remains constant3) Increases4) Decreases

9. A body subjected to strain a number of timesdoes not obey Hooke's law due to1) Yield point 2) Breaking stress3) Elastic fatigue 4) Permanent set

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

11. Gr aph between the appl ied force and thechange in length of a wire within elastic limitis a1) Straight line with negative slope2) Straight line with zero slope3) Straight line with positive slope4) None of the above

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12. Which of the following substances has thehighest elasticity1) Rubber 2) Steel3) Copper 4) Wood

13. A steel wire is stretched by 5 kg wt. I f ther adius of the wire is doubled its Young' smodulus1) Remains unchanged2) Becomes double3) Becomes half4) Becomes 1/4 times

14. A copper wire and a steel wire of the samediameter and length are connected end to endand a force is applied which stretches theircombined length by 1 cm. Then the two wireswill have1) The same stress and strain2) The same stress but different strain3) The same strain but different stresses4) Different stresses and strain

15. The effect of temperature on the value ofmodulus of elasticity for different substancesin general1) Increases with increase in temperature2) Decreases with increase in temperature3) Independent of change in temperature4) Does not change with temperature

16. A stone is suspended in a tub of water withcopper wire. Another stone of equal mass issuspended in kerosene with equal length ofcopper wire then1) Young's modulus is more in the first case2) Young's modulus is more in the second case3) Elongation will be less in the former case4) Elongation will be more in the former case

17. In the given figure if the dimension of thewires are the same and mater ials are differentYoung's modulus is less for1) A

2) B

3) Both4) None

18. A heavy mass is attached to a thin wire and iswhireled in a vertical circle. The wire is mostlikely to break1) When the mass is at the lowest point2) When mass is at the highest point3) When wire is horizontal

4)When mass is at an angle of

2

11cos fromupward vertical

19. Steel is prefer red for making spr ings overcopper because1)Young's modulus of steel is more than that

of copper2) Steel is cheaper3) Young's modulus of copper is more4) Steel is less likely to be oxidised

20. The possible value of Poission's ratio is1) 0.9 2) 0.8 3) 0.4 4) 1

21. The Poisson's ratio can not have the value1) 0.7 2) 0.23) 0.1 4) 0.5

22. The Poission' s r at io (s) should satisfy therelation1) 1 < < 0.52) 0.5 < < 13) 0.5 < < 1 4) 1 < < 0.5

23. The breaking stress of a wire depends on1) Material of a wire2) Shape of cross section3) Length of the wire4) Radius of the wire

24. Breaking stress per unit area of cross sectionof a wire is called1) Yield point2) Breaking stress3) Ductibility4) Tensile stress

25. The proper ty of metals whereby they couldbe drawn into thin wires beyond their elasticlimit without breaking is1) Ductility2) Malleability3) Elasticity4) Hardness

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26. When an elast i c mater ial wi th young' smodulus 'y' is subjected to a stretching stress'S' the elastic energy stored per unit volumeof the mater ial is ---

1) 2YS

2) 2YS2

3) 2YS2

4) 2YS

27. A wire of length 'L ' and cross sectional area'A ' is made up of a mater ial of young' smodulus 'Y' . I f the wire is stretched by 'X'the work done is

1) L

YAX 2) 2

2YAX

3) 2YAX 4) LYAX

2

2

28. Strain energy per unit volume in a stretchedstr ing is

1) strainstress21

2) stress x strain

3) stress x (strain)2 4) strainstress

29. When a wire is stretched to double it's lengtha) Radius is halved b) Strain is unityc) Stress is equal to Youngs modulusd) Stress is unity1) 'a' and 'b' are correct2) 'b' and 'c' are correct3) 'c', 'd' are correct 4) 'd' is correct

30. The formula for strain energy per unit volumea) 1/2 (stress)(strain) b) 1/2 Y (strain)2

c) 21

x y (stress) d) 21

x Y(strain)

1) a, b are correct 2) a, c are correct3) c, d are correct 4) a, b, c are correct

31. Regarding elasticity which of the followingstatements are cor rect?a) the point where the elasticity ends and plasticity star ts is called yield pointb) breaking stress depends on the area ofcross sectionc) the value of poisions ratio can not begreater than 0.5d) air is more elastic than water1) a, b are correct2) a, c are correct3) b, d are correct4) a and d are correct

32. A spr ing of young's modulus 2 1011 pa issuspended vertically and subjected to a loadof 5 kg and elongation is 2 mm. when theloadis doubled match the following (g =9.8 m/s2)Section - A Section - Ba)Elongating force e) 4 mmb)Stress f) Un changedc)Elongation g) 98 Nd)Young's modulus h)Doubled1) ah; bg; cf; de2) ag; bh; ce; df3) ae; bf; cg; dh4) af; be; cg; dh

33. Section - A Section - Ba) Young modulus e) resistance to

change in shapeb) Rigidity modulus f) resistance to change

in linear dimensionc) Shear modulus g) resistance to

change in volumed) Stress h) resistance to

against defermation forces

1) af; bg; ce; dh2) ag; bh; ce; df3) af; be; cg; dh4) ag; be; ch; df

34. Of the following four wires of same materialwhich will have maximum increase in lengthand which will have minimum increase inlength if all are under the action of sameforce? Arrange the wires in the descendingorder of increase in length.i) Natural length=0.5m, diameter=0.5mmii) Natural length=1m, diameter=1mmiii) Natural length=1.5m,diameter=1.5mmiv) Natural length=2.0m, diameter=2mm1) i, ii, iii, iv2) ii, i, iv, iii3) i, iii, iv, ii4) iv, iii, ii, i

ELASTICITY

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35. Four springs of force constants K 1=1000N/m,K 2=1500 N/m, K 3=2500 N/m and K 4=2000N/m ar e subj ected to di f fer ent l oadsproducing same extension. Ar r ange thespr ings with workdone in descending order1) 1,2,3,4 2) 3, 4, 1, 23) 3, 2, 1, 4 4) 3, 4, 2, 1

36. Four wires made of Al, Cu, Fe and steel andof lengths l1 = 100m, l2=150m, l3=250m andl4= 350m, break due to their own weights.Ar r ange these wir es wi th thei r br eakingstresses in ascending orderdAl = 7500 kg/m3 dcu = 7800 kg/m3dFe = 9000 kg/m3 and dsteel = 11000 kg/m31) St, Fe, Al, Ci 2) St, Fe, Cu, Al3) Al, Cu, Fe, St 4) Fe, St, Al, Cu

Instruction For Questions :I n each of the fol l owing quest ions, astatement of Asser tion (A) is followed by acor responding statement of reason (R) justbelow it. Of the statement mark the correctanswer.

1) Both A and R are true and R is thecorrect explanation of A2) Both A and R are true and R is not thecorrect explanation of A3) A is true but R is false4) A is false but R is true

37 . A) : Stress is Restor ing force per unit areaR) : I nter atomic for ces in sol ids ar e

responsible for the property of elasticity.

38. A) : Steel is more elastic than rubber.R) : Under given deforming force, steel is

deformed less than rubber.

39. A) : Bulk modulus of elasticity (k) representsincompressibility of the mater ial.

R) : Bulk modulus of elasticity is proportionalto change in pressure.

40 . A) : A steel wire of natural length ' l' breaks,when elongated by length 'x' . I f or iginallength is halved and elongated by x/2 itbreaks.

R) : Breaking force is directly proportionalto the length of the wire

41 . A) : Silver is a ductile mater ialR) : For a ducti le mater ial yield point and

Breaking polnt are seperated by largerdistance than for Br it tle mater ials onthestressstrain curve.

42. A) : The stretching of a coil is determined byits shear modulus.

R) : Shear modulus change only shape of abody keeping its dimensions unchanged.

43. A) : Strain is a unitless quantity.R) : Strain is equivalent to force.

44. A) : The br idges declared unsafe after a long use.R) : Elastic strength of br idges losses with time.

45. A) : Two identical solid balls, one of ivory andthe other of wet-clay are dropped fromthe same height on the floor. Both the ballswill r ise to same height after bouncing.

R) : Ivory is more elastic than wet-clay.46. A) : Young's modulus for a perfectly plastic

body is zero.R): For a per fectly plastic body, restor ing

force is zero.47. A) : Identical spr ings of steel and copper are

equally stretched. More work will be doneon the steel spr ing.

R) : Youngs modulus of steel is higher thancopper.

48. Consider the statements A and B and identifythe correct answer given below.A) I f the volume of a body r emainsunchanged, When subjected to tensile strain,the value of Possion's ratio is 1/2B) Phosphor bronze has low Young's modulusand high r igidity modulus [2003 M]1) A and B are correct2) A and B ar wrong3) A is correct and B is wrong4) A is wrong and B is right

49. Work done per unit volume is stretching a wire1) is equal to half the product of the stress andstrain2) is equal to the product of the stress and thestrain3) is euqal to the ratio of the stress to the strain4) is equal to the ratio of the strain to stress

AKASH MULTIMEDIA 45


50. The breaking stress of wire depends upon [AIEEE 2002]

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

51. Which of the following affects the elasticityof a substance [AIEEE 2002]1) hammering and annealing2) change in temperature3) impurity in substance4) all of these

52. Consider an ideal monoatomic gas of volumeat pressureP. The bulk modulus at constanttemperature is [UPSEAT 2000]

1) 2p

2) P 3) P 4) PdPdV

53. I f k is the force constant of a spr ing the workdone [against restor ing force] in increasingits extension from 21 l tol will be1) )ll(K 12 2) )ll(2/K 12

3) 2122 llK 4) 2122 ll2K

54. A wire of length L and cross -section 'A' hasyoungs modulus of mater ial Y. I t is stretchedby an amount 'x' . The work done againstrestor ing force is [MP PET 1996]

1) L2YAx

2) L

YAx 2

3) L2

YAx 24)

LYAx2 2

ANSWERS1) 2 2) 2 3) 2 4) 2 5) 26) 2 7) 2 8) 2 9) 3 10) 1

11) 3 12) 2 13) 1 14) 2 15) 216) 3 17) 2 18) 1 19) 1 20) 321) 1 22) 1 23) 1 24) 4 25) 126) 3 27) 4 28) 1 29) 2 30) 131) 2 32) 2 33) 1 34) 1 35) 436) 3 37) 2 38) 1 39) 3 40) 341) 1 42) 1 43) 3 44) 1 45) 446) 1 47) 1 48) 3 49) 1 50) 351) 4 52) 2 53) 4 54) 4

EXERCISE - II(A) (CLASS WORK)YOUNG 'S MODULUS

* 1. A copper wire of 1 mm diameter is stretchedby applying a force on 10N. Find the stress inthe wire.1) 1.273 x 107N/m2 2) 1.373 x 107N/m2

3) 1.473 x 107N/m2 4) 1.573 x 107N/m2

* 2. A tungsten wire of length 20 cm is stretchedby 0.1 cm. Find the strain on the wire.1) 0.002 2) 0.0053) 0.001 4) 0.004

* 3. I n the Sear les method to deter mine theYoungs modulus of a wire, a steel wire oflength 156 cm and diameter 0.054 cm is takenas exper imental wire. The average increasein length for 1 1/2 kg wt is found to be 0.050cm. Then the Youngs modulus of the wire.1) 3.002x1011N/m2 2) 1.002x1011N/m2

3) 2.002x1011N/m2 4) 2.5x1011N/m2

4. The area of cross-section of a wire is 105m2when its length is increased by 0.1% a tensionof 1000N is produced. The Young's modulusof the wire will be ----1) 1012 Nm 2 2) 1011 Nm 23) 109 Nm2 4) 1010 Nm2

5. The following four wires are made of the samemater ial. Which of these will have the largestelongation when the same tension is applied1) l = 50cm and diameter 0.5mm2) l = 100cm and diameter 1.0mm3) l = 200cm and diameter 2.0mm4) l = 300cm and diameter 3.0mm

*6. A copper wire and an aluminium wire havelengths in the ratio 3 : 2, diameters in theratio 2 : 3 and forces applied in the ratio 4 :5. Find the ratio of the increase in length ofthe two wires.(YCu=1.1x1011Nm2, YAl=0.70x1011Nm2)1) 110 : 189 2) 180 : 110

3) 189 : 110 4) 80 : 11

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* 7. There are two wires of same mater ial. Theirradii and lengths are both in the ratio 1 : 2. I fthe extensions produced are equal, what is theratio of the loads ?1) 1 : 2 2) 2 : 13) 1 : 4 4) 4 : 1

8. I f str ess is numer ical ly equal to young' smodulus the elongation will be

1) 41 the original length

2) 21 the original length

3) Equal to the original length4) Twice the original length

9. An aluminium r od has a br eaking str ain0.2%. The minimum cross-sectional area ofthe rod in m2 in order to support a load of104N is if Youngs modulus is 7x109Nm2

1) 1.7x10 4 2) 1.7x10 3

3) 7.1x10 4 4) 1.4x10 4

10. A str ing of cross section 2cm2 is doubled inlength by the application of a longitudinalforce 2x105 dynes. The youngs modulus is -1) 107 dy/cm2 2) 105 dy/cm2

3) 104 dy/cm2 4) 100 dy/cm2

11. A metallic r ing of radius 'r ' and cross sectionalarea A is fitted into a wooden circular disc ofradius R (R > r) . I f the Young's modulus ofthe mater ial of the r ing is Y, the force withwhich the metal r ing expands is : [2004 E]

1) rAYR

2) r)rR(AY

3) Ar)rR(Y

4) ARYR

* 12.When the tension on a wire is 4N its length is

1 . When the tension on the wire is 5N its

length is 2 . Find its natural length.

1) 1 25 4 2) 1 24 5

3) 1 210 8 4) 1 28 10

13. When a metal sphere is suspended at the endof a metal wire its extension is 0.4mm. I fanother metal sphere of the same mater ialwi th i ts r adius hal f as the pr evious i ssuspended then extension would be1) 0.05mm 2) 0.02mm3) 0.01mm 4) 0.32mm

14. A wire whose cross - sectional area is 4 mm2is stretched by 0.1 mm by a certain load. I f asimi lar wi r e of double the ar ea ofcross-section is under the same load, then theelongation would be1)0.5mm 2) 0.05 mm3) 0.005mm 4) 5mm

15. When a tension 'F' is applied, the elongationproduced in uniform wire of length l, radiusr i s e, when tension 2F i s appl ied, theelongation produced in another uniform wireof length 2l and radius 2r made of samemater ial is [ 2000 M ]1) 0.5e 2) 1.0e 3) 1.5e 4) 2.0e

16. Two wires A and B have Young's modulii inthe ratio 1:2 and ratio of lengths is 1:1. Underthe application of same stress the ratio ofelongations is ---1) 1:1 2) 1:2 3) 2:1 4) 1:4

17. A wire is stretched by 0.01m by a cer tainforce 'F' another wire of same mater ial whosediameter and lengths are double to or iginalwire is stretched by the same force then itselongation will be ....1) 0.005m 2) 0.01m 3) 0.02m 4) 0.04m

18. A steel wire is 1m long and 1mm2 in area ofcross-section. I f it takes 200 N to stretch thiswire by 1mm how much force will be requiredto stretch the wire of the same area and ofsame material having length of 10m to 1002 cm1) 100 N 2) 200 N 3) 400 N 4) 2000 N

19. Two bars A and B of circular cross sectionand of same volume made of same mater ialare subjected to tension. I f the diameter of Ais half that of B and if the force applied toboth the rods is the same and it is in the elasticlimit the ratio of extension of A to that of Bwill be ---1) 16 2) 8 3) 4 4) 2

AKASH MULTIMEDIA 47


20. An iron wire of length 4m and diameter 2mmis loaded with a weight of 8kg. I f the Young'smodulus 'Y' for iron is 2x1011Nm 2 then theincrease in length of the wire is ---( g = 10 ms-2)1) 0.2mm 2) 0.5mm3) 2 mm 4) 1 mm

21. Two wires of the same mater ial have massesin the ratio 3:4 . The ratio of their extensionsunder the same load if their lengths are in theratio 9:10 is1) 5 : 3 2) 27 : 403) 6 : 5 4) 27 : 25

22. Two exactly similar wires of steel (Y = 20 x 1011dy/cm2) and copper (Y=12x1011dy/cm2) arestretched by equal forces. I f the total elongationis 1cm elongation of copper wire is1) 3/5 cm 2) 5/3 cm3) 3/8 cm 4) 5/8 cm

23. Two wires of same length and thickness arejoined end to end. Their Young's moduli are15x1010pa and 20x1010pa. I f the combina-tionis stretched by a certain load, the elongationsof these wires will be in the ratio1) 3:4 2) 4:33) 1:1 4) 1:2

* 24.A steel wire of length 20 cm is stretched toincrease its length by 0.2 cm. Find the lateralstrain in the wire if the Poissons ratio for steelis 0.19.1) 0.0019 2) 0.00083) 0.019 4) 0.008

25. Two r ods of di f fer ent mater ial s havingcoefficients of l inear expansion 21 and and youngs moduli i 21 Y andY respectivelyare fixed between two r igid massive walls.Therods are heated such that they undergo thesame increase in temp. There is no bendingof rods. I f 21 : = 2:3 . The thermal stressesdeveloped in the two rods are equal if 21 : YYis equal to1) 4 : 9 2) 3 : 2 3) 9 : 4 4) 2 : 2

* 26.A tungsten wire, 0.5 mm in diameter, is juststr etched between two f i xed points at a

temperature of 400C. Determine the tensionin the wir e when the temperatur e falls to200C. (coef f i cient of l i near expansion oftungsten = 4.5 x 106/0C; Youngs modulus oftungsten = 3.45 x 1010Nm2)1) 6.097N 2) 3.097N3) 5.097N 4) 7.097N

* 27.A uniform steel rod of length 1m and area ofcross section 20 cm2 is hanging from a fixedsupport. Find the increase in the length of therod.

(Ysteel=2.0x1011Nm2, steel =7.85x103kgm3)

1) 1.923 x 105cm 2) 2.923 x 105cm

3) 1.123 x 105cm 4) 3.123 x 105cm

RIGIDITY MODULUS & BULK MODULUS :* 28.A 2.0 cm cube of some substance has its upper

face displaced by 0.15 cm, by a tangentialforce of 0.30N fixing its lower face. Calculatethe r igidity modulus of the substance.1) 1 x 104N/m2 2) 2 x 104N/m2

3) 3 x 104N/m2 4) 0.5 x 104N/m2

* 29.A spher i cal bal l of volume 1000 cm3 i ssubjected to a pressure of 10 atmosphere. Thechange in volume is 102cm3. I f the ball is madeof iron find its bulk modulus. (Atmospher icpressure =1x105Nm2)1) 1 x 1011 N/m2 2) 2 x 1011 N/m2

3) 3 x 1011 N/m2 4) 4 x 1011 N/m2

30. A fr act ional change in volume of oi l is 1per cent when a pressure of 2x107 Nm2 isapplied. Calculate the bulk modulus and itscompressibil ity1) 3x108 Nm 2, 0.33x10 9 m2N 1

2) 5x109 Nm 2, 2x10 10 m2N 1

3) 2x109 Nm 2, 5x10 10 m2N 1

4) 2x109 Nm 2, 5x10 9 m2N 1

* 31.Determine the pressure required to reducethe given volume of water by 2% . Bulkmodulus of water is 2.2x109Nm2.1) 4.4 x 107 N/m2 2) 2.2 x 107 N/m2

3) 3.3 x 107 N/m2 4) 1.1 x 107 N/m2

ELASTICITY

AKASH MULTIMEDIA 48

PHYSICS-1C

32. When 200ml of water i s subj ected to apressure of 2x106 Pa. The decrease in i tsvolume is 0.2ml. The compressibility of wateris ---1) 5x10 8 m2N 1 2) 5x10 10 m2N 13) 5x10 12 m2N 1 4) None

33. A cubical ball is taken to a depth of 200m in asea. The decrease in volume observed to be0.1% . The bulk modulus of the bal l is --( g = 10 ms-2)1) 2x107 Pa 2) 2x106 Pa3) 2x109 Pa 4) 1.2x109 Pa

34. A mater ial has poissions r at io 0.5 . I f auniform rod of it suffers a longitudinal strainof 3102 , then the percentage increase inits volume is1) 0% 2) 10% 3) 20% 4) 5%

35. The increase in length of a wire on strechingis 0.025%. I f its poisson 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

36. A mater ial has normal density and bulkmodulus K. The increase in the density of themater ial when it is subjected to an externalpressure 'p' from all sides is

1) KP 2)

KP3)

KP 4) p

K

37. A wire can be broken by 400 kg.wt. The loadr equir ed to br eak the wir e of double thethickness of the same mater ial will be -1) 800 kg wt 2) 1600 kg wt3) 3200 kg wt 4) 6400 kg wt

38. A cable br eaks i f stretched by mor e than2mm. I t is cut into two equal parts. By howmuch either par t can be stretched withoutbreaking1) 0.25mm 2) 0.5mm3) 1mm 4) 2mm

39. The br eak ing st r ess of steel i s7.9x109 Nm2 and density of steel is 7.9x103kgm3 and g = 10ms2. The maximum lengthof steel wire that can hang vertically withoutbreaking is1) 103 m 2) 104 m3) 102 m 4) 105 m

STRAIN ENERGY40. A metal rod of Young's modulus 2 1010 Nm-2

undergoes an elastic strain of 0.02% the energyper unit volume stored in the rod in joule/m3 is1) 400 2) 800 3) 1200 4) 1600

41. A brass rod has length 0.2m area of crosssection 1.0cm2 and Young' s modulus1011Nm2. I f it is compressed by 5kg. wt alongits length then the change in its energy will be1) An increase of 2.4 105 J2) A decrease of 2.4 105 J3) An increase of 2.4 107 J4) A decrease of 2.4 107 J

* 42.A metal wire of length 2.5 m and area ofcross section 1.5x106m2 is stretched through2 mm. Calculate the wor k done dur ingstretching. (Y=1.25x1011Nm2)1) 0.15 J 2) 0.51 J3) 1.5 J 4) 5.1 J

* 43.A steel wire of length 1.5 m and area of crosssection 1.5 mm2 is stretched by 1.5cm. thenthe wor k done per uni t volume.(Y=2x1011Nm2)1) 1 x 107J/m3 2) 2 x 107J/m33) 3 x 107J/m3 4) 4 x 107J/m3

* 44.When a steel rod of length 2m is compressedits length decreases by 0.1cm. Find the workdone against the compr essive st r ess.(Y

steel=2.0x1011Nm2)1) 2.5 x 104J 2) 1.25 x 104 J3) 5.0 x 104J 4) 7.5 x 104J

45. The youngs modulus of the mater ial of a rodis 20x1010 pascal. When the longi tudinalstrain is 0.04%, The energy stored per unitvolume is1) 4 x 10-3 J/m3 2) 8 x 103 J/m3

3) 16 x 10-3 J/m3 4) 16 x 103 J/m3

46. The work done in stretching a wire by 1mmis 2J. The work necessar y for stretchinganother wire of same mater ial but with doublethe radius of cross-section and half the lengthby 1mm is in Joules [1991 M]1) 16 2) 8 3) 4 4) 1/4

AKASH MULTIMEDIA 49


47. Two wires of same r adius and length aresubjected to the same load. One wire is ofsteel and the other is of copper. I t the Young'smodulus of steel is twice that of copper, theratio of elastic energy stored per unit volumein steel to that of copper wire is

[UPSEAT 2000]1) 1:2 2) 2:1 3) 1:4 4) 4:1

48. A spr ing of force constant 800N/m has anextension of 5cm. The wor k done inextending it from 5cm to 15cm is

[AIEEE 2002]1) 16J 2) 8J 3) 32J 4) 24J

ANSWERS1) 1 2) 2 3) 3 4) 2 5) 16) 3 7) 1 8) 3 9) 3 10) 2

11) 2 12) 1 13) 1 14) 2 15) 216) 3 17) 1 18) 3 19) 1 20) 221) 4 22) 4 23) 2 24) 1 25) 226) 1 27) 1 28) 1 29) 1 30) 331) 1 32) 2 33) 3 34) 1 35) 136) 3 37) 2 38) 3 39) 4 40) 141) 1 42) 1 43) 1 44) 1 45) 446) 1 47) 1 48) 2

EXERCISE - II(B) (HOME WORK)

YOUNG 'S MODULUS

*1. A brass wire of diameter 1 mm and of length2m is stretched by applying a force of 20N.Then the stress in the wire is1) 2.546x107N/m2 2) 3.146x107N/m23) 2.346x107N/m2 4) 2.246x107N/m2

*2. I f an iron wire is stretched by 1% what isthe strain on the wire ?1) 0.01 2) 0.0053) 0.02 4) 0.03

*3. A brass wire of diameter 1 mm and of length2m is stretched by applying a force of 20N.I f the increase in length is 0.51 mm, Thenthe Youngs modulus of the wire is1) 8.848x1010N/m2 2) 7.984x1010N/m23) 6.984x1010N/m2 4) 9.984x1010N/m2

4. Force required to increase the length of wireof area of cross section 'A' by one percent, if'Y' is young's modulus, is1) AY 2)

10AY 3)

100AY 4)

1000AY

5. Two wires of same length and mater ial butwith radii 0.1mm and 0.2mm are loaded withthe same weight. Then1) Thick wire extends more2) Thin wire extends more3) Both undergo same extension4) None

6. A wire of length 'L ' and radius 'r ' is fixed atone end and a force 'F' is applied to the otherend to produce an extension ' l ' . The extensionproduced in another wire of same mater ialof length '2L' and radius '2r' by a force '2F' is --1) l 2) 2l 3) l/2 4) 4l

*7. Two wires of different mater ial have samelengths and areas of cross section. What isthe ratio of their increases in length whenthe forces applied are the same ?(Y1=0.90x1011Nm2, Y2=3.60x1011Nm2)1) 4 : 1 2) 1 : 4 3) 2 : 1 4) 1 : 2

8. The stress required to double the length of awire of Young's modulus 'E' is ---1) E 2) 2E 3) 4E 4) E/2

9. The length of a brass wire is 1m. The strainproduced by a load is 0.001. The change inits length is1) 1cm 2) 1mm 3) 10 cm4) 2 cm

10. The Young's modulus of a mater ial having across-sectional area of 1cm2 is 2x1012 dyne/cm2. I f the length is to be doubled the forcerequired is1) 4x107 N 2) 2x107 N3) 107 N 4) Zero

11. A metal r ing of inner radius r 1 and cross-sectional area 'A' is fitted onto a wooden discof radius r2 > r1. I f 'Y' is the young's modulusof the metal then the tension in the r ing is

1) 1

2

rAYr

2) 1

12

r)rr(AY

3) 1

12

Ar)rr(Y

4) 2

1

ArYr

ELASTICITY

AKASH MULTIMEDIA 50

PHYSICS-1C

12. The length of a metal wire is l when the tensionin it is 'F' and 'xl ' when the tension is 'yF'.Then the natural length of the wire is

1) 1)(

x

lyx2) 1

)(y

lxy

3) 1)(

x

lyx4) 1

)(y

lxy

13. Two wires made of the same mater ial havelengths 3 m & 4 m and masses 18 gm and 16gm respectively . When they are subjected tothe same for ce, then the r at io of thei relongations is1) 1 : 2 2) 4 : 3 3) 16 : 18 4) 9 : 4

14. Two wires of same mater ial and same lengthbut diameters in the ratio 2:3 are stretchedby the same length. The ratio of the forcesapplied on two wires is ----1) 4:3 2) 4:9 3) 2:5 4) 9:4

15. When a uniform wire of radius 'r ' is stretchedby a 2kg weight, the increase in its length is2.00 mm. I f the radius of the wire is r /2 andother condi tions r emaining the same, theincrease in its length is [2000 E]1) 2.00mm 2) 4.00mm3) 6.00mm 4) 8.00mm

16. The extension of a wire by application of loadis 0.3cm. The extension in a wir e of samemater ial but of double the length and half theradius of cross section by the same load willbe in (cm)1) 0.3 2) 0.6 3) 0.2 4) 2.4

17. A wire whose cross sectional area is 2mm2 isstretched by 0.1mm by a cer tain load, I f asimilar wire of tr iple the area of cross sectionis stretched by the same load theelongation of the second wire would be1) 0.33mm 2) 0.033mm3) 3.3mm 4) 0.0033mm

18. A brass wire of length 300 cm when subjectedto a for ce F produces an elongation " a" .Another wire of twice the diameter and ofsame length and mater ial , when subjectedto the force F produces an elongation b. Thenthe value of 'a/b' is1) 1 : 1 2) 4 : 13) 2 : 1 4) 1 : 2

19. A hollow cylinder of inner radius 3 cm andouter r adius 5 cm and a sol id cylinder ofradius 2 cm are subjected to the same force.I f they ar e made of same mater ial and ofsame length, then the r at io of thei relongations is1) 1 : 1 2) 1 : 2 3) 1 : 4 4) 2 : 3

20. A wire of length 10 m and of cross -section 1x 10-4 sq.m is fixed at one end and a load of1200 kg is attached at the other end. Thenthe extension produced is( Y= 9.8 x 1010N/m2)1)12 x 10-3 m 2) 12 x 10-5 m3) 12 x 10-4 m 4) 12 x 10-2 m

21. Two wires of the same mater ial but of lengthsin the ratio 2:3 and diameters in the ratio of3:2 are stretched with equal force.Then the ratio of elongations produced is1) 9 : 4 2) 1 : 13) 8 : 27 4) 9 : 16

22. An iron wire and copper wire having samelength and cross-section are suspended fromsame roof Young's modulus of copper is 1/3rdthat of iron. Then the ratio of the weights tobe added at their ends so that their ends areat the same level is1) 1:3 2) 1:9 3) 3:1 4) 9:1

23. Two wires A and B of same mater ial havelengths in the ratio 1:2 and diameters in theratio 2:1. I f they are stretched by same forcethen the ratio of the increase of their lengthswill be ----1) 2:1 2) 1:4 3) 8:1 4) 1:8

*24.Find the fractional increase in the volume ofa wi r e of ci r cular cr oss sect ion i f i ts

longitudinal strain is 2%. 0.41) 0.004 2) 0.0023) 0.001 4) 0.005

25. Two rods of different mater ials are clampedat their ends r igidly. When they are heatedfor the same r i se in temper atur e, samethermal stresses ar e produced in them. I ftheir Young's modulii are in the ratio x:y thenratio of coefficients of their linear expansion is....

1) x : y 2) y : x 3) x2 : y2 4) y2 : x2

AKASH MULTIMEDIA 51


* 26.A steel wire of cross sectional area 2 mm2 isjust stretched hor izontally between two fixedpoints at a temper atur e of 350C. Thetemperatur e of the wir e changed to 250C.Then (Coefficient of linear expansion of steel= 11x106/0C;Y

steel=2x1011Nm2)1) the thermal strain is 11x105 ;2) the thermal stress is 2.2 x 107 N/m2

3) the tension in the wire at temperature 250C is 44N4) All the above

27. A rope of rubber of density 1.5 x103 Kg/m3 andYoung's modulus 5 x 106 N/m2 , 8m in length ishung from the ceil ing of a room. Then theincrease in length due to its own weight is1) 96 x 10-2 m 2) 12 x 10-4 m3) 96 x 10-3 m 4) 12 x 10-3 m

RIGIDITY MODULUS & BULK MODULUS :* 28.A tangential for ce of 1000N is applied on

upper surface of area 2 mm2 which is 0.05 mfrom its fixed face. Find the shift of the uppersur face wi th respect to its f ixed face if i tsmodulus of r igidity is 0.30 x 1011 Nm2.1) 0.833 mm 2) 0.734 mm3) 0.433 mm 4) 0.533 mm

* 29.A copper cube of each side of length 1 cm issubjected to a pressure of 100 atmospheres.Find the change in its volume if the bulkmodulus of copper is 1.4x1011Nm2. (Atmospher icpr essur e=1x105Nm2)1) 0.7143 x 1010 m3 2) 0.5143 x 1010 m3

3) 0.4143 x 1010 m3 4) 0.8243 x 1010 m3

30. I f a rubber ball is taken down to a 100 m deeplake i ts volume is decr eased by 0.1% . I fg = 10 m/sec2 . Then the bulk modulus ofelasticity is1) 109Pa 2) 1010Pa3)1011Pa 4)3x109Pa

* 31.Find the pressure to be applied to increasethe density of water by 1% if the bulk modulusof water is 2.2x109Nm2.1) 2.2 x 107 N/m2 2) 1.2 x 107 N/m2

3) 3.2 x 107 N/m2 4) 4.2 x 107 N/m2

32. A sphere contracts in volume by 0.01% whentaken to the bottom of sea 1km deep. I f thedensity of water is 1gm /cc Then the bulkmodulus of water is1) 9.8 105 Nm2 2) 9.8 108 Nm23) 9.8 1010 Nm2 4) 9.8 106 Nm2

33. A ball falling in a lake of depth 200m shows adecrease of 0.1% in its volume at the bottom.What is the bulk modulus [AFMC 1997]1) 19.6 108N/m2

2) 19.6 10-10N/m2

3) 19.6 1010N/m-2

4) 19.6 10 -8N/m-2

34. A uni for m cube i s subj ected to volumecompression. I f each side is decreased by 1%.Then the bulk strain is1) 0.01 2) 0.06 3) 0.02 4) 0.03

35. The longitudinal strain in a metal bar is 0.05.I f the Poission's ratio for the metal is 0.25,the lateral strain is1) 0.0125 2) 0.4 3) 0.6 4) 0.025

* 36.Estimate the change in the density of waterin ocean at a depth of 500 m below thesur face. The density of water at the surface= 1030 kgm3 and the bulk modulus of water= 2.2x109Nm21) 2.363 kg/m32) 1.363 kg/m33) 4.363 kg/m3 4) 3.363 kg/m3

37. The breaking force of a wire is 'F' . When thethickness of the wire is doubled, the breakingforce will be1) 2F 2) 4F 3) F 4) 8F

38. A wir e can suppor t a load M g wi thoutbreaking. I t is cut into two equal parts. Themaximum load that each part can suppor t is1) Mg/4 2) Mg/2 3) Mg 4) 2Mg

39. Breaking stress for steel is 8x106 Nm 2. Thedensity of steel is 8x103 kgm 3 and g = 10ms 2.What will be the maximum length of steel wirewhich will hang without breaking under itsload.1) 200m 2) 100m 3) 50m 4) 25m

ELASTICITY

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PHYSICS-1C

STRAIN ENERGY40. A rod of Young's modulus 20 GPa undergoes

a linear strain of 6x10 4. Then increase in itsenergy density is ---1) 3600 Jm 3 2) 7200 Jm 33) 1800 Jm 3 4) None

41. A wire is suspended ver tically from one ofit's ends is stretched by attaching a weight of200N to the lower end. The weight stretchesthe wire by 1mm. Then the elastic energystored in the wire is [AIEEE 2003]1) 0.2J 2) 10J 3) 20J 4) 0.1J

* 42.A brass wire of length 1m and area of crosssection 1 mm2 is stretched by applying a forceof 20N. then the work done per unit volumeof the wire is(Ybr ass=0.91x1011Nm2)1) 2198 J 2) 1049 J3) 4422 J 4) 4521 J

* 43.When a copper wire of area of cross section1 mm2 is stretched by a force of 10N. Thenthe wor k done in str etching the wi r e i s(Y=1.2x1011Nm2)1) 416.7 J 2) 208.35 J3) 833.4 J 4) 316.7 J

* 44.A steel wire of length 80 cm and of area ofcross section 1.5 mm2 is stretched. Find thework done in increasing its length from 80.2cm to 80.3 cm. (Y=2.0x1011Nm2)1) 0.9375 J 2) 0.5375 J3) 0.6374 J 4) 0.8695 J

45. A metal wire 4m long and 2x107 sq.m incross-section is stretched by a force of 30N.I f the work done in stretching that wire is4.5x102J the young's modulus of the wire is1) 2x1011 Pa 2) 4x1011 Pa3) 2x1012 Pa 4) 4x1012 Pa

46. A long spr ing when str etched by 2cm itspotential energy is U. I f the same spr ing isstretched by 8cm the potential energy of thespr ing is ----1) 8U 2) 16U 3) 4U 4) U

47. Two wires of same mater ial and size but withlengths in the ratio 5:3 are stretched by thesame force. The ratio of work done in twocases is1) 5:8 2) 8:5 3) 5:3 4) 3:5

48. A man of mass 100 kg gets into a car. I f thecent r e of gr avi ty of the car system isdepressed by 5mm. Spr ing constant of thesystem is ---1) 1.96x105 Nm 1 2) 104 Nm 1

3) 1.2 Nm 1 4) Infinity

ANSWERS

1) 1 2) 1 3) 4 4) 3 5) 26) 1 7) 1 8) 1 9) 2 10) 2

11) 2 12) 2 13) 1 14) 2 15) 416) 4 17) 2 18) 2 19) 3 20) 121) 3 22) 3 23) 4 24) 1 25) 226) 4 27) 3 28) 1 29) 1 30) 131) 1 32) 3 33) 1 34) 4 35) 136) 1 37) 2 38) 3 39) 2 40) 141) 4 42) 1 43) 1 44) 1 45) 146) 2 47) 3 48) 1

EXERCISE - III(A)(CLASS WORK)

* 1. What mass must be suspended from the freeend of a tungsten wire of length 50 cm anddiameter 0.25 mm to stretch it by 1 mm (Y oftungsten = 3.448 x 1011 Nm2)1) 3.455 kg 2) 4.445 kg

3) 1.445 kg 4) 2.445 kg

2. The length of a rubber cord is l1 meters whenthe tension is 4N and l2 meters when the thetension is 5N. The length in meters when thetension is 9 N is1) 5l1 - 4l2 2)5l2 -4l13) 9l1- 8l2 4) 8l2

- 5l1

3. A wir e of length 1m and r adius 1mm issubjected to a load. The extension is 'x' . Thewire is melted and then drawn into a wire ofsquare cross-section of side 1mm. What is itsextension under the same load?1) x2 2) 2x3) x 4) x/

AKASH MULTIMEDIA 53


* 4. A steel wire of length 1 m and cross sectionalarea 1.5 mm2 is hung from a r igid support,wi th a stone of volume 2000 cm3 hangingfrom the other end. Find the decrease in thelength of the wi r e, when the stone i scompletely immer sed in water.

(Ysteel=2x1011Nm2, water =103kgm3)

1) 0.06533 mm 2) 0.02533 mm3) 0.04533 mm 4) 0.01533 mm

5. Two wires of different mater ials, each 2mlong and of diameter 2mm, are joined in ser iesto form a composite wire. What force wil lproduce a total extension of 0.9 mm.( Y1 = 2 x 1011pa & Y2 = 6 x 1011 Pa)1)282.6 N 2) 212 N3) 319.8 N 4) 382.6 N

* 6. An aluminium wire and a copper wire of thesame length and cross section are joined endto end. The composite wire is hung from ar igid support and a load is suspended fromthe free end. if the increase in length of thecomposite wire is 1.8mm, find the increase inthe length of each wire.(YAl=7x1010Nm2, YCu=1.1x1011Nm2)1) 1.1 mm, 0.7 mm 2) 0.7 mm, 1.1 mm3) 2.2 mm, 1.4 mm 4) 1.4 mm, 2.2 mm

7. Two wires A & B are identical in shape andsize and are stretched by same magnitude offorce. Then the extensions are found to be0.2% and 0.3% respectively. Find the ratioof their Young's modulii1) 2 : 3 2) 3 : 2 3) 4 : 9 4) 9 : 4

8. One end of a uniform wire of length L and ofweight W is attached r igidly to a point in theroof and a weight W1 is suspended from itslower end. I f 'S' is the area of cross- sectionof the wire, then find the stress in the wire atheight (3L /4) from its lower end?

1)

S

w/43w1 2)

S

/4w3w 1

3) 4ww1 4)

S

w/43w1

9. The adjacent graph shows the extension of a wire of length 1 m suspended from thetop of a r oof at one end wi th a load Wconnected to the other end. I f the cr osssectional area of the wire is 106m2, calculatethe young's modulus of the mater ial of thewir e

1) 11 22 10 /N m

2) 11 22 10 /N m

W(N)

l x 1

0-4 m

1234

20 40 60 803) 12 23 10 /N m4) 13 22 10 /N m

10. A lift is tied with thick wires and the mass ofthe l ift is one metr ic ton. I f the maximumacceler at ion of the l i f t is 1ms2 and themaximum stressof the wire is 1.4x108Nm2.I f g = 10ms 2 the minimum diameter of thatwire is ----1) 10m 2) 10 1m3) 10 2m 4) None

11. A light rod of length 200cm is suspended fromthe cei l ing hor izontal ly by means of twovertical wires of equal length tied to its ends.One of the wires is made of steel and is ofcr oss-section 0.1sq.cm and the other is ofbr ass of cr oss-sect ion 0.2 sq.cm. Find theposition along the rod at which a weight maybe hung to produce (i) equal stresses in bothwires and (ii) equal strains in both wires.Ybrass = 10x1011 dyne/cm2 andY

steel = 20 x 1011 dyne/cm2

1) 133.3cm; 100cm 2) 167cm; 50cm3) 200cm; 100cm 4) none

12. A wire 3m in length and 1mm in diameter at300 c and kept in a low temper atur e at1700c and is stretched by hanging a weight of10kg at one end calculate the change in thelength of the wir e. Given =1.2x105/0CY=2x1011 Nm2 g=10ms21) 2x103 m 2) 7.2x103 m3) 95.6 x 105 m 4) 5.2 m

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13. A long steel wire of length 'L ' is suspendedfrom the ceiling of a room. A sphere of mass'm' and radius 'r 1' is attached to the lowerend of the wire. the height of the ceil ing is(L+2r1+l). When the sphere is made to oscillateas a pendulum, its lowest point just touchesthe floor. The velocity of the sphere at thelowest point will be ( L >> r1,l and r is theradius of the wire)

1)2r ye

Lgm

2)

2r yLg

me

3)2y r e

m

4) None

* 14.A block of mass 2 kg is attached to one end ofa wire of cross sectional area 1 mm2 and iswhir led in a vertical circle of radius 40 cm.The speed of the block at the bottom of thecircle 5ms1. Find the elongation of the wirewhen the block i s at the bot tom.(Y=2x1011Nm2)1) 0.2892 mm 2) 0.054 mm3) 0.1446 mm 4) 0.0017 mm

* 15.A stone of mass 2 kg is fastened to one end ofa steel wire of cross sectional area 2 mm2 andis whir led in a hor izontal circle of radius 25cm.I f the breaking stress of steel is 1.2x109Nm2find the maximum number of revolutions thestone can make per minute without the str ingbreaking.1) 661.5 rpm 2) 331.5 rpm3) 221.5 rpm 4) 441.5 rpm

16. A metal cube of side length 8.0cm has i tsupper sur face displaced with respect to thebottom by 0.10 mm when tangential force of4 104N is applied at the top with bottomsurface fixed. The r igidity modulus of themater ial of the cube is [2001 M]1) 4 109Nm-2 2) 5 109Nm-23) 8 109Nm-2 4) 108Nm-2

17. A metal r ope of densi ty 6000kgm-3 hasbreaking stress 9.8x108 Nm 2. This rope isused to measure the depth of the sea. Thenthe depth of the sea that can be measuredwithout breaking is -----1) 10x103 m 2) 20x103 m

3) 30x103 m 4) 40x103 m

18. The upper end of a wire of radius 4mm andlength 100cm is clamped and its other end istwisted through an angle of 300 then the angleof shear is1) 0.0120 2) 0.120 3) 1.20 4) 120

19. A uniform metal rod of 2mm2 cross section isheated from 00C to 200C. The coefficient ofl i near expansion of the r od i s12 106/C. I ts youngs modulus of elasticityis 1011 Nm2. The ener gy stor ed per uni tvolume of the rod is1) 2880 Jm3 2) 1500 Jm33) 5760 Jm3 4) 1440 Jm3

20. Two spr ings of spr ing constants 1500N/m and3000 N/m respectively are stretched by thesame force. The potential energy possessedby the two will be in the ratio [MP PMT 1998]1) 4:1 2) 1:4 3) 2:1 4) 1:2

21. A boy has a catapult made of a rubber cordof length 42cm and diameter 6.0mm. The boystr etches the cor d by 20cm to catapul t apebble of mass 20gm. The pebble flies off witha speed of 20m/sec. Find the young's modulusfor rubber. Ignore the change in the cross-section of the cord in stretching

Hint : 12 Fe =

12 mV

2

1) 5.97 x 106 N/m2 2) 2.97 x 106 N/m23) 0.97 x 106 N/m2 4) 9.7 x 106 Nm2

22. One end of a long metallic wire of length L istied to the ceiling. The other end is tied tomassless spr ing of spr ing constant K.A mass(m) hangs freely fr om the fr ee end of thespr ing. Th area of crossection and youngsmodulus of the wire are A and Y respectively.I f the mass is slightly pulled down and releasedit will oscillate with a time per iod T equal to

1) km

2 2) YAK

)KLYA(m2

3) KL

mYA2 4)

YAmL

2

23. On loading a metal wir e of cr oss section106 m2 and length 2m by a mass of 210 kg itextends by 16mm and suddenly broke fromthe point of support. I f density of that metali s 8000 kgm3 and i ts speci f i c heat i s420 Jkg1 K 1 the r ise in temperature of wireis ( g = 10 ms-2)1) 2.50c 2) 50c 3) 60c 4) 100c

AKASH MULTIMEDIA 55


ANSWERS1) 1 2) 2 3) 1 4) 1 5) 26) 1 7) 2 8) 1 9) 1 10) 3

11) 1 12) 3 13) 1 14) 1 15) 116) 2 17) 2 18) 2 19) 1 20) 321) 2 22) 2 23) 1

EXERCISE - III(B)(HOME WORK)

* 1. A steel wire of length 1.5 m and diameter 0.5mm is stretched by applying a weight of 1 kgwt. Then find the increase in the length of wire(Y=2 x 1011 Nm2)1) 0.2743 mm 2) 0.3743 mm3) 0.1743 mm 4) 0.4743 mm

2. The length of a rubber cord is l1 meters whenthe tension 4N and l2 meters when the thetension is 5N. The or iginal length of wire inmeters is1) 5l1 - 4l2 2)5l2 -4l13) 9l1- 8l2 4) 8l2 - 5l1

3. The elongation of a steel wire, strectched bya force is e. I f a wire of the same mater ial,double the length and half the diameter issubjected to double the force, its elongationwill be [1999 E]1) 16e 2)4e 3)(1/4)e 4) (1/16)e

* 4. A brass wire of cross sectional area 2 mm2 issuspended from a r igid support and a body ofvolume 100 cm3 is attached to its other end.I f the decrease in the length of the wire is0.11mm, when the body i s completelyimmersed in water, find the natural length ofthe wire.(Ybr ass=0.91x1011Nm2, water =103kgm3)1) 2.043 m 2) 1.043 m3) 4.043 m 4) 3.043 m

5. Two wires of the same length and radius arestretched by the same load and the differenceof their elongations produced is found to be0.25 cm . I f their modulus of elasticity are inthe ratio 5:3. Then the individual elongations are1) 0.375cm, 0.625 cm 2) 0.625 cm, 0.650 cm3) 0.325 cm, 0.350cm4) 0.675 cm, 0.325cm

* 6. An aluminium wire and a steel wire of thesame length and cross section are joined endto end. The composite wire is hung from ar igid support and a load is suspended fromthe free end. I f the increase in length of thecomposite wire is 1.35 mm, find the ratio ofthe i) stress in the two wires and ii) strains inthe two wires.(YAl=7x1010Nm2, Ysteel=2x1011Nm2)1) 1 : 1, 20 : 7 2) 1 : 1, 10 : 73) 20 : 7, 1 : 1 4) 10 : 7, 1 : 1

7. An Aluminium wire and a steel wire of thesame length and cross section are joined endto end. The composite wire is hung from ar igid support and a load is suspended fromthe free end. The young's modulus of steel 20/7 times that of the Aluminium. The ratio ofincrease of length of steel to Aluminium is ....1) 20/7 2) 400/49 3) 7/20 4) 49/400

8. The stress versus strain graphs for wires oftwo mater ials A and B are as shown in thefigure. I f YA and YB are the Young's moduliiof the mater ials, then1) YB = 2YA2) YA = YB

030

060

AB

X

Y

Stre

ssStrainO

3) YB = 3YA4) YA = 3YB

9. One end of a uniform wire of length andweight W is attached to a point in the roofand weight W/2 is suspended from the lowerend. I f A is the area of cross section of thewire then stress in the wire at its mid point is1) W/2A 2) 2W/A 3) 3W/2A 4) W/A

10. When a mass is suspended from the end of awire the top end of which is attached to theroof of the lift the extension is 'e' when thelift is stationary. I f the lift moves up with aconstant acceleration g/2 the extension of thewire would be ....1) 2e/3 2) 3e/2 3) 2e 4) 3e

11. A steel wire of diameter 0.8mm and length1m is clamped firmly at two points A and Bwhich ar e 1m apar t and in the samehor izontal plane. A body is hung from themiddle point of the wire such that the middlepoint sags 1cm lower f r om the or iginalposition. Calculate the mass of the body. GivenY = 2 x 1012 dyne/cm21) 60 gr 2) 82 gr 3) 100gr 4) 112 gr

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12. A 20kg load is suspended from the lower endof a wire 20cm and 1mm2 in cross sectionalarea. The upper half of the wire is iron andthe lower half is aluminiumFind the elongation in the wire(Y iron= 20x1010 Nm2 YA1=7x1010 Nm2 )1) 18.9x103m 2) 18.9x105m3) 1.78x103m 4) None

13. A mass m kg is whir led in a vertical plane bytying it at the end of a flexible wire of lengthl and area of cross section 'A' . When the massis at its lowest position the strain produced inthe wire is, if Youngs modulus of the wire is'Y' (if V= g5 )

1) mgAY6 2) AY

mg63)

AYmg5

4) mgAY5

* 14.A block of mass 2 kg is attached to one end ofa wire of cross sectional area 1 mm2 and iswhir led in a vertical circle of radius 40 cm.The speed of the block at the bottom of thecircle 5ms1. Find the elongation of the wirewhen the block is at the top of the ci rcle.(Y=2x1011Nm2)1) 0.2892 mm 2) 0.054 mm3) 0.1446 mm 4) 0.0017 mm

15. The body of mass 5kg is attached to a wire0.2m long its breaking strength is 5x 107 Nm-2I t can be rotated in a hor izontal cir cle withmaximum angular velocity 5rads-1, then thearea of cross-section of the wire is1) 5 x10-7m2 2) 5 x 10-6m2

2) 5 x 10-4m2 4) 25 x10-6m2

16. A rectangular block of size 10 cm x 8 cm x 5cm is kept i n thr ee di f fer ent posi t i ons(I ) 10 cm x 8 cm on ground.(I I ) 5 cm x 8 cm on ground.(I I I ) 10 cm x 5 cm on ground.In each case, the shadded area is r igidly fixedand a definite force F is applied tangentiallyto the opposite face to deform the block. Thedisplacement of the upper face will be1) Same in all the three cases2) Maximum in (I) position3) Maximum in (II) position4) Maximum in (III) position

17. A metal r ope of densi ty 5000kgm-3 hasbreaking stress 19.6x108 Nm 2. This rope isused to measure the depth of the sea. Thenthe depth of the sea that can be measuredwithout breaking is -----1) 50x103 m 2) 20x103 m3) 30x103 m 4) 40x103 m

18. A 2 m long rod of radius 1 cm which is fixedfrom one end is given a twist of 0.8 radians atthe other end. The shear strain developed willbe1) 0.002 2) 0.004 3) 0.008 4) 0.016

19. A steel wire of mass 3.16Kg is stretched to atensile strain of 1 10-3. What is the elasticdeformation energy if density =7.9g/cc andY=2 1011N/m2 ? [ROORKEE 1998]1) 4KJ 2) 0.4KJ

3) 0.04KJ 4) 4J

20. A spring of force constant 'K' is cut into two equalparts. The force constant of each part is1) 2K 2) K/2 3) 4K 4) K/4

21. The rubber cord of a catapult has a crosssectional area 1mm2 and total unstr etchedlength 10cm. I t is stretched to 12cm and thenreleased to proj ect a missile of mass 5gm.Taking Y for rubber 5x108 Nm2 calculate thevelocity of projection ...1) 10 ms1 2) 15 ms1

3) 20 ms1 4) 25 ms1

22. Two spr ings A and B have thei r for ceconstants K 1 and K 2 (K 1 > K2) when they arestretched by the same force work done is1) More in case of A2) More in case of B3) Same in both the cases4) None

23. The limiting stress of a typical human bone is0.9x108 Nm2 whi le young' s modulus i s1.4x1010 Nm2. How much ener gy can beabsorbed by two leg bones without breakingif each has a typical length of 50cm and anaverage cross sectional area of 5cm2 ?1) 145 J 2) 290 J3) 490 J 4) 435 J

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ANSWERS1) 2 2) 1 3) 1 4) 1 5) 16) 1 7) 3 8) 4 9) 4 10) 2

11) 2 12) 1 13) 2 14) 2 15) 116) 4 17) 1 18) 2 19) 3 20) 121) 3 22) 2 23) 1

COMPREHENSION QUESTIONS

Comprehension - I (1 - 3) :A wi r e of cr oss-sect ion A i s st r etchedhor izontally between two clamps located 2lmetre apart. A weight W kg is suspended fromthe mid point of the wire. I f the mid point sagsvertically through a distance x

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ADDITIONAL MATERIAL FOR AIEEE

1. Relation among elastic contants Y, n, k,

1) 9 1 3y k n 2) y = 2n (1 + )

3) y = 3k (1 2 ) 4) = 3k 2n

2 n 3k

EXERCISE IV

1. The Young's modulus and r igidity modulus ofa metal are 40 GPa and 20 GPa respectively.Then the bulk modulus is1) 13.33GPa 2) 26.67GPa

3) 6.67GPa 4) 120GPa

2. For a given mater ial, young's modulus is 2.6times the r igidity modulus. The poisson's ratiois---1) 0.5 2) 0.3 3) 0.2 4) 0.6

3. I f the Poissions ratio of a substance is 0.25then the ratio of its Youngs modulus andr igidity modulus is1) 2/5 2) 5/2 3) 4/5 4) 5/4

4. The Young's modulus of steal is 210 GPa. I tspoissions ratio is 0.3. Then its r igidity modulusis1) 8.8GPa 2) 80.8GPa

3) 19.6GPa 4) 6.6GPa

5. For a given mater ial the Youngs modulus is2.4 t imes that of i ts r igidi ty modulus i tspoission's ratio is1) 2.4 2) 1.2 3) 0.4 4) 0.2

6. I f the Poission's ratio of a solid is 2/5. Findthe ratio of its Young's modulus to the r igiditymodulus1) 5 / 4 2) 7 / 15

3) 14 / 9 4) 14 / 5

7. I f the shear modulus and bulk modulus of amater ial ar e 0.42x1011Nm2 and0.56 x 1011 Nm 2 respectively then young'smodulus of that mater ial is ---1) 1010 Nm 2 2) 1.33x1011 Nm 2

3) 0.75x1011 Nm 2 4) 1011 Nm 2

8. Y,K ,n represent respectively the young' smodulus, bulk modulus and r igidity modulusof a body. I f r igidity modulus is twice the bulkmodulus, then [1999 M]

1) K185y 2) K5

9y

3) 59y 4) K5

18y

9. I f for mater ial Young's modulus is 6.61010N/m2 and bulk modulus is 111010N/m2 then thePossion's ratio is ...1) 0.4 2) 0.6 [ 1993 E ]2) 1 4) infinity

10. For a given mater ial the Young's modulus is2.4times that of r igidity modulus. I ts Possion'sradio is [1990 M]1) 2.4 2) 1.2 3) 0.4 4) 0.2

ANSWERS1) 1 2) 2 3) 2 4) 2 5) 46) 4 7) 4 8) 4 9) 1 10) 4

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Elasticity 2