Elasticity 2

PHYSICS - I C

4.

5.

ELASTICITY

Elasticity : The property of a body by virtue of which it regains its original size and shape when deformation forces are removed is called elasticity.

i) L ongitudinal str ess : If the restoring forces are perpendicular to the area of cross section and are along the length of the wire, the stress is called longitudinal stress.

Elasticity is a molecular phenomenon.

During longitudinal stress, the body undergoes change in length but no change in shape and volume.

El asti c body : A body which shows elastic behaviour is called elastic body Eg. Steel, rubber Nearest approach to perfectly elastic body is quartz.

6.

Pl astic body : A body which does not show elastic behaviour is called plastic body. Eg : Putty, chewing gum, lead solder, wax.

7.

A body which is more difficult to strain is more elastic.

Eg: i) Steel is more elastic than rubber ii) Springs are made of steel but not of copper because steel is more elastic than copper.

8. 9.

Hammering, rolling and addition of impurity increases elasticity. Annealing and rise in temperature decreases elasticity.

ii) Tangential stress (or shear ing stress) : If the restoring forces are parallel to the surface, the stress is called shearing stress. During tangential stress the shape of the body changes, size and volume remain same. iii) Bulk str ess (or volume str ess) : If a body is subjected to equal forces normally on all the faces the stress involved is called bulk stress.

13. Str ain : The deformation produced per unit magnitude is called strain l

change in length

a) longitudinal strain = original length = l change in volume

v v

10. For invar steel, the elastic property does not change with temperature.

b) Bulk strain = originalvolume =

11. Str ess : The restoring force developed per unit area of the deformed body is called stress

lateral displacement between two layers perpendicular distance between the two layers

Re storing force F Stress = cross sec tional area  A

c) Shearing strain =  =

change in radius

d) Transverse strain = original radius =

r r

e) Shearing strain = 2 x longitudinal strain F = External force applied

f) Bulk strain = 3 x longitudinal strain

A = r2 = area of cross section

g) Longitudinal strain : shearing strain : Bulk strain = 1 : 2 : 3

r = radius of the wire Unit :

14. The maximum value of the stress within which

N dyne 2 or pascal , m cm2 1

-1

Dimensional Formula : M L T Stress is a tensor

12. Str ess is of thr ee types : i) Longitudinal stress

-2

the body regain its original size and shape completely is called elastic limit.

15. Hooke’s law : Within the elastic limit, stress is directly proportional to strain.

ii) Tangential stress

stress = E = constant strain

iii) Bulk stress

E is modulus of elasticity

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37

PHYSICS-1C

ELASTICITY Unit of E :

N dyne or pascal , 2 m2 cm

Thermal stress =

i) within the elastic limit, stress - strain graph is a straight line passing through the origin Slope of the straight line is E ii) A spring balance works on the principle of Hooke’s law. iii) Modulus of elasticity does not depend upon the dimensions of the body but it is a property of the material of the body. iv) Within the elastic limit, the load-extension graph is a straight li ne passing through the origin.

16. Young’s modulus : Y= longitudinal stress longitudinal strain Y=

FI Ae Mgl

If load is M, then F = Mg and y= pr 2e where r is radius of the wire. i) The stress required to double the length of a wire or to produce 100% longitudinal strain is equal to Young’s modulus of the wire. ii) Y of a perfectly elastic material is infinite and that of a perfectly inelastic material is zero. iii) If a load ‘M’ produces an elongation e in a wire, the rise in temperature required to produce the same elongation is  t . Then Mg  YAa t

where A is area of cross section of the wire and  is coefficient of linear expansion of the material of the wire. iv) when a metal bar is fixed between two walls has a rise in temperature, the bar tries to expand and exerts force on the walls. This force is called thermal force. Thermal force F = Y A a q where  is co-efficient of linear expansion of the bar  is rise in temperature. Y is young’s modulus, A is area of cross section. Thermal force is independent of length of the bar. v) Ther mal str ess : Thermal force per unit area is called thermal stress

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Thermal force area

=

YA A

= Y

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

 2d g e 2Y l =length of the wire, d=density of the wire Y = young’s modulus of the material of the wire g = acceleration due to gravity vii)Two wires made of same material having lengths l 1 & l 2 and radii r1 and r2 are subjected to the same f or ce. Then the ratio of their elongations is e1 l1 r22 l   (since e  2 ) e2 l2 r12 r viii) Two wires made of same mater i al having lengths l 1 and l 2 and masses m1 and m2 are subjected to the same force. Then the ratio of their elongations is e1 l12 m2 l2  2 (since e  ) e2 l2 m1 m ix) Two wires of same mater i al and same volume having areas of cross section A1 & A2 are subjected to the same force. Then the ratio of their elogations is e1 A22 1  2 (since e  2 ) e2 A1 A x) Two wires of same mater i al and same volume having radii r 1 and r 2 are subjected to the same f or ce. Then the ratio of their elongations is e1 r24 1  4 (since e  4 ) e2 r1 r 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 immersed in a non-viscous liquid of density dL then its new elongation is

e1 = e(1–

dL dB ) 38

PHYSICS - I C

ELASTICITY

ix) Two wires of same length and radius have youngs modulii Y1 & Y2 are subjected to the same force. Then the ratio of their elongations is

e1 Y2  e 2 Y1

(since e 

l ) Y

x) Two wires of same length and radius are joined end to end and loaded. If the youngs modulii of the materials of the wires are Y1 & Y2 and the combination behaves as a single wire then its young's modulus is

xi) Two wires of same length and radius are joined in parallel and loaded. If the youngs moduli of the materials of the wires are Y1 & Y2 and the combination behaves as a single wire then its young's modulus is

Y1  Y2 2

17. Rigidity modulus, n =

20. Bulk modulus K =

volume stress volume strain

=

VP V

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 its expansion, P  K g q

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, then

2Y Y Y 1 2 Y1  Y2

Y

1  2 where  is in radians. W

F Tangential stress = A shearing strain

where  is shearing strain i) If n is low for a wire then it can be twisted very easily. ii) Since phosphorbronze has very low rigidity modulus, it is used as suspension wire in moving coil galvanometer. iii) For a fluid (liquid or gas) the rigidity modulus is zero. 18. A rod of length 'l ' and radius 'r' is fixed at one end. If the other free end is twisted through an angle '' then the angle of shear  is given by

f  r q

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

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decrease in its volume  V 

h d gV 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 = P where  is the ratio of two specific heats. 23. Compr essi bi l i t y : The reciprocal of bulk modulus is called compressibility, C = 1/K For incompressible substances , C = 0; K =  . 24. Possion’s ratio () =

lateral contraction strain  r/r  =  longitudinal elongation strain  l /l  i) Poisson’s ratio has no units and no dimensions ii) Theoretical limits of  : -1 to 0.5 iii) Practical limit of  : 0 to 0.5 iv) If  = 0.5 then the substance is perfectly incompressible. v) The Poisson’s ratio of a given material is '  '. When the load is applied on it, the longitudinal l elongation strain is . Then the fractional l change in its volume is V l  1  2s  V l

25. i) If large deformation occurs between the elastic limit and the breaking point, the material is ductile. 39

PHYSICS-1C

ELASTICITY

Ex : copper, silver, gold etc., ii) If the deformation between the elastic limit and the breaking point is very small or if the wire breaks immediately after crossing the elastic limit, the material is brittle. Ex : glass, ceramic etc. 26. Br eaking str ess : i) The breaking stress of a wire is the maximum stress the material can withstand. ii) Breaking stress =

Breaking Force initial area of cross section

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

i) For a perfectly elastic body, elastic after effect is zero. ii) For a perfectly inelastic body, elastic after effect is infinite. 29. Work done in stretching a wire, w= i) w =

Breaking Stress dg

27. Elastic fatigue : a) The state of temporary loss of elastic nature due to continuous strain is called elastic fatigue. b) Due to elastic fatigue i) 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 the removal of deforming force by a material is called its elastic after effect.

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x stretching force x extension.

1 2

Fe =

1 YAe 2 2 l

1 F2l 1 F2l = 2 AY 2 r 2 y

=

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

work done  volume

(a strain energy per unit volume

or elastic potential energy per unit volume) = 1 2

x stress x strain

2

1  Stress = x Y x (strain) = 2 Y iv) The total work done per unit volume in stretching an object upto the limits of perfect elasticity is known as its resiliance. v) Area under F-e graph gives the work done or the strain energy stored in the wire. Area = 1 F.e = W 2 1 2

2

30. Spr ings : i) A spring that obeys Hooke’s law has force constant or spring constant

The maximum length of a wire that can hung without breaking under its own weight is l

1 2

Y=

Fl Ax

; F=

YA x l

and F = Kx

x is elongation Where 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 =

YA l

1 l iii) If a spring (or a wire) of force constant K is cut into ‘n’ equal parts, the force constant of each part of the wire is ‘nk’. iv) Potential energy of a stretched spring = ii) K Y, K A, K 

1 2

Fx =

1 2

Kx2 =

1 F2 2 K

v) Two springs have force constant K1 and K2 a) When they are stretched by the same force and if their elastic energies are E

E1 and E2 then

E

1 2

K =

K

2 1

40

PHYSICS - I C

ELASTICITY

b) When they are extended by the same length, E E

1 2

K =

K

When a spr ing is str etched by a for ce, the r esultant str ain in known as 1) Longitudinal strain2) Sheariing strain 3) Volume strain 4) Bulk strain

4.

For a gas elastic limit 1) Exists 2) Does not exist 3) Exists only at absolute zero 4) Exists for a perfect gas

5.

Hooke' s law states that 1) Stress is inversely proportional to strain 2) Stress is directly proportional to strain 3) Stress is independent of strain 4) Stress is proportional to elastic modulus

6.

Within elastic limit, if str ess is doubled the str ain becomes 1) Half 2) Twice 3) Thrice 4) None

7.

The law which gover ns the wor king of a spr ing balance is 1) Kepler's law 2) Robert Hooke's law 3) Newton's law 4) Young's law

8.

Accor ding to Hooke' s law of elasticity the r atio of str ess to str ain 1) Does not remain constant 2) Remains constant 3) Increases 4) Decreases

9.

A body subj ected to str ain a number of times does not obey Hooke's law due to 1) Yield point 2) Breaking stress 3) Elastic fatigue 4) Permanent set

1 2

c) When they are extended till their F

energies are same then

1

F

=

2

K1 K2

d) The potential energy of a spring increases, whether it is stretched or compressed. e) Springs in series, Keff =

K K

1 2

K +K 1

f)

3.

2

Springs in parallel , Keff = K + K 1

2

g) The reciprocal of spring constant is called compliance.

31. i) When a spring is stretched strain involved is shearing strain ii) When a wire is stretched modulus of elasticity involved is Young’s modulus iii) When a wire is twisted modulus of elasticity involved is rigidity modulus.

32. Torsion of a cylinder : When one end of a cylinder of length 'l' and radius 'r' is clamped and the other end is given a twist '  ', then the restoring torque is given by

 nr   2l Where 'n' is the modulus of rigidity of the material of the cylinder.

EXERCISE - I 1.

2.

THEORI TI CAL QUESTI ONS A spir al spr ing is str etched by a for ce the r esultant str ain pr oduced in the spr ing is 1) Volume strain 2) Shearing strain 3) Tensile strain 4) None of the above Reason for the deformation of a regular body 1) Bulk strain 2) Shearing strain 3) Linear strain 4) Lateral strain

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10. The modulus of elasti city is dimensionally equivalent to 1) Stress 2) Surface tension 3) Strain 4) Coefficient of viscosity 11. Gr aph between the appl ied for ce and the change in length of a wir e within elastic limit is a 1) Straight line with negative slope 2) Straight line with zero slope 3) Straight line with positive slope 4) None of the above 41

ELASTICITY

12. Which of the following substances has the highest elasticity 1) Rubber 2) Steel 3) Copper 4) Wood 13. A steel wir e is str etched by 5 kg wt. I f the r adius of the wi r e is doubl ed its Young' s modulus 1) Remains unchanged 2) Becomes double 3) Becomes half 4) Becomes 1/4 times 14. A copper wir e and a steel wir e of the same diameter and length ar e connected end to end and a for ce is applied which str etches their combined length by 1 cm. Then the two wir es will have 1) The same stress and strain 2) The same stress but different strain 3) The same strain but different stresses 4) Different stresses and strain 15. The effect of temper atur e on the value of modulus of elasticity for differ ent substances in gener al 1) Increases with increase in temperature 2) Decreases with increase in temperature 3) Independent of change in temperature 4) Does not change with temperature 16. A stone is suspended in a tub of water with copper wire. Another stone of equal mass is suspended in ker osene with equal length of copper wir e then 1) Young's modulus is more in the first case 2) Young's modulus is more in the second case 3) Elongation will be less in the former case 4) Elongation will be more in the former case 17. I n the given figur e if the dimension of the wir es ar e the same and mater ials ar e differ ent Young's modulus is less for 1) A 2) B 3) Both 4) None

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PHYSICS-1C

18. A heavy mass is attached to a thin wir e and is whir eled in a ver tical cir cle. The wir e is most likely to br eak 1) When the mass is at the lowest point 2) When mass is at the highest point 3) When wire is horizontal 4) When mass is at an angle of upward vertical

 1   cos 1   2

from

19. Steel is pr efer r ed for making spr ings over copper because 1) Young's modulus of steel is more than that of copper 2) Steel is cheaper 3) Young's modulus of copper is more 4) Steel is less likely to be oxidised 20. The possible value of Poission's r atio is 1) 0.9 2) 0.8 3) 0.4 4) 1 21. The Poisson's r atio can not have the value 1) 0.7 2) 0.2 3) 0.1 4) – 0.5 22. The Poission' s r atio (s) shoul d satisfy the r elation 1) –1 <  < 0.5 2) – 0.5 < < 1 3) 0.5 < < 1 4) – 1 < < – 0.5 23. The br eaking str ess of a wir e depends on 1) Material of a wire 2) Shape of cross section 3) Length of the wire 4) Radius of the wire 24. Br eaking str ess per unit ar ea of cr oss section of a wir e is called 1) Yield point 2) Breaking stress 3) Ductibility 4) Tensile stress 25. The pr oper ty of metals wher eby they could be dr awn into thin wir es beyond their elastic limit without br eaking is 1) Ductility 2) Malleability 3) Elasticity 4) Hardness 42

PHYSICS - I C

ELASTICITY

26. W hen an el ast i c mat er i al wi t h young' s modulus 'y' is subj ected to a str etching str ess 'S' the elastic ener gy stor ed per unit volume of the mater ial is ---

YS 1) 2

S2Y 2) 2

S2 3) 2Y

4)

S 2Y

27. A wir e of length 'L ' and cr oss sectional ar ea ' A ' i s made up of a mat er i al of young' s modulus 'Y'. I f the wir e is str etched by 'X' the wor k done is 1)

YAX L

3) YAX2

YAX2 2 YAX2 4) 2L

2)

28. Str ain ener gy per unit volume in a str etched str ing is 1)

1  stress strain 2

3) stress x (strain)2

2) stress x strain 4)

stress strain

29. When a wir e is str etched to double it's length a) Radius is halved b) Str ain is unity c) Stress is equal to Youngs modulus d) Str ess is unity 1) 'a' and 'b' are correct 2) 'b' and 'c' are correct 3) 'c', 'd' are correct 4) 'd' is correct 30. The for mula for str ain ener gy per unit volume a) 1/2 (stress)(str ain) b) 1/2 Y (str ain)2 (strain) 1 1 c) x y (str ess) d) x 2 2 Y 1) a, b are correct 2) a, c are correct 3) c, d are correct 4) a, b, c are correct 31. Regar ding elasticity which of the following statements ar e cor r ect? a) the point wher e the elasticity ends and plasticity star ts is called yield point b) br eaking str ess depends on the ar ea of cr oss section c) the value of poisions r atio can not be gr eater than 0.5 d) air is mor e elastic than water 1) a, b are correct 2) a, c are correct 3) b, d are correct 4) a and d are correct

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32. A spr ing of young' s modulus 2  1011 pa is suspended ver tically and subj ected to a load of 5 kg and elongation is 2 mm. when theload is doubled – match the following (g =9.8 m/s2) Section - A Section - B a)Elongating for ce e) 4 mm b)Str ess f) Un changed c)Elongation g) 98 N d)Young's modulus h)Doubled 1) a–h; b–g; c–f; d–e 2) a–g; b–h; c–e; d–f 3) a–e; b–f; c–g; d–h 4) a–f; b–e; c–g; d–h 33. Section - A a) Young modulus

Section - B e) r esistance to change in shape b) Rigidity modulus f) r esistance to change in linear dimension c) Shear modulus g) r esistance to change in volume d) Str ess h) r esistance to against defermation forces 1) a–f; b–g; c–e; d–h 2) a–g; b–h; c–e; d–f 3) a–f; b–e; c–g; d–h 4) a–g; b–e; c–h; d–f

34. Of the following four wir es of same material

which will have maximum increase in length and which will have minimum incr ease in length if all ar e under the action of same force? Ar r ange the wires in the descending or der of incr ease in length. i) Natur al length=0.5m, diameter=0.5mm ii) Natur al length=1m, diameter =1mm iii) Natur al length=1.5m,diameter=1.5mm iv) Natural length=2.0m, diameter =2mm 1) i, ii, iii, iv 2) ii, i, iv, iii 3) i, iii, iv, ii 4) iv, iii, ii, i 43

ELASTICITY

35. Four springs of force constants K 1=1000N/m, K 2=1500 N/m, K 3=2500 N/m and K 4=2000 N/m ar e subj ect ed t o di f f er ent l oads pr oduci ng same extensi on. A r r ange t he spr ings with wor kdone in descending or der 1) 1,2,3,4 2) 3, 4, 1, 2 3) 3, 2, 1, 4 4) 3, 4, 2, 1 36. Four wires made of Al, Cu, Fe and steel and of lengths l 1 = 100m, l 2=150m, l 3=250m and l 4= 350m, br eak due to their own weights. Ar r ange these wi r es wi th thei r br eak i ng str esses in ascending or der dAl = 7500 kg/m3 dcu = 7800 kg/m3 dFe = 9000 kg/m3 and dsteel = 11000 kg/m3 1) St, Fe, Al, Ci 2) St, Fe, Cu, Al 3) Al, Cu, Fe, St 4) Fe, St, Al, Cu I nstr uction For Questions : I n each of t he f ol l owi ng quest i ons, a statement of Asser tion (A) is followed by a cor r esponding statement of r eason (R) j ust below it. Of the statement mar k the cor r ect answer. 1) Both ‘A’ and ‘R’ are true and ‘R’ is the correct explanation of ‘A’ 2) Both ‘A’ and ‘R’ are true and ‘R’ is not the correct explanation of ‘A’ 3) ‘A’ is true but ‘R’ is false 4) ‘A’ is false but ‘R’ is true 37 . A) : Str ess is Restor ing for ce per unit ar ea R) : I nt er at omi c f or ces i n sol i ds ar e responsible for the proper ty of elasticity. 38. A) : Steel is more elastic than r ubber. R) : Under given defor ming for ce, steel is defor med less than r ubber. 39. A) : Bulk modulus of elasticity (k) r epr esents incompr essibility of the mater ial. R) : Bulk modulus of elasticity is pr opor tional to change in pr essur e. 40 . A) : A steel wir e of natur al length 'l' br eaks, when elongated by length 'x'. I f or iginal length is halved and elongated by x/2 it br eaks. R) : Br eaking for ce is dir ectly pr opor tional to the length of the wir e

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PHYSICS-1C

41 . A) : Silver is a ductile mater ial R) : For a ductile mater ial yield point and Br eaking polnt ar e seper ated by lar ger di stance than for Br ittle mater ials on thestr ess–str ain cur ve. 42. A) : The str etching of a coil is deter mined by its shear modulus. R) : Shear modulus change only shape of a body keeping its dimensions unchanged. 43. A) : Str ain is a unitless quantity. R) : Str ain is equivalent to for ce. 44. A) : The bridges declared unsafe after a long use. R) : Elastic strength of bridges losses with time. 45. A) : Two identical solid balls, one of ivor y and the other of wet-clay ar e dr opped fr om the same height on the floor. Both the balls will r ise to same height after bouncing. R) : I vor y is more elastic than wet-clay. 46. A) : Young's modulus for a per fectly plastic body is zer o. R): For a per fectly plastic body, r estor ing for ce is zer o. 47. A) : I dentical spr ings of steel and copper ar e equally str etched. Mor e wor k will be done on the steel spr ing. R) : Youngs modulus of steel is higher than copper. 48. Consider the statements A and B and identify the cor rect answer given below. A ) I f t he vol ume of a body r emai ns unchanged, When subj ected to tensile str ain, the value of Possion's r atio is 1/2 B) Phosphor bronze has low Young's modulus and high r igidity modulus [2003 M ] 1) A and B are correct 2) A and B ar wrong 3) A is correct and B is wrong 4) A is wrong and B is right 49. Wor k done per unit volume is stretching a wire 1) is equal to half the product of the stress and strain 2) is equal to the product of the stress and the strain 3) is euqal to the ratio of the stress to the strain 4) is equal to the ratio of the strain to stress 44

PHYSICS - I C

ELASTICITY

50. The br eaking str ess of wir e depends upon [AI EEE 2002] 1) length of wire 2) radius of wire 3) material of wire 4) shape of cross section

EXERCISE - II(A) (CLASS WORK)

YOUNG 'S MODULUS

51. Which of the following affects the elasticity of a substance [AI EEE 2002] 1) hammering and annealing 2) change in temperature 3) impurity in substance 4) all of these

* 1. A copper wir e of 1 mm diameter is str etched by applying a for ce on 10N. Find the str ess in the wir e.

52. Consider an ideal monoatomic gas of volume at pressureP. The bulk modulus at constant temper ature is [UPSEAT 2000]

* 2. A tungsten wir e of length 20 cm is str etched by 0.1 cm. Find the str ain on the wir e. 1) 0.002 2) 0.005 3) 0.001 4) 0.004 * 3. I n the Sear l e’s method to deter mi ne t he Young’s modulus of a wir e, a steel wir e of length 156 cm and diameter 0.054 cm is taken as exper imental wir e. The aver age incr ease in length for 1 1/2 kg wt is found to be 0.050 cm. Then the Young’s modulus of the wire. 1) 3.002x1011 N/m2 2) 1.002x1011 N/m2 3) 2.002x1011 N/m2 4) 2.5x1011 N/m2

1)

p 2

2) P

3)  P

4)

P dP dV

53. I f k is the for ce constant of a spr ing the wor k done [against r estor ing for ce] in incr easing its extension fr om l1to l 2 will be 1) K(l 2  l1 )



3) K l 2 2  l12

2) K / 2(l 2  l1 )



4)

K 2 2 l 2  l1 2





54. A wir e of length L and cr oss -section 'A' has youngs modulus of mater ial Y. I t is stretched by an amount ' x' . The wor k done against r estor ing for ce is [M P PET 1996]

YAx 2 L 2YAx 2 4) L ANSWERS

YAx 2L YAx 2 3) 2L 1)

1) 1.273 x 107N/m2 2) 1.373 x 107N/m2 3) 1.473 x 107N/m2 4) 1.573 x 107N/m2

4.

The ar ea of cr oss-section of a wir e is 10–5m2 when its length is incr eased by 0.1% a tension of 1000N is produced. The Young's modulus of the wir e will be ---1) 1012 Nm– 2 2) 1011 Nm– 2 9 –2 3) 10 Nm 4) 1010 Nm–2

5.

The following four wir es ar e made of the same mater ial. Which of these will have the lar gest elongation when the same tension is applied 1) l = 50cm and diameter 0.5mm 2) l = 100cm and diameter 1.0mm 3) l = 200cm and diameter 2.0mm 4) l = 300cm and diameter 3.0mm

2)

1) 2

2) 2

3) 2

4) 2

5) 2

6) 2

7) 2

8) 2

9) 3

10) 1

11) 3

12) 2

13) 1

14) 2

15) 2

16) 3

17) 2

18) 1

19) 1

20) 3

21) 1

22) 1

23) 1

24) 4

25) 1

26) 3

27) 4

28) 1

29) 2

30) 1

31) 2

32) 2

33) 1

34) 1

35) 4

36) 3

37) 2

38) 1

39) 3

40) 3

41) 1 46) 1 51) 4

42) 1 47) 1 52) 2

43) 3 48) 3 53) 4

44) 1 49) 1 54) 4

45) 4 50) 3

AKASH MULTIMEDIA

* 6. A copper wir e and an aluminium wir e have lengths in the r atio 3 : 2, diameter s in the r atio 2 : 3 and for ces applied in the r atio 4 : 5. Find the r atio of the incr ease in length of the two wir es. (Y Cu=1.1x1011Nm–2, Y Al =0.70x1011Nm–2) 1) 110 : 189

2) 180 : 110

3) 189 : 110

4) 80 : 11 45

PHYSICS-1C

ELASTICITY

* 7. Ther e ar e two wir es of same mater ial. Their r adii and lengths ar e both in the r atio 1 : 2. I f the extensions pr oduced ar e equal, what is the r atio of the loads ? 1) 1 : 2 2) 2 : 1 3) 1 : 4 4) 4 : 1 8.

I f str ess i s numer i cal l y equal to young' s modulus the elongation will be 1)

1 the original length 4

2)

1 the original length 2

3) Equal to the original length 4) Twice the original length

9.

An al umi ni um r od has a br eak ing str ai n 0.2% . The minimum cr oss-sectional ar ea of the r od in m2 in or der to suppor t a load of 104N is if Youngs modulus is 7x109Nm–2 1) 1.7x10– 4 2) 1.7x10– 3 3) 7.1x10– 4 4) 1.4x10– 4

10. A str ing of cr oss section 2cm2 is doubled in length by the application of a longitudinal for ce 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 r adius 'r ' and cr oss sectional area A is fitted into a wooden circular disc of r adius R (R > r ) . I f the Young's modulus of the mater ial of the r ing is Y, the force with which the metal r ing expands is : [2004 E] 1)

AYR r

2)

AY( R  r ) r

3)

Y (R  r) Ar

4)

YR AR

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

1 . When the tension on the wir e is 5N its length is  2 . Find its natur al length.

1) 51  4 2

2) 41  5 2

3) 101 82

4) 81 102

AKASH MULTIMEDIA

13. When a metal spher e is suspended at the end of a metal wir e its extension is 0.4mm. I f another metal spher e of the same mater ial wi t h i t s r adi us hal f as t he pr evi ous i s suspended then extension would be 1) 0.05mm 2) 0.02mm 3) 0.01mm 4) 0.32mm 14. A wir e whose cr oss - sectional ar ea is 4 mm2 is str etched by 0.1 mm by a cer tain load. I f a si mi l ar wi r e of doubl e t he ar ea of cr oss-section is under the same load, then the elongation would be 1)0.5mm 2) 0.05 mm 3) 0.005mm 4) 5mm 15. When a tension 'F' is applied, the elongation pr oduced in unifor m wir e of length l, r adius r i s e, when t ensi on 2F i s appl i ed, t he elongation pr oduced in another unifor m wir e of l ength 2l and r adius 2r made of same mater 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 in the r atio 1:2 and r atio of lengths is 1:1. Under the application of same str ess the r ati o of elongations is --1) 1:1 2) 1:2 3) 2:1 4) 1:4 17. A wir e is str etched by 0.01m by a cer tain for ce 'F' another wir e of same mater ial whose diameter and lengths ar e double to or iginal wir e is str etched by the same for ce then its elongation will be .... 1) 0.005m 2) 0.01m 3) 0.02m 4) 0.04m 18. A steel wir e is 1m long and 1mm2 in ar ea of cr oss-section. I f it takes 200 N to str etch this wir e by 1mm how much for ce will be r equir ed to str etch the wir e of the same ar ea and of same material having length of 10m to 1002 cm 1) 100 N 2) 200 N 3) 400 N 4) 2000 N 19. Two bar s A and B of cir cular cross section and of same volume made of same mater ial are subj ected to tension. I f the diameter of A is half that of B and if the for ce applied to both the r ods is the same and it is in the elastic limit the r atio of extension of A to that of B will be --1) 16 2) 8 3) 4 4) 2 46

PHYSICS - I C

20. An ir on wir e of length 4m and diameter 2mm is loaded with a weight of 8kg. I f the Young's modulus 'Y' for ir on is 2x1011Nm– 2 then the increase in length of the wire is ---( g = 10 ms-2) 1) 0.2mm 2) 0.5mm 3) 2 mm 4) 1 mm 21. Two wires of the same mater ial have masses in the r atio 3:4 . The r atio of their extensions under the same load if their lengths ar e in the r atio 9:10 is 1) 5 : 3 2) 27 : 40 3) 6 : 5 4) 27 : 25 22. Two exactly similar wires of steel (Y = 20 x 1011 dy/cm2) and copper (Y=12x1011dy/cm2) ar e stretched by equal forces. If the total elongation is 1cm elongation of copper wire is 1) 3/5 cm 2) 5/3 cm 3) 3/8 cm 4) 5/8 cm 23. Two wires of same length and thickness are j oined end to end. Their Young's moduli are 15x1010pa and 20x1010pa. I f the combina-tion is str etched by a cer tain load, the elongations of these wir es will be in the r atio 1) 3:4 2) 4:3 3) 1:1 4) 1:2 * 24. A steel wir e of length 20 cm is str etched to incr ease its length by 0.2 cm. Find the later al str ain in the wire if the Poisson’s r atio for steel is 0.19. 1) 0.0019 2) 0.0008 3) 0.019 4) 0.008 25. Two r ods of di f f er ent mat er i al s havi ng coefficients of linear expansion 1 and 2 and youngs modulii Y1 and Y2 r espectively ar e fixed between two r igid massive walls.The r ods ar e heated such that they under go the same incr ease in temp. Ther e is no bending of r ods. I f  1 :  2 = 2:3 . The ther mal str esses developed in the two r ods ar e equal if Y1 : Y2 is equal to 1) 4 : 9 2) 3 : 2 3) 9 : 4 4) 2 : 2 * 26. A tungsten wire, 0.5 mm in diameter, is j ust str etched bet ween two f i xed poi nt s at a

AKASH MULTIMEDIA

ELASTICITY

temper atur e of 400C. Deter mine the tension in the wir e when the temper atur e fall s to 200C. (coef fi ci ent of l i near expansi on of tungsten = 4.5 x 10–6/0C; Young’s modulus of tungsten = 3.45 x 1010Nm–2) 1) 6.097N 2) 3.097N 3) 5.097N 4) 7.097N * 27. A unifor m steel r od of length 1m and ar ea of cr oss section 20 cm2 is hanging fr om a fixed suppor t. Find the incr ease in the length of the r od. (Y steel =2.0x1011Nm–2, steel =7.85x103kgm–3) 1) 1.923 x 10 –5 cm

2) 2.923 x 10 –5 cm

3) 1.123 x 10 –5 cm

4) 3.123 x 10 –5 cm

RI GI DI TY M ODUL US & BUL K M ODUL US : * 28. A 2.0 cm cube of some substance has its upper face displaced by 0.15 cm, by a tangential for ce of 0.30N fixing its lower face. Calculate the 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 vol ume 1000 cm 3 i s subj ected to a pr essur e of 10 atmospher e. The change in volume is 10–2cm3. I f the ball is made of ir on find its bulk modulus. (Atmospher ic pr essur e =1x105Nm–2) 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 actional change i n vol ume of oi l i s 1 per cent when a pr essur e of 2x107 Nm–2 i s applied. Calculate the bulk modulus and its compr essibility 1) 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. Deter mine the pr essur e r equir ed to r educe the gi ven vol ume of water by 2% . Bul k modulus of water is 2.2x109Nm–2. 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 47

PHYSICS-1C

ELASTICITY

32. W hen 200ml of wat er i s subj ect ed t o a pr essur e of 2x106 Pa. T he decr ease in i ts volume is 0.2ml. The compr essibility of water is --1) 5x10– 8 m2N– 1 2) 5x10– 10 m2N– 1 – 12 2 – 1 3) 5x10 mN 4) None 33. A cubical ball is taken to a depth of 200m in a sea. The decr ease in volume obser ved to be 0.1% . T he bul k modul us of the bal l i s -( g = 10 ms-2) 1) 2x107 Pa 2) 2x106 Pa 9 3) 2x10 Pa 4) 1.2x109 Pa 34. A mat er i al has poi ssi ons r ati o 0.5 . I f a unifor m r od of it suffer s a longitudinal str ain of 2  103 , then the per centage incr ease in its volume is 1) 0% 2) 10% 3) 20% 4) 5% 35. The incr ease in length of a wir e on str eching is 0.025% . I f its poisson r atio is 0.4, then the per centage decr ease in the diameter is : [2004 M ] 1) 0.01 2) 0.02 3) 0.03 4) 0.04 36. A mater ial has nor mal density  and bulk modulus K . The incr ease in the density of the mater ial when it is subj ected to an exter nal pr essur e 'p' fr om all sides is 1)

P K

2)

KP 

3)

P K

K

4) p

37. A wir e can be br oken by 400 kg.wt. The load r equi r ed to br eak the wi r e of doubl e the thickness of the same mater ial will be 1) 800 kg wt 2) 1600 kg wt 3) 3200 kg wt 4) 6400 kg wt

STRAI N ENERGY 40. A metal rod of Young's modulus 2 1010 Nm-2 undergoes an elastic strain of 0.02% the energy per unit volume stored in the r od in joule/m3 is 1) 400

2) 800

3) 1200

4) 1600

41. A br ass r od has length 0.2m ar ea of cr oss secti on 1.0cm 2 and Young' s modul us 1011Nm–2. If it is compressed by 5kg. wt along its length then the change in its energy will be 1) An increase of 2.4  10–5 J 2) A decrease of 2.4  10–5 J 3) An increase of 2.4  107 J 4) A decrease of 2.4  107 J

* 42. A metal wi r e of length 2.5 m and ar ea of cr oss section 1.5x10–6m2 is str etched thr ough 2 mm. Cal cul at e t he wor k done dur i ng str etching. (Y=1.25x1011Nm–2) 1) 0.15 J

2) 0.51 J

3) 1.5 J

4) 5.1 J

* 43. A steel wir e of length 1.5 m and ar ea of cr oss section 1.5 mm2 is str etched by 1.5cm. then t he wor k done per uni t vol ume. (Y=2x1011Nm –2) 1) 1 x 107J/m3 2) 2 x 107J/m3 7 3 3) 3 x 10 J/m 4) 4 x 107J/m3 * 44. When a steel r od of length 2m is compr essed its length decr eases by 0.1cm. Find the wor k done agai nst t he compr essi ve st r ess. (Y steel =2.0x1011Nm –2) 1) 2.5 x 104J 2) 1.25 x 104 J 4 3) 5.0 x 10 J 4) 7.5 x 104J

38. A cabl e br eaks i f str etched by mor e than 2mm. I t is cut into two equal par ts. By how much either par t can be str etched without br eaking 1) 0.25mm 2) 0.5mm 3) 1mm 4) 2mm

45. The youngs modulus of the mater ial of a r od is 20x1010 pascal. W hen the l ongi tudi nal str ain is 0.04% , The ener gy stor ed per unit volume is 1) 4 x 10-3 J/m3 2) 8 x 103 J/m3

39. T he br eak i ng st r ess of st eel i s 7.9x109 Nm–2 and density of steel is 7.9x103 kgm–3 and g = 10ms–2. The maximum length of steel wir e that can hang ver tically without br eaking is 1) 103 m 2) 104 m 3) 102 m 4) 105 m

46. The wor k done in str etching a wir e by 1mm is 2J. The wor k necessar y for str etchi ng another wir e of same mater ial but with double the r adius of cr oss-section and half the length by 1mm is in Joules [1991 M ] 1) 16 2) 8 3) 4 4) 1/4

AKASH MULTIMEDIA

3) 16 x 10-3 J/m3

4) 16 x 103 J/m3

48

PHYSICS - I C

ELASTICITY

47. Two wir es of same r adi us and l ength ar e subj ected to the same load. One wir e is of steel and the other is of copper. I t the Young's modulus of steel is twice that of copper, the r atio of elastic ener gy stor ed per unit volume in steel to that of copper wir e is [UPSEAT 2000] 1) 1:2

2) 2:1

3) 1:4

4.

1) AY

2) 2 7) 1 12) 1 17) 1 22) 4 27) 1 32) 2 37) 2 42) 1 47) 1

3) 3 8) 3 13) 1 18) 3 23) 2 28) 1 33) 3 38) 3 43) 1 48) 2

4) 2 9) 3 14) 2 19) 1 24) 1 29) 1 34) 1 39) 4 44) 1

5) 1 10) 2 15) 2 20) 2 25) 2 30) 3 35) 1 40) 1 45) 4

EXERCISE - II(B) (HOME WORK)

YOUNG 'S MODULUS

* 1. A brass wire of diameter 1 mm and of length 2m is str etched by applying a for ce of 20N. Then the str ess in the wir e is 1) 2.546x107N/m2 2) 3.146x107N/m2 3) 2.346x107N/m2 4) 2.246x107N/m2 * 2. I f an ir on wir e is str etched by 1% what is the str ain on the wir e ? 1) 0.01 2) 0.005 3) 0.02 4) 0.03 * 3. A brass wire of diameter 1 mm and of length 2m is str etched by applying a for ce of 20N. I f the incr ease in length is 0.51 mm, Then the Young’s modulus of the wire is 1) 8.848x1010N/m2 2) 7.984x1010N/m2 3) 6.984x1010N/m2 4) 9.984x1010N/m2 AKASH MULTIMEDIA

AY 10

3)

AY 100

4)

AY 1000

Two wir es of same length and mater ial but with r adii 0.1mm and 0.2mm ar e loaded with the same weight. Then 1) Thick wire extends more 2) Thin wire extends more 3) Both undergo same extension 4) None

6.

A wir e of length 'L ' and r adius 'r ' is fixed at one end and a for ce 'F' is applied to the other end to pr oduce an extension 'l'. The extension pr oduced in another wir e of same mater ial of length '2L' and radius '2r' by a force '2F' is -1) l 2) 2l 3) l /2 4) 4l

ANSWERS 1) 1 6) 3 11) 2 16) 3 21) 4 26) 1 31) 1 36) 3 41) 1 46) 1

2)

5.

4) 4:1

48. A spr ing of for ce constant 800N/m has an ext ensi on of 5cm. T he wor k done i n extending it fr om 5cm to 15cm is [AI EEE 2002] 1) 16J 2) 8J 3) 32J 4) 24J

For ce r equir ed to incr ease the length of wir e of ar ea of cr oss section 'A' by one per cent, if 'Y' is young's modulus, is

* 7. Two wires of different mater ial have same lengths and ar eas of cr oss section. What is the r atio of their incr eases in length when the for ces applied ar e the same ? (Y 1=0.90x1011Nm–2, Y 2=3.60x1011Nm–2) 1) 4 : 1 2) 1 : 4 3) 2 : 1 4) 1 : 2 8.

The str ess r equir ed to double the length of a wire of Young's modulus 'E' is --1) E 2) 2E 3) 4E 4) E/2

9.

The length of a br ass wir e is 1m. The str ain pr oduced by a load is 0.001. The change in its length is 1) 1cm 2) 1mm 3) 10 cm 4) 2 cm

10. The Young's modulus of a mater ial having a cr oss-sectional ar ea of 1cm2 is 2x1012 dyne/ cm2. I f the length is to be doubled the for ce r equir ed is 1) 4x107 N 2) 2x107 N 3) 107 N 4) Zero 11. A metal r ing of inner r adius r 1 and cr osssectional ar ea 'A' is fitted onto a wooden disc of r adius r 2 > r 1. I f 'Y' is the young's modulus of the metal then the tension in the r ing is 1)

AYr2 r1

2)

AY(r2  r1 ) r1

3)

Y(r2  r1 ) Ar1

4) Ar 2

Yr1

49

PHYSICS-1C

ELASTICITY

12. The length of a metal wir e is l when the tension in it is 'F' and 'xl' when the tension is 'yF'. Then the natur al length of the wire is ( y  x) l ( x  y) l 1) 2) y 1 x 1 ( y  x) l ( x  y) l 3) 4) y 1 x 1

19. A hollow cylinder of inner r adius 3 cm and outer r adius 5 cm and a sol id cylinder of r adius 2 cm ar e subj ected to the same for ce. I f they ar e made of same mater ial and of same l engt h, t hen t he r at i o of t hei r elongations is 1) 1 : 1 2) 1 : 2 3) 1 : 4 4) 2 : 3

13. Two wir es made of the same mater ial have lengths 3 m & 4 m and masses 18 gm and 16 gm r espectively . When they ar e subj ected to t he same f or ce, t hen t he r at i o of t hei r elongations is 1) 1 : 2 2) 4 : 3 3) 16 : 18 4) 9 : 4

20. A wir e of length 10 m and of cr oss -section 1 x 10-4 sq.m is fixed at one end and a load of 1200 kg is attached at the other end. Then the extension pr oduced is ( Y= 9.8 x 1010N/m2) 1)12 x 10-3 m 2) 12 x 10-5 m 3) 12 x 10-4 m 4) 12 x 10-2 m

14. Two wires of same mater ial and same length but diameter s in the r atio 2:3 ar e str etched by the same length. The r atio of the for ces applied on two wir es is ---1) 4:3 2) 4:9 3) 2:5 4) 9:4 15. When a unifor m wir e of r adius 'r ' is str etched by a 2kg weight, the incr ease in its length is 2.00 mm. I f the r adius of the wir e is r /2 and other condi ti ons r emai ni ng the same, the incr ease in its length is [2000 E] 1) 2.00mm 2) 4.00mm 3) 6.00mm 4) 8.00mm 16. The extension of a wir e by application of load is 0.3cm. The extensi on in a wir e of same mater ial but of double the length and half the r adius of cr oss section by the same load will be in (cm) 1) 0.3 2) 0.6 3) 0.2 4) 2.4 17. A wir e whose cr oss sectional ar ea is 2mm2 is str etched by 0.1mm by a cer tain load, I f a similar wir e of tr iple the ar ea of cr oss section is str etched by the same load the elongation of the second wir e would be 1) 0.33mm 2) 0.033mm 3) 3.3mm 4) 0.0033mm 18. A br ass wir e of length 300 cm when subj ected to a for ce F pr oduces an el ongation " a" . Another wir e of twice the diameter and of same length and mater ial , when subj ected to the force F produces an elongation b. Then the value of 'a/b' is 1) 1 : 1 3) 2 : 1

2) 4 : 1 4) 1 : 2

AKASH MULTIMEDIA

21. Two wires of the same mater ial but of lengths in the r atio 2:3 and diameter s in the r atio of 3:2 ar e str etched with equal for ce. Then the r atio of elongations pr oduced is 1) 9 : 4 2) 1 : 13) 8 : 27 4) 9 : 16 22. An ir on wir e and copper wir e having same length and cr oss-section ar e suspended fr om same roof Young's modulus of copper is 1/3r d that of ir on. Then the r atio of the weights to be added at their ends so that their ends ar e at the same level is 1) 1:3 2) 1:9 3) 3:1 4) 9:1 23. Two wires A and B of same mater ial have lengths in the r atio 1:2 and diameter s in the r atio 2:1. I f they ar e str etched by same for ce then the r atio of the incr ease of their lengths will be ---1) 2:1 2) 1:4 3) 8:1 4) 1:8 *24. Find the fr actional incr ease in the volume of a wi r e of ci r cul ar cr oss sect i on i f i t s

longitudinal str ain is 2% .   0.4 1) 0.004 3) 0.001

2) 0.002 4) 0.005

25. Two rods of different mater ials are clamped at their ends r igidly. When they ar e heated f or t he same r i se i n temper at ur e, same ther mal str esses ar e pr oduced in them. I f their Young's modulii are in the r atio x:y then ratio of coefficients of their linear expansion is .... 1) x : y 2) y : x 3) x2 : y2 4) y2 : x2 50

PHYSICS - I C

* 26. A steel wir e of cr oss sectional ar ea 2 mm2 is j ust str etched hor izontally between two fixed poi nt s at a t emper at ur e of 350C. T he temper atur e of the wir e changed to 250C. Then (Coefficient of linear expansion of steel = 11x10–6/0C;Y steel =2x1011Nm –2) 1) the thermal strain is 11x10–5 ; 2) the thermal stress is 2.2 x 107 N/m2 3) the tension in the wire at temperature 250C is 44N 4) All the above 27. A r ope of rubber of density 1.5 x103 Kg/m3 and Young's modulus 5 x 106 N/m2 , 8m in length is hung fr om the ceiling of a r oom. Then the incr ease in length due to its own weight is 1) 96 x 10-2 m 2) 12 x 10-4 m 3) 96 x 10-3 m 4) 12 x 10-3 m RI GI DI TY M ODUL US & BUL K M ODUL US : * 28. A tangential for ce of 1000N i s appli ed on upper sur face of ar ea 2 mm2 which is 0.05 m fr om its fixed face. Find the shift of the upper sur face wi th r espect to its fixed face if its modulus of r igidity is 0.30 x 1011 Nm–2. 1) 0.833 mm 2) 0.734 mm 3) 0.433 mm 4) 0.533 mm * 29. A copper cube of each side of length 1 cm is subj ected to a pr essur e of 100 atmospher es. Fi nd the change in its vol ume if the bulk modulus of copper is 1.4x1011Nm–2. (Atmospher ic pr essur e=1x105Nm –2) 1) 0.7143 x 10–10 m3 2) 0.5143 x 10–10 m3 3) 0.4143 x 10–10 m3 4) 0.8243 x 10–10 m3 30. I f a r ubber ball is taken down to a 100 m deep l ak e i ts vol ume i s decr eased by 0.1% . I f g = 10 m/sec2 . Then the bulk modulus of elasticity is 1) 109Pa 2) 1010 Pa 3)10 11 Pa 4)3x10 9 Pa * 31. Find the pr essur e to be applied to incr ease the density of water by 1% if the bulk modulus of water is 2.2x109Nm–2. 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

AKASH MULTIMEDIA

ELASTICITY

32. A spher e contr acts in volume by 0.01% when taken to the bottom of sea 1km deep. I f the density of water is 1gm /cc Then the bulk modulus of water is 1) 9.8  105 Nm–2 2) 9.8  108 Nm–2 3) 9.8  1010 Nm–2 4) 9.8  106 Nm–2 33. A ball falling in a lake of depth 200m shows a decr ease of 0.1% in its volume at the bottom. What is the bulk modulus [AFM C 1997] 8 2 1) 19.6  10 N/m 2) 19.6  10-10 N/m2 3) 19.6  1010 N/m-2 4) 19.6  10 -8 N/m-2

34. A uni f or m cube i s subj ect ed to vol ume compr ession. I f each side is decr eased by 1% . Then the bulk str ain is 1) 0.01 2) 0.06 3) 0.02 4) 0.03 35. The longitudinal str ain in a metal bar is 0.05. I f the Poission' s r atio for the metal is 0.25, the later al str ain is 1) 0.0125 2) 0.4 3) 0.6 4) 0.025 * 36. Estimate the change in the density of water

in ocean at a depth of 500 m below the sur face. The density of water at the surface = 1030 kgm–3 and thebulk modulus of water = 2.2x109Nm–2 1) 2.363 kg/m3 2) 1.363 kg/m3 3) 4.363 kg/m3 4) 3.363 kg/m3 37. The br eaking for ce of a wir e is 'F'. When the thickness of the wir e is doubled, the br eaking for ce will be 1) 2F 2) 4F 3) F 4) 8F 38. A wi r e can suppor t a l oad M g wi thout br eaking. I t is cut into two equal par ts. The maximum load that each par t can suppor t is 1) Mg/4 2) Mg/2

3) Mg

4) 2Mg

39. Br eaking str ess for steel is 8x106 Nm– 2. The density of steel is 8x103 kgm– 3 and g = 10ms– 2. What will be the maximum length of steel wir e which will hang without br eaking under its load. 1) 200m 2) 100m 3) 50m 4) 25m 51

PHYSICS-1C

ELASTICITY

STRAI N ENERGY 40. A rod of Young's modulus 20 GPa under goes a linear str ain of 6x10– 4. Then incr ease in its ener gy density is --1) 3600 Jm– 3 2) 7200 Jm– 3 – 3 3) 1800 Jm 4) None

48. A man of mass 100 kg gets into a car. I f the cent r e of gr avi t y of t he car syst em i s depr essed by 5mm. Spr ing constant of the system is --1) 1.96x105 Nm– 1 2) 104 Nm– 1 3) 1.2 Nm– 1 4) Infinity

ANSWERS

41. A wir e is suspended ver tically fr om one of it's ends is str etched by attaching a weight of 200N to the lower end. The weight str etches the wir e by 1mm. Then the elastic ener gy stor ed in the wir e is [AI EEE 2003] 1) 0.2J 2) 10J 3) 20J 4) 0.1J * 42. A br ass wire of length 1m and area of cross section 1 mm2 is str etched by applying a force of 20N. then the work done per unit volume of the wir e is(Y br ass=0.91x1011Nm–2) 1) 2198 J 2) 1049 J 3) 4422 J 4) 4521 J * 43. When a copper wir e of ar ea of cr oss section 1 mm2 is str etched by a for ce of 10N. Then t he wor k done i n str etchi ng t he wi r e i s (Y=1.2x1011Nm –2) 1) 416.7 J 2) 208.35 J 3) 833.4 J 4) 316.7 J * 44. A steel wir e of length 80 cm and of ar ea of cr oss section 1.5 mm2 is str etched. Find the wor k done in incr easing its length from 80.2 cm to 80.3 cm. (Y=2.0x1011Nm–2) 1) 0.9375 J 2) 0.5375 J 3) 0.6374 J 4) 0.8695 J 45. A metal wir e 4m l ong and 2x10–7 sq.m in cr oss-section is str etched by a for ce of 30N. I f the wor k done in str etching that wir e is 4.5x10–2J the young's modulus of the wir e is 1) 2x1011 Pa 2) 4x1011 Pa 12 3) 2x10 Pa 4) 4x1012 Pa 46. A l ong spr i ng when str etched by 2cm its potential ener gy is U. I f the same spr ing is str etched by 8cm the potential ener gy of the spr ing is ---1) 8U 2) 16U 3) 4U 4) U 47. Two wires of same mater ial and size but with lengths in the r atio 5:3 ar e str etched by the same for ce. The r atio of wor k done in two cases is 1) 5:8 2) 8:5 3) 5:3 4) 3:5

AKASH MULTIMEDIA

1) 1

2) 1

3) 4

4) 3

5) 2

6) 1

7) 1

8) 1

9) 2

10) 2

11) 2

12) 2

13) 1

14) 2

15) 4

16) 4

17) 2

18) 2

19) 3

20) 1

21) 3

22) 3

23) 4

24) 1

25) 2

26) 4

27) 3

28) 1

29) 1

30) 1

31) 1

32) 3

33) 1

34) 4

35) 1

36) 1

37) 2

38) 3

39) 2

40) 1

41) 4

42) 1

43) 1

44) 1

45) 1

46) 2

47) 3

48) 1

EXERCISE - III(A) (CLASS WORK)

* 1. What mass must be suspended fr om the fr ee end of a tungsten wir e of length 50 cm and diameter 0.25 mm to str etch it by 1 mm (Y of tungsten = 3.448 x 1011 Nm–2) 1) 3.455 kg 2) 4.445 kg 3) 1.445 kg

2.

The length of a r ubber cor d is l 1 meter s when the tension is 4N and l 2 meter s when the the tension is 5N. The length in meter s when the tension is 9 N is 1) 5l 1 - 4l 2 2)5l 2 -4l 1 3) 9l1- 8l2

3.

4) 2.445 kg

4) 8l2 - 5l1

A wi r e of l ength 1m and r adi us 1mm i s subj ected to a load. The extension is 'x' . The wir e is melted and then dr awn into a wir e of squar e cr oss-section of side 1mm. What is its extension under the same load? 1)  2 x 3) x

2) x 2 4)  / x 52

PHYSICS - I C

ELASTICITY

(Y steel =2x1011Nm –2, water =103kgm–3) 1) 0.06533 mm 3) 0.04533 mm

5.

2) 0.02533 mm 4) 0.01533 mm

Two wir es of differ ent mater ial s, each 2m long and of diameter 2mm, ar e j oined in ser ies to for m a composite wir e. What for ce will pr oduce a total extension of 0.9 mm. ( Y 1 = 2 x 1011pa & Y 2 = 6 x 1011 Pa) 1)282.6 N 2) 212 N 3) 319.8 N 4) 382.6 N

* 6. An aluminium wir e and a copper wir e of the same length and cr oss section ar e j oined end to end. The composite wir e is hung fr om a r igid suppor t and a load is suspended fr om the fr ee end. if the incr ease in length of the composite wir e is 1.8mm, find the incr ease in the length of each wir e. (Y Al =7x1010Nm –2, Y Cu=1.1x1011Nm–2) 1) 1.1 mm, 0.7 mm 2) 0.7 mm, 1.1 mm 3) 2.2 mm, 1.4 mm 4) 1.4 mm, 2.2 mm 7.

Two wires A & B are identical in shape and size and ar e str etched by same magnitude of for ce. Then the extensions ar e found to be 0.2% and 0.3% respectively. Find the r atio of their Young's modulii 1) 2 : 3 2) 3 : 2 3) 4 : 9 4) 9 : 4

8.

One end of a unifor m wir e of length L and of weight W is attached r igidly to a point in the r oof and a weight W 1 is suspended fr om its lower end. I f 'S' is the ar ea of cr oss- section of the wir e, then find the str ess in the wir e at height (3L /4) fr om its lower end?

 w1   3w/4   1)  S   3)

w1  w 4

AKASH MULTIMEDIA

 w   3w1/4   S  

2) 

 3w1   w/4   4)  S  

9.

The adj acent gr aph shows the extension  of a wir e of length 1 m suspended fr om the top of a r oof at one end wi th a l oad W connected to the other end. I f the cr oss sectional ar ea of the wir e is 10–6m2, calculate the young' s modulus of the mater ial of the wir e 1) 2 1011N / m 2 2) 2 1011N / m 2

l x 10-4m

* 4. A steel wir e of length 1 m and cr oss sectional ar ea 1.5 mm2 is hung fr om a r igid suppor t, wi th a stone of vol ume 2000 cm3 hangi ng fr om the other end. Find the decr ease in the l engt h of t he wi r e, when t he st one i s compl et el y i mmer sed in wat er.

4 3 2 1

3) 3 1012 N / m 2 13

20 40 60 80

W(N) 4) 2 10 N / m 10. A lift is tied with thick wir es and the mass of the lift is one metr ic ton. I f the maximum accel er at i on of the l i f t i s 1ms–2 and the maximum str essof the wir e is 1.4x108Nm–2. I f g = 10ms– 2 the minimum diameter of that wir e is ---1) 10m 2) 10– 1m 3) 10– 2m 4) None 2

11. A light r od of length 200cm is suspended fr om the cei l i ng hor i zontal l y by means of two ver tical wir es of equal length tied to its ends. One of the wir es is made of steel and is of cr oss-secti on 0.1sq.cm and the other i s of br ass of cr oss-section 0.2 sq.cm. Find the position along the r od at which a weight may be hung to pr oduce (i) equal str esses in both wir es and (ii) equal str ains in both wir es. Y br ass = 10x1011 dyne/cm2 and Y steel = 20 x 1011 dyne/cm2 1) 133.3cm; 100cm 2) 167cm; 50cm 3) 200cm; 100cm

4) none

12. A wir e 3m in length and 1mm in diameter at 300 c and k ept i n a l ow temper at ur e at 1700c and is str etched by hanging a weight of 10kg at one end calculate the change in the l ength of the wi r e. Gi ven  =1.2x10–5/ 0C Y=2x1011 Nm–2 g=10ms–2 1) 2x10–3 m 2) 7.2x10–3 m 3) 95.6 x 10–5 m

4) 5.2 m 53

PHYSICS-1C

ELASTICITY

13. A long steel wir e of length 'L ' is suspended from the ceiling of a room. A sphere of mass ' m' and r adius 'r 1' is attached to the lower end of the wir e. the height of the ceiling is (L +2r 1+l). When the spher e is made to oscillate as a pendulum, its lowest point j ust touches the floor. The velocity of the spher e at the lowest point will be ( L >> r 1,l and r is the r adius of the wir e) 1)

r2 ye  Lg m

2)

r 2 y  Lg me

yr2e 4) None m * 14. A block of mass 2 kg is attached to one end of a wir e of cr oss sectional ar ea 1 mm2 and is whir led in a ver tical cir cle of r adius 40 cm. The speed of the block at the bottom of the cir cle 5ms–1. Find the elongation of the wir e when t he bl ock i s at t he bot t om. (Y=2x1011Nm –2) 1) 0.2892 mm 2) 0.054 mm 3) 0.1446 mm 4) 0.0017 mm

3)

* 15. A stone of mass 2 kg is fastened to one end of a steel wir e of cr oss sectional ar ea 2 mm2 and is whir led in a hor izontal cir cle of r adius 25cm. I f the br eaking str ess of steel is 1.2x109Nm–2 find the maximum number of r evolutions the stone can make per minute without the str ing br eaking.

1) 661.5 rpm 3) 221.5 rpm

2) 331.5 rpm 4) 441.5 rpm

16. A metal cube of side l ength 8.0cm has i ts upper sur face displaced with r espect to the bottom by 0.10 mm when tangential for ce of 4  104N i s appli ed at the top with bottom sur face fixed. The r igidity modulus of the mater ial of the cube is [2001 M ] 1) 4  109 Nm-2 2) 5  109 Nm-2 3) 8  109 Nm-2 4) 108Nm-2 17. A met al r ope of densi t y 6000k gm -3 has br eaking str ess 9.8x108 Nm– 2. This r ope is used to measur e the depth of the sea. Then the depth of the sea that can be measur ed without br eaking is ----1) 10x103 m 2) 20x103 m 3) 30x103 m

4) 40x103 m

AKASH MULTIMEDIA

18. The upper end of a wir e of r adius 4mm and length 100cm is clamped and its other end is twisted thr ough an angle of 300 then the angle of shear is 1) 0.0120 2) 0.120 3) 1.20 4) 120 19. A unifor m metal r od of 2mm2 cr oss section is heated fr om 00C to 200C. The coefficient of l i near expansi on of t he r od i s 12  10–6/ºC. I ts youngs modulus of elasticity i s 1011 Nm –2. T he ener gy stor ed per uni t volume of the r od is 1) 2880 Jm–3 2) 1500 Jm–3 3) 5760 Jm–3 4) 1440 Jm–3 20. Two spr ings of spr ing constants 1500N/m and 3000 N/m r espectively ar e str etched by the same for ce. The potential ener gy possessed by the two will be in the r atio [MP PMT 1998] 1) 4:1 2) 1:4 3) 2:1 4) 1:2 21. A boy has a catapult made of a r ubber cor d of length 42cm and diameter 6.0mm. The boy str etches the cor d by 20cm to catapul t a pebble of mass 20gm. The pebble flies off with a speed of 20m/sec. Find the young's modulus for r ubber. I gnor e the change in the cr osssection of the cor d in str etching Hint :

1 1 Fe = mV2 2 2

1) 5.97 x 106 N/m2 2) 2.97 x 106 N/m2 3) 0.97 x 106 N/m2 4) 9.7 x 106 Nm2 22. One end of a long metallic wir e of length L is tied to the ceiling. The other end is tied to massless spr ing of spr ing constant K .A mass (m) hangs fr eely fr om the fr ee end of the spr ing. Th ar ea of cr ossection and youngs modulus of the wire are A and Y respectively. I f the mass is slightly pulled down and r eleased it will oscillate with a time per iod T equal to 1) 2

m k

2) 2

m( YA  KL ) YAK

3) 2

mYA KL

4) 2

mL YA

23. On l oadi ng a metal wi r e of cr oss secti on 10–6 m2 and length 2m by a mass of 210 kg it extends by 16mm and suddenly br oke fr om the point of suppor t. I f density of that metal i s 8000 k gm –3 and i t s speci f i c heat i s 420 Jkg–1 K –1 the r ise in temper atur e of wir e is ( g = 10 ms-2) 1) 2.50c 2) 50c 3) 60c 4) 100c 54

PHYSICS - I C

ELASTICITY

1) 1 6) 1 11) 1 16) 2 21) 2

2) 2 7) 2 12) 3 17) 2 22) 2

3) 1 8) 1 13) 1 18) 2 23) 1

4) 1 9) 1 14) 1 19) 1

5) 2 10) 3 15) 1 20) 3

EXERCISE - III(B) (HOME WORK) * 1. A steel wir e of length 1.5 m and diameter 0.5 mm is str etched by applying a weight of 1 kg wt. Then find the incr ease in the length of wir e (Y=2 x 1011 Nm–2) 1) 0.2743 mm 2) 0.3743 mm 3) 0.1743 mm 4) 0.4743 mm 2.

The length of a r ubber cor d is l 1 meter s when the tension 4N and l 2 meter s when the the tension is 5N. The or iginal length of wir e in meter s is 1) 5l 1 - 4l 2 2)5l 2 -4l 1 3) 9l1- 8l2 4) 8l2 - 5l1

3.

The elongation of a steel wir e, str ectched by a for ce is e. I f a wir e of the same mater ial, double the length and half the diameter is subj ected to double the for ce, its elongation will be [1999 E] 1) 16e 2)4e 3)(1/4)e 4) (1/16)e

* 4. A br ass wir e of cr oss sectional ar ea 2 mm2 is suspended fr om a r igid suppor t and a body of volume 100 cm3 is attached to its other end. I f the decr ease in the length of the wir e is 0.11mm, when t he body i s compl et el y immer sed in water, find the natur al length of the wir e. (Y br ass=0.91x1011Nm –2, water =103kgm–3) 1) 2.043 m 2) 1.043 m 3) 4.043 m 4) 3.043 m 5.

Two wires of the same length and r adius are str etched by the same load and the differ ence of their elongations pr oduced is found to be 0.25 cm . I f their modulus of elasticity ar e in the ratio 5:3. Then the individual elongations are 1) 0.375cm, 0.625 cm 2) 0.625 cm, 0.650 cm 3) 0.325 cm, 0.350cm 4) 0.675 cm, 0.325cm

AKASH MULTIMEDIA

* 6. An aluminium wir e and a steel wir e of the same length and cr oss section ar e j oined end to end. The composite wir e is hung fr om a r igid suppor t and a load is suspended fr om the fr ee end. I f the incr ease in length of the composite wir e is 1.35 mm, find the r atio of the i) str ess in the two wir es and ii) str ains in the two wir es. (Y Al =7x1010Nm–2, Y steel =2x1011Nm –2) 1) 1 : 1, 20 : 7 2) 1 : 1, 10 : 7 3) 20 : 7, 1 : 1 4) 10 : 7, 1 : 1 7. An Aluminium wir e and a steel wire of the same length and cr oss section ar e j oined end to end. The composite wir e is hung fr om a r igid suppor t and a load is suspended fr om the fr ee end. The young's modulus of steel 20/ 7 times that of the Aluminium. The r atio of increase of length of steel to Aluminium is .... 1) 20/7 2) 400/49 3) 7/20 4) 49/400 8. The str ess ver sus str ain gr aphs for wir es of two mater ials A and B are as shown in the figure. I f Y A and Y B are the Young's modulii of the mater ials, then Y A 1) YB = 2YA B 2) YA = YB 3) YB = 3YA

Stress

ANSWERS

600

300

X O Strain 9. One end of a unifor m wir e of length  and weight W is attached to a point in the r oof and weight W/2 is suspended fr om the lower end. I f A is the area of cross section of the wir e then str ess in the wire at its mid point is 1) W/2A 2) 2W/A 3) 3W/2A 4) W/A 10. When a mass is suspended fr om the end of a wir e the top end of which is attached to the r oof of the lift the extension is 'e' when the lift is stationar y. I f the lift moves up with a constant acceler ation g/2 the extension of the wir e would be .... 1) 2e/3 2) 3e/2 3) 2e 4) 3e 4) YA = 3YB

11. A steel wir e of diameter 0.8mm and length 1m is clamped fir mly at two points A and B whi ch ar e 1m apar t and i n t he same hor izontal plane. A body is hung fr om the middle point of the wir e such that the middle poi nt sags 1cm l ower f r om the or i gi nal position. Calculate the mass of the body. Given Y = 2 x 1012 dyne/cm2 1) 60 gr 2) 82 gr 3) 100gr 4) 112 gr 55

PHYSICS-1C

ELASTICITY

12. A 20kg load is suspended fr om the lower end of a wir e 20cm and 1mm2 in cr oss sectional ar ea. The upper half of the wir e is ir on and the lower half is aluminium Find the elongation in the wir e (Y iron= 20x1010 Nm–2 Y A1=7x1010 Nm–2 ) 1) 18.9x10–3m 2) 18.9x10–5m 3) 1.78x10–3m 4) None 13. A mass m kg is whir led in a ver tical plane by tying it at the end of a flexible wir e of length l and ar ea of cr oss section 'A'. When the mass is at its lowest position the str ain pr oduced in the wire is, if Youngs modulus of the wire is 'Y' (if V= 5g ) 1)

AY 6mg

2)

6 mg AY

3)

5mg AY

4)

AY 5mg

* 14. A block of mass 2 kg is attached to one end of a wir e of cr oss sectional ar ea 1 mm2 and is whir led in a ver tical cir cle of r adius 40 cm. The speed of the block at the bottom of the cir cle 5ms–1. Find the elongation of the wir e when the block is at the top of the ci r cle. (Y=2x1011Nm –2) 1) 0.2892 mm 2) 0.054 mm 3) 0.1446 mm 4) 0.0017 mm 15. The body of mass 5kg is attached to a wir e 0.2m long its br eaking strength is 5x 107 Nm-2 I t can be r otated in a hor izontal cir cle with maximum angular velocity 5r ads-1, then the ar ea of cr oss-section of the wir e is 1) 5 x10-7 m2 2) 5 x 10-6m2 2) 5 x 10-4m2 4) 25 x10-6 m2 16. A r ectangular block of size 10 cm x 8 cm x 5 cm i s k ept i n t hr ee di f f er ent posi t i ons (I ) 10 cm x 8 cm on gr ound. (I I ) 5 cm x 8 cm on gr ound. (I I I ) 10 cm x 5 cm on gr ound. I n each case, the shadded ar ea is r igidly fixed and a definite for ce F is applied tangentially to the opposite face to defor m the block. The displacement of the upper face will be 1) Same in all the three cases 2) Maximum in (I) position 3) Maximum in (II) position 4) Maximum in (III) position

AKASH MULTIMEDIA

17. A met al r ope of densi t y 5000k gm -3 has br eaking str ess 19.6x108 Nm– 2. This r ope is used to measur e the depth of the sea. Then the depth of the sea that can be measur ed without br eaking is ----1) 50x103 m 2) 20x103 m 3 3) 30x10 m 4) 40x103 m 18. A 2 m long r od of r adius 1 cm which is fixed fr om one end is given a twist of 0.8 r adians at the other end. The shear str ain developed will be 1) 0.002 2) 0.004 3) 0.008 4) 0.016 19. A steel wir e of mass 3.16K g is str etched to a tensile str ain of 1  10-3. What is the elastic defor mation ener gy if density  =7.9g/cc and Y=2  1011N/m2 ? [ROORK EE 1998] 1) 4KJ

2) 0.4KJ

3) 0.04KJ

4) 4J

20. A spring of force constant 'K' is cut into two equal parts. The force constant of each part is 1) 2K 2) K/2 3) 4K 4) K/4 21. The r ubber cor d of a catapult has a cr oss sectional ar ea 1mm2 and total unstr etched length 10cm. I t is str etched to 12cm and then r el eased to pr oj ect a missile of mass 5gm. Taking Y for r ubber 5x108 Nm–2 calculate the velocity of pr oj ection ... 1) 10 ms–1 2) 15 ms–1 –1 3) 20 ms 4) 25 ms–1 22. Two spr i ngs A and B have t hei r for ce constants K 1 and K 2 (K 1 > K 2) when they are stretched by the same force work done is 1) More in case of A 2) More in case of B 3) Same in both the cases 4) None 23. The limiting str ess of a typical human bone is 0.9x108 Nm –2 whi l e young' s modul us i s 1.4x1010 Nm –2. H ow much ener gy can be absor bed by two leg bones without br eaking if each has a typical length of 50cm and an aver age cr oss sectional ar ea of 5cm2 ? 1) 145 J 2) 290 J 3) 490 J 4) 435 J 56

PHYSICS - I C

ELASTICITY

Compr ehension - I I I (6 - 7) :

ANSWERS 1) 2 6) 1 11) 2 16) 4 21) 3

2) 1 7) 3 12) 1 17) 1 22) 2

3) 1 8) 4 13) 2 18) 2 23) 1

4) 1 9) 4 14) 2 19) 3

5) 1 10) 2 15) 1 20) 1

A unifor m of length L aqnd mass M is pulled horozontally on a smooth surfce with a force F.

6.

COMPREHENSION QUESTIONS Compr ehension - I (1 - 3) :

7.

A wi r e of cr oss-sect i on A i s st r et ched hor izontally between two clamps located 2l metre apar t. A weight W kg is suspended from the mid point of the wir e. I f the mid point sags ver tically thr ough a distance x