FACTORS EFFECTING ELASTICITY

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Marjory Knight
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1 Synopsis : ELASTICITY. Defoming foce : The extena foce acting on a body on account of which its size o shape o both change is defined as the defoming foce.. Restoing foce : The foce which estoes the size and shape of the body when defomation foces ae emoved is caed estoing foce. Defoming foce and estoing foce ae not action eaction pai. Restoing foce opposes the change in the size and shape of a body. 3. Rigid body : A body whose shape and size cannot be changed howeve age the appied foce is caed igid body. Thee is no pefecty igid body in natue. 4. : The popety of a body by vitue of which it egains its oigina size and shape immediatey afte the defomation foces ae emoved is caed easticity. is a moecua phenomenon. It is because of cohesive foces. 5. Eastic body : A body which shows eastic behaviou is caed eastic body. E.g. stee, ubbe. Quatz is vey neay pefecty eastic body. 6. Pastic body : A body which does not show eastic behaviou is caed pastic body. E.g. putty, cay, mud, wax, ead, dough, chewing gum, butte wax etc. 7. Out of the given mateias, a body in which it is moe difficut to poduce stain is moe eastic. OR The body which equies geate defoming foce to poduce a cetain change in dimension is moe eastic. a) stee is moe eastic than ubbe b) gass is moe eastic than ubbe c) wate is moe eastic than ai d) spings ae made of stee but not of coppe because stee is moe eastic than coppe. 8. By the pocess of hammeing o oing the body easticity inceases. 9. By the pocess of anneaing, the eastic popety of a body is educed. 0. o inva stee (e-64%, Ni-36%) the eastic popety is constant iespective of change in tempeatue. (used in making penduum cocks). ACTORS EECTING ELASTICITY : a. Effect of tempeatue : In genea as the tempeatue inceases the eastic popety of a mateia deceases. b. Effect of impuities : Addition of impuity to meta may incease o decease the easticity. If the impuity has moe easticity than the mateia to which it is added, it inceases the easticity. If the impuity is ess eastic than the mateia it deceases the easticity.. Stess : The estoing foce deveoped pe unit aea of coss-section of the defomed body is caed stess. Restoing foce Stess Coss sec tiona aea A N dyne Unit,, pasca m cm Dimensiona fomua : M L T 3. i) Pessue is aways noma to the aea, whie stess can be eithe noma o tangentia. ii) Pessue on a body is aways compessive, whie stess can be compessive o tensie.

2 iii) Pessue is a scaa, whie stess is a tenso. Stess is of thee types : i) Longitudina stess : If the estoing foces ae pependicua to the aea of coss-section and ae aong the ength of the wie, the stess is caed ongitudina stess. Duing ongitudina stess, the body undegoes change in ength but not in shape and voume. ii) Tangentia stess (o sheaing stess) : If the estoing foces ae paae to the suface, the stess is caed sheaing stess. iii) Buk stess (o voume stess) : If a body is subjected to equa foces nomay on a the faces, the stess invoved is caed buk stess. 4. Stain : The defomation poduced pe unit magnitude is caed stain. change in ength a) ongitudina stain oigina ength e b) sheaing stain θ Δ atea dispacement between two ayes pependicua distance between the two ayes change in voume c) Buk stain oigina voume Δv v d) Tansvese stain o atea stain change in adius Δ oigina adius e) sheaing stain x ongitudina stain f) buk stain 3 x ongitudina stain g) ongitudina stain : sheaing stain : buk stain ::3 Shea stain is equivaent to two equa ongitudina eongation and compessiona stains in mutuay pependicua diections. The maximum vaue of the stess within which the body egains its oigina size and shape is caed eastic imit. 5. Hooke s aw : Within the eastic imit of a body, stess is diecty popotiona to stain. stess E constant stain dyne newton Unit of E : CGS :, SI : o pasca cm m i) Within the popotionaity imit stess-stain gaph is a staight ine passing though the oigin. ii) A sping baance woks on the pincipe of Hooke s aw. iii) Moduus of easticity does not depend upon the dimensions of the body but is a popety of the mateia of the body. iv) Within the popotionaity imit, the oad extension gaph is a staight ine passing though the oigin. 6. Behaviou of a wie unde the action of a oad :

3 Y S b Stess B A C D E O O I Stain X A Popotionaity imit B Eastic imit C Yieding point D Beaking point S b Utimate tensie stength a) Stess is popotiona to stain upto a imit, which is caed popotionaity imit. A is the imit of popotionaity. Upto this imit, Hooke s aw is obeyed. b) The smaest vaue of stess which poduces a pemanent change in the body is caed eastic imit. c) If the wie is oaded beyond the eastic imit, a stage is eached whee the wie begins to fow with no incease in the oad and this point is caed yied point. d) Beyond the yied point, if the oad is inceased futhe the extension inceases apidy and the wie becomes naowe and finay beaks. The point at which the wie beaks is caed beaking point. e) Maximum stess equied to beak the wie is caed utimate tensie stength. f) The capacity of a mateia to withstand age stesses without pemanent set is caed esiience. g) The wie egains its oigina ength if the eastic imit is not exceeded. h) The wie does not obey Hooke s aw between the popotionaity imit and eastic imit. But wie egains its oigina ength when the oad is emoved. i) A pemanent set (OP) is poduced in the wie beyond eastic imit. j) The stess equied to each the beaking point is caed beaking stess. k) If the gap between eastic imit and beaking point (BD) of a meta is age, it is caed a ductie meta. ) If the wie beaks soon afte exceeding imit, the meta is said to be bitte. (If the gap BD is sma). 7. Types of modui of easticity : Thee ae thee modui of easticity. ) Young moduus Y ) Rigidity moduus n 3) Buk moduus K ) Young s moduus : Young s moduus is the atio of ongitudina stess to ongitudina stain within the eastic imit of a body. ongitudina stess Y A x ongitudina stain e A e 3

4 When a mass M is attached to the owe end, Y but Mg, Aπ Ae Mg M g. Y., Y e e π π i) The stess equied to doube the ength of a wie (o to poduce 00% ongitudina stain) is equa to Young s moduus of the wie. ii) Y of a pefecty eastic mateia is infinite and that of a pefecty ineastic mateia is zeo. ) Rigidity moduus : Rigidity moduus is the atio between sheaing stess and Δ sheaing stain within the eastic imit of a body. sheaing stess η A sheaing stain tan θ Aθ θ (fo sma vaues of θ, tanθ θ) (o) n. A Δ. i) If η is ow fo a wie, it can be twisted vey easiy. ii) Since phospho-bonze has vey ow igidity moduus, it is used as a suspension fibe in moving coi gavanometes. Rigidity (shea) moduus is used to cacuate the stain poduced in a od unde twisting stess. It is aso used to cacuate the estoing toque when a wie o a cyinde is twisted. Toque C poduced pe unit twist 4 πη of a wie of ength and adius is given by C whee η is igidity moduus. iii) If a od of ength and adius is fixed at one end and the othe end is twisted by an ange θ, then φ θ. whee φ is ange of shea. 3) Buk moduus : Buk moduus is the atio between voume stess and voume stain within the eastic imit of a body. voume stess K / A. voume stain Δv v ( sign indicates the decease in voume) Pv K Δv i) If a bock of coefficient of cubica expansion γ is heated such that the ise in tempeatue is θ, the pessue to be appied on it to pevent its expansionkγθ whee K is its buk moduus. ii) When a ubbe ba of voume V buk moduus K is taken to a depth h in wate, decease in its voume is hdgv Δ V (d density of mateia). K A moduii of easticity Y, n, K have same units and dimensions -> [M L T ] N.m i) Soids possess Y, n and K. ii) Liquids and gases possess ony K. iii) Buk moduus of gases is vey ow, whie that of iquids and soids is vey high. 4

5 iv) Isothema buk moduus of a gaspessue of the gas (P) v) Adiabatic buk moduus of a gas γp whee γ atio of two specific heats. 4) Compessibiity : The ecipoca of buk moduus is caed compessibiity C K o incompessibe substances C 0, K atea contaction stain 8. Poisson s atio (σ) ongitudina eongation stain o tansvese stain Δ ongitudina stain Δ i) Poisson s atio has no unit and has no dimensions. ii) Theoetica imits of σ to 0.5. iii) Pactica imit of σ 0 to 0.5 iv) If σ 0.5 the substance is pefecty incompessibe. 9. Reation among eastic constants Y, η, K, σ : i) ii) Y η ( +σ ) Y K η 3K η iii) Y 3K( σ) iv) σ ( η+ 3K) 0. Appications of Y : i) A ong wie suspended veticay can eongate due to its own weight. dg ii) Eongation of a wie due to its own weight e ; is ength of the wie, d is density of the wie, Y YYoung s moduus of the mateia of the wie, gacceeation due to gavity iii) A vey ong wie suspended veticay can beak due to its own weight. iv) Maximum ength of the wie that can be hung veticay without beakings/dg whee s is beaking stess. v) Beaking stess : a) The beaking stess of a wie is the maximum stess the mateia can withstand. beaking foce b) Beaking stess initia aea of coss - section vi) Beaking foce Beaking stess x aea of coss-section vii) Beaking foce : ) is independent of ength of the wie ) depends on the aea of coss-section and natue of mateia of the wie. 3) beaking foce α aea of coss-section. 4) If we cut a cabe that can suppot a maximum oad of W into two equa pats, then each pat can suppot a maximum oad of W.. Eastic hysteesis is the esut of eastic afte effect. Thee is a ag between stess and stain. The ag is known as eastic hysteesis. 5

6 . Eastic atigue : a) The state of tempoay oss of eastic natue due to continuous stain is caed eastic fatigue. b) Due to eastic fatigue : i) a wie can be boken within the eastic imit ii) a wie can be cut into pieces without using instuments iii) aiway tacks and bidges ae decaed unsafe afte ong use iv) sping baances show wong eadings afte ong use. 3. Stain enegy : stain enegy of a stetched wie xe oce x eongation E Potentia enegy stoed pe unit voume in a stained body is caed stain enegy density. Potentia enegy stoed in a wie due to twisting τθ. 5. Stain enegy density : wok x stess x stain voume (stess) x y x (stain) x y If K is the foce constant, enegy stoed fo extension e is given by E Ke 6. Laws of eongation : i) e α ; eongation is popotiona to ength of wie ii) e α ; eongation is popotiona to foce appied iii) e α /A o / eongation is invesey popotiona to aea of coss-section o squae of the adius. iv) eongation is invesey popotiona to Young s moduus. e A v) o two wies made of same mateia,.. e A vi) o two wies made of same mateia, when same foce is appied on them 6 e.. e vii) o two wies, made of same mateia, and of same voume when same foce is appied, eongations atio is given by e e 4 A 4 A (since, V A constant A A A ) A viii) If and ae the ength of a wie unde tension T and T, the actua ength of the wie 7. Spings : i) o a sping that obeys Hooke s aw, equivaent foce constant o sping constant is K T T T T YA.

7 ii) K α Y, K α A, K α / iii) If a sping (o a wie) of foce constant K is cut into n equa pats, the foce constant of each pat of the wie is nk. ( m + n)k ( m + n)k iv) If a sping (o a wie) of foce constant k is cut in the atio of m:n, k m ; k n m n v) Potentia enegy of a stetched sping x Kx K vi) Two spings have foce constants K and K a) When they ae stetched by the same foce and if thei eastic enegies ae E and E. b) When they ae extended by the same ength E K E K c) When they ae extended ti thei enegies ae same, d) The potentia enegy of a sping inceases, whethe it is stetched o compessed. KK e) Spings in seies K eff K + K K K E K E K f) Spings in paae K eff K +K g) The ecipoca of sping constant is caed compiance. vii) When a spia sping is stetched, stain invoved is ongitudina stain. (thickness is sma) viii) When a heica sping is stetched, stain invoved is ongitudina and sheaing stain. (thickness is age) ix) When a wie is stetched, moduus of easticity invoved is Young s moduus x) When a wie is twisted, moduus of easticity invoved is igidity moduus. xi) Inte atomic foce constant k Y. Young s moduus x (inte atomic distance) 8. Thema foce : i) When a meta ba is fixed between two was and the tempeatue is aised, the ba ties to expand and exets a foce on the was. This foce is caed thema foce given by YAαθ αco-efficient of inea expansion of the ba θ ise in tempeatue Y Young s moduus, Aaea of coss-section Thema foce is independent of the ength of ba. ii) Thema stess : thema foce YAαθ Thema stess Yαθ aea A iii) If a oad M poduces an eongation e in a wie the ise in tempeatue equied to poduce the same Mg e eongation is (since αδ t ) YAα whee A aea of coss-section of the wie and α coefficient of inea expansion of the mateia of the wie. 7

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