ME – MECHANICAL ENGINEERING GATE : 2000

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SECTI ON A

1. Fol l ow ing 25 quest i ons car r y 1 mar k each

1.1 I f z = f (x, y) dz is equal to

(a) f / x)dx + (f / y)dy

(b) (f / y)dx + (f / x)dy

(c) (f / y)dx – (f / x)dy

(d) (f / y)dx – (f / x)dy

1.2x

xx

t

FHG

IKJ1

11

2

L–

– is

(a) (b) 0

(c) 2 (d) 1

1.3 The solu t i on of the di f fer ent ial equat i on

d y

dx

dydx

2

2 + y = 0

(a) Aex + Be– x

(b) ex (Ax + B)

(c) e– x {A cos ( 3 /2) x + B cos ( 3 /2) x}

(d) e– x/2 {A cos ( 3 /2) x + B cos ( 3 /2) x}

1.4 The t h r ee char act er i st i c r oot s of t hefol lowing matr ix A,

A =

1 2 30 2 30 0 2

are

(a) 1, 2, 3 (b) 1, 2, 2

(c) 1, 0, 0 (d) 0, 2, 3

1.5 Avai labi l i ty of a system at any given stateis

(a) a pr oper ty of the system

(b) the maximum work obtainable as thesystem goes to dead state

(c) the total ener gy of the system

(d) the maximum usefu l wor k obtainableas the system goes to dead state

1.6 A steam tur bine receives steam steadilyat 10 bar wi th an enthalpy of 3000 kJ/kgand discharges at 1 bar wi th an enthaply of2700 kJ/kg. The work output is 250 kJ/kg.The changes i n k i net i c and pot en t i alener gies ar e negl igible. The heat t r ansferf r om t he t u r bi ne casi ng t o t hesur roundings is equal to

(a) 0 kJ (b) 50 kJ

(c) 150 kJ (d) 250 kJ

1.7 I n a vapou r compr ession r efr i ger at i onsystem, l iquid to suct ion heat exchanger isused to

(a) Keep the COP constant

(b) pr event t he l i qu i d r ef r i ger ant f r omenter ing the compressor

(c) subcool the l iqu id r efr iger ant leavingthe condenser

(d) subcool the vapour r efr igerant from theevaporator

1.8 A steel steam pipe 10 cm inner diameterand 11 cm outer diameter is covered withan insulat ion having a thermal conduct ivityof 1 W/mK. I f the convect ive heat t r ansfercoefficient betweenthe sur face of insulat ionand the sur rounding ai r is 8 W/m2K , thencr it ical r adius of insulat ion is

(a) 10 cm (b) 11 cm

(c) 12.5 cm (d) 15 cm

1.9 Whi ch of t he fol l ow i ng i s a pr essu r ecompounded turbine?

(a) Parsons (b) Curt is

(c) Rateau (d) al l the thr ee

1.10 When the speed of a cent r i fugal pump isdoubled, then power requi red to dr ive thepump wil l

(a) increase 8 t imes

(b) increase 4 t imes

(c) double

(d) r emain the same

GATE 2000 MECHANICAL ENGINEERING

GATE : 2000 ME – MECHANICAL ENGINEERING

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1.11 Global warming is caused by

(a) ozone

(b) carbon dioxide

(c) nit r ogen

(d) carbon monoxide

1.12 Nav ier St ok e's equat i on r epr esent s t heconser vat ion of

(a) energy

(b) mass

(c) pressure

(d) momentum

1.13 I n a bi-axial st r ess pr oblem, the str esses inx and y di r ect i ons ar e x = 200 M Paand y= 100 MPa. The maximum pr incipalst ress in MPa, is

(a) 50 (b) 100

(c) 150 (d) 200

1.14 The r at io of tension on t he t i ght side tothat on the slack side in a flat belt dr ive is

(a) pr opor t i onal t o t he pr oduct ofcoefficient of fr ict ion and lap angle

(b) an exponent ial funct ion of the pr oductof coefficient of fr ict ion and lap angle.

(c) pr opor t ional to the lap angle

(d) pr opor t ional to the coefficient of fr ict ion

1.15 The nat ur al f r equency of an undampedvibr at i ng syst em is 100 r ad/s. A damperwith a damping factor of 0.8 is int r oducedinto the system. The fr equency of vibr at ionof the damped system, in rad/s, is

(a) 60 (b) 75

(c) 80 (d) 100

1.16 A steel shaft 'A' of diameter 'd ' and length'l 'is subjected to a torque 'T' Another shaft 'B'made of aluminium of the same diameter'd ' and length 0.5 l is also subjected to thesame t or que 'T '. The shear modu lus ofst eel i s 2.5 t imes t he shear modulus ofaluminium. The shear st r ess in the steelshaft is 100 MPa. The shear st r ess in thealuminium shaft , in MPa, is

(a) 40 (b) 50

(c) 100 (d) 250

1.17 A 1.5 kW motor is running at 1440 rev/min.I t is to be connected to a st i r r er running at 36 r ev/mi n. The gear i ng ar r angementsuitable for this applicat ion is

(a) different ial gear

(b) hel ical gear

(c) spur gear

(d) wor m gear

1.18 A steel wheel of 600 mm diameter rol ls ona hor izontal steel r ai l . I t car r ies a load of500 N. The coefficient of rol l ing resistanceis 0.3 mm. The for ce in N, necessary to r ol lthe wheel along the rai l is

(a) 0.5 (b) 5

(c) 15 (d) 150

1.19 Abr asive mater ial used in gr inding wheelselected for gr inding fer rous al loys is

(a) si l icon carbide

(b) diamond

(c) aluminium oxide

(d) boron carbide

1.20 Cast steel cr ankshaft sur face is har denedby

(a) nit r iding (b) normal ising

(c) carbur ising (d) induct ion heat ing

1.21 Disposable pat terns are made of

(a) wood (b) rubber

(c) metal (d) polystyr ene

1.22 Deep hole dr i l l ing of smal l diamet er, say0.2 mm is done wi th EDM by select ing thetool mater ial as

(a) copper wire

(b) tungsten wire

(c) brass wire

(d) tungsten carbide

1.23 I n computer aided dr aft ing pract ice, an ar cis defined by

(a) two end points only

(b) centre and radius

(c) radius and one end point

(d) two end points and cent re

ME – MECHANICAL ENGINEERING GATE : 2000

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1.24 I n a t ime study exer cise, the t ime obser vedfor an act i v i t y was 54 seconds. Theoperator had a per formance rat ing of 120. Aper sonal t ime al lowance of 10% is gi ven.The st andar d t i me for t he act i v i t y, i nseconds, is

(a) 54

(b) 60.8

(c) 72

(d) 58.32

1.25 Cellular manufactur ing is suitable for

(a) a single pr oduct in lar ge volumes

(b) one-off pr oduct ion of several var iet ies

(c) products wi th simi lar featur es made inbatches

(d) l ar ge var i et y of pr oduct s i n l ar gevolumes

2. Fol l ow ing 25 quest i ons car r y 2 mar k s each

2.1 The Laplace t r ansform of the funct ion sin2

2t is

(a)12 2 162s

ss

–( )

LNM

OQP

(b)s

s2 16

(c) 142s

ss

–( )

(d) s

s2 4

2.2 The max i mum val ue of t he di r ect i onalder ivat ive of the funct ion

= 2x2 + 3y2 + 5z2 at a point (1, 1, – 1) is

(a) 10

(b) – 4

(c) 152

(d) 152

2.30

2

0

2 / /z z sin (x + y) dx dy is

(a) 0 (b)

(c) /2 (d) 2

2.4 I n a manufactur ing plant , the probabil i ty ofmak ing a defect ive bol t is 0.1. The meanand standar d deviat ion of defect ive bol ts ina total of 900 bolt s are respect ively

(a) 90 and 9 (b) 9 and 90

(c) 81 and 9 (d) 9 and 81

2.5 I n given figur e, i f t he pressur e of gas inbulb A is 50 cm Hg vacuum and patm = 76 cmHg, then height of column H is equal to

A

patmH

Hg

(a) 26 cm (b) 50 cm

(c) 76 cm (d) 126 cm

2.6 For a compressible fluid, sonic velocit y is

(a) a proper ty of the fluid

(b) always given by (RT)1/2 where , R andT ar e respect ively the rat io of speci ficheats, gas constant and temperature in K

(c) always given by (p/)s1/2 . Where p, and s are r espect ively pr essure, densit yand entropy.

(d) always gr eater than the velocity of fluidat any locat ion.

2.7 When an ideal gas wi th constant speci ficheat s i s t h r ot t l ed adi abat i cal l y, w i t hnegligible changes in k inet ic and potent ialener gies

(a) h = 0, T = 0

(b) h > 0, T = 0

(c) h > 0, S > 0

(d) h = 0, S > 0

wher e h , T and S r epr esent r espect ively,en t hal py, t emper at u r e and en t r opy,temperature and entr opy.

2.8 When atmospher ic air is heated at constantpr essure, i ts

(a) humidity rat io does not change

(b) relat ive humidity increases

(c) dew point temperature does not change

(d) wet bulb temper ature increases

GATE : 2000 ME – MECHANICAL ENGINEERING

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2.9 Consi der ai r st andar d Ot t o and D ieselcycles, both having the same state of air atthe star t of compr ession. I f t he maximumpr essur e in both t he cycles i s the same,t hen compr essi on r at i o ' r ' and t heefficiency '' ar e related by

(a) r Diesel > r Ot to (b) r Diesel > r Ot to

(c) Otto > Diesel (d) Otto > Diesel

2.10 Air enters a counter flow heat exchanger at70C and leaves at 40C. Water enter s at30C and l eaves at 50C. The L M TD indegr ee C is

(a) 5.65 (b) 4.43

(c) 19.52 (d) 20.17

2.11 A fuel r epr esented by the formula C8H 16 isused i n an I .E . Engi ne. Gi ven t hatthe molecular weight of ai r is 29 and that4.76 kmols of ai r contain 1 kmol of oxygenand 3.76 k mols of ni t r ogen, the Ai r /Fuelrat io by mass is

(a) 11.47 (b) 12.78

(c) 14.79 (d) 19.52

2.12 For t he fol l ow ing "M at ch i ng" exer ci se,choose t he cor r ect one f r om among t healternat ives a, b, c and d.

Gr oup 1

1. M ar ine Diesel Engine

2. Air condi t ioning

3. Steam Power Plant

4. Gas Tur bine Power Plant

Gr oup 2

A. Two str oke engine

B. Four st r oke engine

C. Rotary engine

D. Cool ing and dehumidi ficat ion

E. Cooling tower

F. Brayton cycle

G. Rankine cycle

H. D - sl ide valve

(a) (b) (c) (d)

1. B C C A

2. E F F D

3. F E G G

4. H G E F

2.13 An automobi l e of weight W i s shown i nFigur e. A pul l 'P' is applied as shown. Thereact ion at the fr ont wheels (locat ion A) is

wp

bAB

a a

(a) W/2 – pb/2a (b) W/2 + pb/2a

(c) W/2 – pa/2b (d) W/2

2.14 A simply suppor ted beam car r ies a load 'P'through a bracket , as shown in Figur e. Themaximum bending moment in the beam is

al/2

l

(a) Pl /2 (b) Pl /2 + aP/2

(c) Pl /2 + aP (d) Pl /2 – aP

2.15 I n an exper iment to find the veloci ty andaccelerat ion of a par t icular cam r otat ing at10 r ad/s, the values of displacements andvel oci t i es ar e r ecor ded. The sl ope ofdi splacement cur ve at an angle of '' i s1.5 m/s and the slope of veloci t y curve atthe same angle is – 0.5 m/s2. The veloci t yand acceler at ion of the cam at the instantar e respect ively

(a) 15 m/s and – 5m/s2

(b) 15 m/s and 5 m/s2

(c) 1.2 m/s and – 0.5 m/s2

(d) 1.2 m/s and 0.5 m/s2

2.16 As shown in Figur e, a mass of 100 kg isheld bet ween t wo spr i ngs. The nat ur alf r equency of v i br at ion of the system, i ncycles/s, is

100 kg

20 kN/m

20 kN/m

(a)1

2(b)

5

(c)10

(d)20

ME – MECHANICAL ENGINEERING GATE : 2000

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2.17 The l i fe of a bal l bear ing at a load of 10 kNis 8000 hour s. I t s l i fe in hour s, i f t he load isi ncr eased t o 20 k N , k eepi ng al l ot hercondit ions the same, is

(a) 4000

(b) 2000

(c) 1000

(d) 500

2.18 A fi t is specified as 25 H8/e8. The tolerancevalue for a nominal diameter of 25 mm inI T8 i s 33 mi cr ons and fundamen t aldeviat ion for the shaft is – 40 micr ons. Themaximum clearance of the fi t in microns is

(a) – 7 (b) 7

(c) 73 (d) 106

2.19 A 1.5 mm thick sheet is subject to unequalbiaxial st r etching and the t r ue st r ains inthe di r ect ions of st r etching ar e 0.05 and0.09. The fi nal t hi ckness of the sheet inmm is

(a) 1.414

(b) 1.304

(c) 1.362

(d) 289

2.20 Fr om an equilateral t r iangular plate of side'a', a square plate of maximum size has tobe cut . The side of such a squar e plate is

(a) 3 a/2 (b) (2 + 3 ) a/4

(c) (1 + 3 ) a/8 (d) 3 a / (2 + 3 )

2.21 The str ess-st rain behaviour of a mater ial isshown i n F i gu r e. I t s r esi l i ence andtoughness, in Nm/m3, are r espect ively

150

1209070

30Stre

ss (M

Pa)

0.004 0.004 0.012Strain (mm, mm)

(a) 28 104, 76 104

(b) 28 104, 48 104

(c) 14 104, 90 104

(d) 76 104, 104 104

2.22 A slot is to be mil led cent r al ly on a blockw i t h a di mensi on of 40 ± 0.05 mm. Ami l l ing cut ter of 20 mm width i s locatedwi t h r efer ence t o t he si de of t he blockwithin ± 0.02 mm. The maximum offset inmm between the cent r e l i nes of t he slotand the block is

(a) ± 0.070 (b) 0.070

(c) ± 0.020 (d) 0.045

2.23 I n f i n i sh mach i n i ng of an i sl and on acast ing wi th CNC mil l ing machine, an endmi l l with 10 mm diameter is employed. Thecor ner points of the island ar e r epresentedby (0,0), (0,30), (50, 30), and (50, 0). Byapplying cut ter r adius r ight compensat ion,the t r ajectory of the cut ter wil l be

(a) (– 5, 0), (– 5, 35), (55, 35), (55, – 5), (– 5, – 5)

(b) (0, – 5), (55, – 5), (55, 35), (– 5, 35), (– 5, – 5)

(c) (5, 5), (5, 25), (45, 25), (45, 5), (5, 5)

(d) (5, 5), (45, 5), (45, 25), (5, 25), (5, 5)

2.24 I n a si ngl e ser ve i n f i n i t e popu l at i onqueuing model , ar r ivals fol low a Poissondistr ibut ion with mean = 4 per hour. Theser vice t imes ar e exponent ial wi th meanser v i ce t ime equal t o 12 mi nu t es. Theexpected length of the queue wi l l be

(a) 4 (b) 3.2

(c) 1.25 (d) 5

2.25 I n a t i me ser i es for ecast i ng model , t hedemand for five t ime per iods was 10, 13,15, 18 and 22. A l i near r egr essi on f i tr esu l t ed i n an equat i on F = 6.9 + 2.9 twher e F is the for ecast for per i od t . Thesum of absolute deviat ions for the five datais

(a) 2.2 (b) 0.2

(c) – 1.2 (d) 24.3

(SECTI ON B)

F ol l ow i n g 15 qu est i on s car r i es 5 m ar k seach

3. Est imate the root of the equat ion (e– x – x) = 0to four decimal accur acy by employing theNewton – Raphson method star t ing with aninit ial guess of x0 = 0.

GATE : 2000 ME – MECHANICAL ENGINEERING

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4. A cer t ai n mass of a pu r e subst anceunder goes an i r r ever si bl e pr ocess for mstate 1 to state 2, the path of the pr ocessbeing a str aight l ine on the T– s diagr am.Calculate T1 = 330K, T2 = 440K;

U1 = 170 kJ, U 2 = 190 kJ; H 1 = 220 kJ,H 2 = 247 k J and S1 = 0.23 k J /K andS2 = 0.3 k J /K wher e T, U , H and Sr epr esent temper atur e, int er nal ener gy,enthalpy and entropy respect ively.

5. A t h i n met al pl at e i s exposed t o sol arradiat ion. The air and the sur roundings areat 30C. The heat t r ansfer coef fici ent byfr ee convect ion fr om the upper sur face ofthe plate is 17.4 W/m2K . The plate has anabsor pt ivi t y of 0.9 at solar wavelength andan emissivi ty of 0.1 at the long wavelength.Neglect i ng any heat loss f r om t he l owersu r face, det er mi ne t he i nci den t sol arr adi at i on i n t ensi t y i n k W/m 2, i f t hemeasur ed equi l ibr ium temperatur e of theplate is 50C. Stefan Bolt zmann constant is5.67 10– 8 W/m2K 4.

6. I n an ideal air -standar d Gas tur bine cycle,the minimum and maximum temper aturesare respect ively 310K and 1100K . Dr awthe cycle on a T-S diagram and calculatethe opt imal pressure r at io of the cycle formaximum wor k out pu t . Assume for ai r( – 1)/ is the r at io of specific heats.

7. A Diesel engine develops a Brake power of4.5 kW. I t s indicated thermal efficiency is30% and the mechanical efficiency is 85% .Take the calor if ic value of the fuel as 40000k J /k g and cal cu l at e (a) t he fuelconsumpt ion in kg/h and (b) the indicatedspecific fuel consumpt ion.

8. A simple impulse tur bine expands steamfr ict ionlessly f r om 12 bar, 250C wi t h anenthalpy of 2935 kJ/kg to an enthalpy of2584 kJ/kg at 0.1 bar. Assuming that thenozzle makes an angle of 20 wi th the blademot ion, and that the blades are symmetr ical,f i nd t he blade veloci t y t hat pr oducesmaximum efficiency for a tur bine speed of3600 rev/min. Assume that the steam entersthe nozzle with negligible velocity.

9. I n a 5 kW cool i ng capaci t y r efr iger at i onsyst em oper at i ng on a simpl e vapou rcompr ession cycle, the r efr iger ant enteer sthe evapor ator with an enthalpy of 75 kJ/kgand leaves wi th an enthalpy of 183 kJ/kg.The ent hal py of t he r efr i ger ant af t ercompression is 210 kJ/kg. Show the cycle onT-S or P-h diagram. Calculate the following:(a) COP (b) power input to compr essor and(c) rate of heat t ransfer at the condenser.

10. A single act ing single cyl inder reciprocat ingair compressor r unning at 7.5 r ev/s, takesin ai r at 100 k Pa, 27C. The compr essordelivers air at 600 kPa at a flow rate of 0.12m3/s measur ed at suct ion condit ions. Giventhat the percentage clear ance is 4 and thatthe index of compr ession and expansion is1.2, cal cu l at e (a) t he i deal vol umet r i ceff i ciency and (b) the cyl i nder bor e andst roke, assuming the bor e/ st roke rat io is0.8

11. A single device shown in gi ven figur e,der ive the r elat ionship between for ces Pand al l t erms of the angle .

P

l l

l

F Xel

12. A steel bol t having a nominal diameter of20 mm and a pi t ch of 2.4 mm is used toconnect t wo plat es of 10 mm t h i ck nesseach. An aluminium tube of inner diameter22 mm and out er di amet er of 40 mm i ssepar at ing the plates as shown in Figure.The nut is pul led snug (just t ight ) and thengiven a one-thi r d addi t ional tur n. Find ther esul t ing str esses in the bolt and the tubeneglect i ng the defor mat ion of the plat es.Young's modulus of steel and aluminium are207103 MPa and 67.5103 MPa respect ively.

350 mm

10 mm

10 mm

ME – MECHANICAL ENGINEERING GATE : 2000

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13. A single block brake with a 300mm diameterbr ak e dr um shown i n F i g. i s used t oabsorb a tor que of 75 Nm. The coefficient offr ict ion between the dr um and the l ining is0.35. The pressure on the block is uni form.Calculate the for ce P.

200 mm 250 mm

50 mm

300 mm

P

14. A Punch pr ess, f i t t ed w i t h a f l ywheelhaving a r adius of gyrat ion of 0.5 m, r unsat 260 r ev/min. The pr ess i s capable ofpunch i ng 600 hol es per hou r . Eachpunching oper at ion takes 1.5 seconds andrequires a work of 10,000 Nm. The rat ing ofthe electr ic motor used is 2 kW. Determinethe mass of the flywheel , i f t he speed of theflywheel should not drop below 240 rev/min.

15. A fu l l jour nal bear ing wi t h a jour nal of75 mm diameter and a bear ing of 75 mml engt h i s subject ed t o a r adial l oad of2500 N at 400 r ev /mi n . The l ubr i can ti s SAE 30 at 75C having a viscosi t y of16.5 10 – 3 k g/ms. Radi al cl ear ance i s0.03 mm. Eccent r i ci t y of t he bear i ng i s0.27. Find the Sommer feld number and theminimum fi lm thickness.

16. A ci r cu l ar r od 50 mm in di amet er and200 mm long is welded to a plate by fi l letwelding al l ar ound t he ci r cumfer ence asshown in Figur e. The size of the weld is15 mm. The sect ion modulus of the weld is22000 mm3. Deter mine the r esultant in theweld.

50 mm200 mm

10 kN

17. Two cast i ngs, a cube and a sl ab of t hesame mat er i al sol i di f y under i den t i calmou l d condi t i ons. The vol umes of t hecast ings ar e equal but the slab dimensionsare in the rat io of 1 : 2 : 4. Find the rat io ofthe sol idi f icat ion t ime of the cube to that ofthe slab.

18. A cyl indr ical bi l let of 100 mm diameter isfor ged f r om 50mm hei gh t t o 40 mm at1000C. The mater ial has a constant flowst r ess of 80M Pa. (a) F i nd t he wor k ofdefor mat ion. (b) I f a 10 kN drop-hammer isused to complete the r educt ion in one blow,then what wil l be the height of the fal l?

19. A 15 mm diameter HSS dr i l l is used at acut t ing speed of 20 m/ min and a feed r ateof 0.2 mm/r ev. Under these condi t ions, thedr i l l l i fe is 100 min. The dr i l l ing length ofeach hole is 45 mm and the t ime taken foridle mot ions is 20s. The tool change t ime is300s. Calculate (a) number of holes producedusing one dr i l l and (b) aver age pr oduct iont ime per hole.

20. A convent ional lathe and a CNC lathe ar eunder considerat ion for machining a givenpar t . The r elevant data ar e shown below:

Prepar at ion Pr oduct ion MachineCost t ime per par t oper ator cost(Rs) (min) (Rs)

Convent ional Lathe30 30 75

CNC Lathe150 15 120

F i nd t he br eak - even quan t i t y, abovewhich CNC lathe is economical.

21. Solve the fol l owing l i near pr ogr ammingpr oblem by simplex method

Maximize 4x1 + 6x2 + x3

Subject to 2x1 – x2 + 3x3 5

x1, x2, x3 0

(a) What i s t he sol u t i on t o t he aboveproblem?

(b) Add the constr aint x2 2 to the simplextable of par t (a) and find the solut ion.

22. A company places orders for supply of twoi tems A and B. The order cost for each ofthe i t ems is Rs.300/or der. The inventor ycar r ying cost is 18% of the unit pr ice peryear per unit . The uni t pr ices of the i temsar e Rs.40 and Rs.50 r espect i vel y. Theannual demands ar e 10,000 and 20,000r espect ively. (a) Find the economic or derquant i t i es and t he min imum t ot al cost ,(b) A supplier is wi l l ing to give 1% discounton pr ice, i f both the i tems are or dered fromhim and i f t he or der quant i t i es for eachi tem is 1000 uni ts or more. I s i t pr ofi tableto avai l the discount?