MECHANICAL ENGINEERING GATE : 2002

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SECTI ON - A

1. This quest ion consi sts of TWENTY FI VEsub-quest ions (1.1 -1.25) of ONE mar ks each.For each of these sub-quest ions, four possibleanswers (a, b, c and d) are given, out of whichonly one is cor rect . Answer each sub-quest ionby darkening the appr opr iate bubble on theOBJECTIVE RESPONSE SHEET (ORS) usinga soft HB pencil . Do not the ORS for any roughwor k. You may l ike to use the Answer Bookfor any rough work , i f needed. (25 1 = 25)

1.1 Two dice are thrown. What is the probabil i tythat the sum of the numbers on the two dice iseight?

(a) 5/36 (b) 5/18

(c) 1/4 (d) 1/3

1.2 Whi ch of t he fol l ow i ng funct i ons i s notdifferent iable in the domain [– 1, 1}?

(a) f (x) = x2

(b) f (x) = x– 1

(c) f (x) = 2

(d) f (x) = maximum (x, – x)

1.3 A regression model is used to express a var iableY as a funct ion of another var iable X.

(a) there is a causal relationship between Y and X

(b) a value of X may be used to est imate a valueof Y

(c) values of X exact ly determine values of Y

(d) there is no causal relat ionship between Yand X

1.4 The pr inciples of mot ion economy are most lyused while conduct ing

(a) a method study on an operat ion

(b) a t ime study on an operat ion

(c) a financial apprat isal of an operat ion

(d) a feasi bi l i t y st udy of t he pr oposedmanufactur ing plant

1.5 The st andar d t ime of an oper at ion whi l econduct ing a t ime study is

(a) mean observed t ime + al lowances

(b) nor mal t ime + al lowances

(c) mean obser ved t ime r at i ng factor +allowances

(d) nor mal t ime rat ing factor + al lowances

1.6 I n car r ying out a wor k sampling study in amachine shop, i t was found that a par t icularlathe was down for 20% of the time. What wouldbe the 95% confidence interval of this est imateif 100 observat ions were made?

(a) (0.16, 0.24) (b) (0.12, 0.28)

(c) (0.08, 0.32) (d) none of these

1.7 An item can be purchased for Rs 100. Theorder ing cost is Rs. 200 and the inventor ycar r ying cost is 10% of the i tem cost annum. I ft he annual demand i s 4000 uni t s, t heneconomic order quant i ty (in units) is

(a) 50 (b) 100

(c) 200 (d) 400

1.8 The minimum number of teeth on the pinionto operate without interference in standard ful lheight involute teeth gear mechanism with 20°pressure angle is

(a) 14 (b) 12

(c) 18 (d) 32

1.9 The minimum number of l inks in a singledegree-of-freedom planar mechanism with bothhigher and lower k inemat ic pairs is

(a) 2 (b) 3

(c) 4 (d) 5

1.10 The Cor iol is component of acceler at ion ispresent in

(a) 4-bar mechanisms with 4 turning pairs

(b) shaper mechanism

(c) sl ider-crank mechanism

(d) Scotch Yoke mechanism

GATE 2002MECHANICAL ENGINEERING

GATE : 2002 MECHANICAL ENGINEERING

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1.11 The total area under the stress-strain curve ofa mild steel specimen tested up to fai lure undertension is a measure of

(a) duct il ity (b) ult imate strength

(c) sti ffness (d) toughness

1.12 The number of components in a stress tensordefining stress at a point in thr ee dimensionsis

(a) 3 (b) 4

(c) 6 (d) 9

1.13 A lead-screw with hal f nuts in a lathe, free torotate in both di rect ions has

(a) V-threads

(b) Whitwor th threads

(c) But t ress threads

(d) Acme threads

1.14 The pr imary purpose of a sprue in a cast ingmould is to

(a) feed the cast ing at a rate consistent withthe rate of sol i f icat ion

(b) act as a reservoir for molten metal

(c) feed molen metal from the pour ing basinto the gate

(d) help feed the cast ing unt i l al l sol idificat iontakes place

1.15 Hot rol l ing of mi ld steel is car r ied out

(a) at recrystal l isat ion temperature

(b) between 100C to 150C

(c) below recrystal l isat ion temperature

(d) above recrystal l isat ion temperature

1.16 Which of the fol lowing arc welding processesdoes not use consumable electrodes?

(a) GMAW

(b) GTAW

(c) Submerged Arc Welding

(d) none of these

1.17 Trepanning is per formed for

(a) finishing a dr i l led hole

(b) producing a large hole without dr i l l ing

(c) t ruing a hole for al ignment

(d) enlarging a dr i l led hole

1.18 The hardness of a gr inding wheel is determinedby the

(a) hardness of abrasive grains

(b) abi l i ty of the bond to retain abrasives

(c) har dness of the bond

(d) abi l i ty of the gr inding wheel to penetr atethe wor k piece

1.19 A posit ive value to Joule-Thomson coefficientof a fluid means

(a) temperature drops dur ing throt t l ing

(b) t emper at ur e r emains const ant dur i ngthrott l ing

(c) temper ature r ises dur ing throt t l ing

(d) none of these

1.20 I f t her e ar e m physical quant i t i es and nfundament al dimensions i n a par t i cu l arpr ocess, t he number of non -dimensionalpar ameters is

(a) m + n

(b) m n

(c) m - n

(d) m / n

1.21 I f x is the distance measured from the leadingedge of a flat plate, then laminar boundary layerthickness var ies as

(a) 1/x

(b) x4/5

(c) x2

(d) x1/2

1.22 Flow separat ion in flow past a sol id object iscaused by

(a) a r educt ion of pressure to vapour pressure

(b) a negat ive pressure gradient

(c) a posit ive pressure gradient

(d) the boundary layer thickness reducing tozero

1.23 A cor r ect ly designed conver gent -diver gentnozzle working at a designed load is

(a) always isentropic

(b) always choked

(c) never choked

(d) never isentropic

MECHANICAL ENGINEERING GATE : 2002

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1.24 The value of Biot number i s ver y smal l(less than 0.01) when

(a) the convect ive r esistance of the fluid isnegligible

(b) the conduct ive resistance of the fluid isnegligible

(c) the conduct ive resistance of the sol id isnegligible

(d) none of these

1.25 For the same inlet and out let temperatures ofhot and cold fluids, the Log Mean TemperatureDifference (LMTD) is

(a) gr eater for paral lel flow heat exchangerthan for counter flow heat exchanger.

(b) gr eater for counter flow heat exchangerthan for paral lel flow heat exchanger.

(c) same for both parallel and counter flow heatexchangers.

(d) dependent on the proper t ies of the fluids.

2 This quest ion consists of TWENTY FI VE sub-quest ions (2.1 - 2.25) of TWO marks each. Foreach of t hese sub-quest i ons four possibl eanswers (A, B, C and D) are given, out of whichonly one is cor rect . Answer each sub-quest ionby darkening the appr opr iate bubble on theOBJECTIVE RESPONSE SHEET (ORS) usinga soft HB pencil . Do not use the ORS for anyrough work. You may l ike to use the AnswerBook for any rough wor k, i f needed.

(25 2 = 50)

2.1 The fol lowing set of equat ions has

3x + 2y + z = 4

x – y + z = 2

– 2x + 2z = 5

(a) no solut ion

(b) a unique solut ion

(c) mult iple solut ions

(d) an inconsistency

2.2 The funct ion f(x, y) = 2x2 + 2xy – y3 has

(a) only one stat ionar y point at (0, 0)

(b) two stationary points at (0, 0) and (1/6, – 1/3)

(c) two stat ionary points at (0, 0) and (1, – 1)

(d) no stat ionary point

2.3 Manish has to t ravel fr om A to D changingbuses at stops B and C enroute. The maximumwai t ing t ime at either stop can be 8 minuteseach, but any t ime of wait ing up to 8 minutesis equally l ikely at both places. He can affordup to 13 minutes of total wait ing t ime, i f he isto ar r ive at D on t ime. What is the probabil i tythat Manish wil l ar r ive late at D?

(a) 8/13

(b) 13/64

(c) 119/128

(d) 9/128

2.4 Arr ivals at a telephone both are considered tobe Poisson, with an average t ime of 10 minutesbetween successive ar r ivals. The length of aphone cal l is dist r ibuted exponent ial ly withmean 3 minutes. The probability that an ar r ivaldoes not have to wait befor e service is

(a) 0.3

(b) 0.5

(c) 0.7

(d) 0.9

2.5 The supply at thr ee sources is 50, 40 and 60units respect ively whilst the demand at the fourdest inations is 20, 30, 10 and 50 units. In solvingthis t r anspor tat ion problem

(a) a dummy source of capacit y 40 uni ts isneeded

(b) a dummy dest inat ion of capacity 40 units isneeded

(c) no sol u t i on ex i st s as t he pr obl em i sinfeasible

(d) none solu t i on exists as t he pr oblem isdegenerate

2.6 A project consists of three paral lel paths withdur at ions and var iances of (10, 4), (12, 4) and(12, 9) respect ively. According to the standardPERT assumpt ions, the dist r ibut ion of theproject durat ion is

(a) Beta with mean 10 and standard deviat ion 2

(b) Beta with mean 12 and standard deviat ion 2

(c) N or mal wi t h mean 10 and st andar ddeviat ion 3

(d) N or mal wi t h mean 12 and st andar ddeviat ion 3

GATE : 2002 MECHANICAL ENGINEERING

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2.7 The coupling used to connect two shafts withlarge angular misal ignment is

(a) a Flange coupling

(b) an Oldham’s coupling

(c) a Flexible bush coupling

(d) a Hooke’s joint

2.8 A stat ic load is mounted at the centre of a shaftrotat ing at uniform angular velocity. This shaftwil l be designed for

(a) the maximum compr essive str ess (stat ic)

(b) the maximum tensi le st ress (stat ic)

(c) the maximum bending moment (stat ic)

(d) fat igue loading

2.9 Large speed reductions (greated than 20) in onestage of a gear t rain are possible through

(a) Spur gear ing

(b) Wor m gear ing

(c) Bevel gear ing

(d) Helical gear ing

2.10 I f the wire diameter of a closed coi l hel icalspr i ng subject ed t o compr essi ve l oad i sincreased from 1 cm to 2 cm, other parametersremaining same, then deflect ion wil l decr easeby a factor of

(a) 16 (b) 8

(c) 4 (d) 2

2.11 The relat ionship between Young’s modulus (E),Bulk modulus (K) and Poisson’s rat io () isgiven by

(a) E = 3 K (1– 2) (b) K = 3 E (1– 2)

(c) E = 3 K (1– ) (d) K = 3 E (1– )

2.12 The rat io of Euler ’s buckl ing loads of columnswith the same parameters having (i ) both endsfixed, and (i i ) both ends hinged is

(a) 2 (b) 4

(c) 6 (d) 8

2.13 I f the length of the cant i lever beam is halved,then natural frequency of the mass M at theend fo this cant i lever beam of negligible massis increased by a factor of

(a) 2 (b) 4

(c) 8 (d) 8

2.14 I n centr i fugal cast ing , the impur it ies are

(a) uniformly dist r ibuted

(b) for ced towards the outer sur face

(c) t rapped near the mean radius of the cast ing

(d) col lected at the cent re of the cast ing

2.15 The duct ili ty of a mater ial with work hardening

(a) increases

(b) decreses

(c) remains unaffected

(d) unpredictable

2.16 The temper ature of a carbur ising flame in gaswelding is ...... that of a neutral or an oxidisingflame.

(a) lower than (b) higher than

(c) equal to (d) unr elated to

2.17 I n a blank ing oper at ion, t he clear ance i sprovided on

(a) the die

(b) both the die and the punch equally

(c) the punch

(d) br i t t le the punch nor the die

2.18 A buil t -up-edge is formed while machining

(a) duct i le mater ials at high speed

(b) duct i le mater ials at low spped

(c) br i t t le mater ials at high speed

(d) br i t t le mater ials at low speed

2.19 The t ime taken to dr i l l a hole through a 25mm t hick plate wi th the dr i l l r otat ing at300 r.p.m. and moving at a feed rate of 0.25mm/revolut ion is

(a) 10 sec (b) 20 sec

(c) 60 sec (d) 100 sec

2.20 The proper t ies of mer cury at 300K are:density = 13529 kg/m3, specific heat at constantpressure = 0.1393 kJ/kg-K, dynamic viscosity= 0.1523 10-2 N.s/m2 and thermal conductivity= 8.540 W/m-K . The Prandt l number of themer cury at 300K is

(a) 0.0248

(b) 2.48

(c) 24.8

(d) 248

MECHANICAL ENGINEERING GATE : 2002

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2.21 What is the value of the view factor for twoincl ined flat plates having common edge ofequal width, and with an angle of 20 degrees?

(a) 0.83

(b) 1.17

(c) 0.66

(d) 1.34

2.22 A Carnot cycle is having an efficiency of 0.75. I ft he temper at ur e of the high t emper atur ereservoir is 727C, what is the temperature oflow temperature reservoir?

(a) 23C

(b) – 23C

(c) 0C

(d) 250C

2.23 An i deal ai r st andar d Ot t o cycl e has acompression rat io of 8.5. I f t he rat io of thespecific heats of () is 1.4, then what is thethermal efficiecy (in percentage) of the Ottocycle?

(a) 57.5

(b) 45.7

(c) 52.5

(d) 95

2.24 What is the speed of sound in Neon gas at atemperature of 500K (Gas constant of Neon is0.4210 kJ/kg-K)?

(a) 492 m/s

(b) 460 m/s

(c) 592 m/s

(d) 543 m/s

2.25 The efficiency of superheat Rank ine cycle ishigher t han that of simple Rank ine cyclebecause

(a) the enthalpy of main steam is higher forsuperheat cycle

(b) the mean temperature of heat addit ion ishigher for superheat cycle

(c) the temperature of steam in the condenseris high

(d) the qual i ty of stem in the condenser is low

SECTI ON - B

This sect ion consists of TWENTY quest ions of FIVEmar ks each. ANY FI FTEEN out of these quest ionshave to be answer ed on the Answer Book provided.

3. Using Laplace t ransforms, solve

2

2

d ydt

+ 4y = 12 t

given that t = 0 and dy/dt = 9 at t = 0

4. A tube of 35 mm outside diameter was tur nedon a lathe and the fol lowing data was obtained:

Rake angle = 35

Feed rate = 0.1 mm/rev

Cutt ing force = 2000 N

Cutt ing speed = 15 m/min

Length of cont inuous chip in one revolut ion

= 60 mm

Feed force = 800 N

Calculate the chip thickness, shear plane angle,velocity of chip along tool face and coefficientof fr ict ion.

5. Whi le measur ing the effect ive diameter of anexternal metr ic scr ew thr ead gauge of 3.5 mmpitch, a 30.5 mm diameter cyl indr ical standardand 2 mm diameter wi r es wer e used. t hemicr ometer r eading over the standard andwir es was 13.3768 mm. The cor r espondingreading over the thread gauge and wi res was12.2428 mm. Cal cu lat e t he t hr ead gaugeeffect ive diameter.

6. The arc length-voltage character ist ic of a D.C.arc is given by the equat ion V = 24 + 4L whereV is the arc voltage in volt s and L is the arcl en gt h i n mm. The st at i c vol t amper echar act er i st i c of t he pow er sou r ce i sapproximated by a straight l ine with a no loadvol tage of 80 Vol ts and shor t circui t cur rent of600 Amper es. Deter mine the opt imum ar elength for maximum power.

7. Est imate the metal removal rate (in cc/hr ) ofan alloy containing 18% Cobalt, 62% Nickel and20% Chr omium dur i ng Elect r o-ChemicalM ach i n i ng (E CM ) w i t h a cu r r en t of500 Amper es. The densi t y of t he al l oy i s8.28 gm/cc. The fol lowing data is available :

GATE : 2002 MECHANICAL ENGINEERING

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Metal Gr am At omic Wei gh t Valency

Cobalt 58.93 2

Nicket 58.71 2

Chromium 51.99 6

Assume Faraday’s constant as 96.500 Coulombs/mole

8. A chalk st ick is twisted to fai led to fai lure byapplying opposite torques T at the two ends.Take a square element ABCD with two sidespar al lel to the longi tudinal axis of the st ick.

(a) Show the free body diagram with pr incipalst resses.

(b) Find out the pr incipal stresses 1 and 2 andthe pr incipal planes.

(c) Show the plane on which fai lure/fracturewil l take place.

9. A cant i lever AB of length ‘L ’ has fixed end Aand free end B. I t is loaded by applying aconcentr ated load W at the mid point C of thecant i lever.

(a) Determine the deflect ion and slope at pointsC and B.

(b) Show def l ect i ons and sl opes on t hecant i lever.

10. I n a single degree-of-freedom, 4-l ink, revolutejointed Grashoff mechanism, ‘I ’ is the longestl ink, ‘s’ is the shor test l ink and p, q are the tworemaining l inks. These l inks may be in anyorder.

(a) Wr ite down the condi t ion for feasibi l ty ofGrashoff mechanism.

(b) When will you get crank-crank, crank-rockerand rocker -rocker type of mechanisms byfixing l inks, l , p or q espect ively?

(c) Draw these mechanisms showing mot ionsof crank and rocker l inks.

11. A toy cyl inder of weight w, length l , and radiusr rol ls without sl ipping on the inner cyl indr icalsur face of radius R, which is fixed.

(a) Deter mine t he di ffer ent ial equat ion ofmot ion of the toy for large osci l lations aboutits lowest point using energy method.

(b) Wr ite down the equat ion of mot ion andcir cular frequency of osci l lat ion of the toyfor small osci l lat ions.

The cent re of gravity of the joy is assumed tobe at the centre of the toy cyl inder.

12. A hollow mild steel shaft is subjected to torsionaland bending moments T and M respect ively.The outside and inside diameters of the shaftare D and d respect ively.

(a) Der ive the expressions for the maximumshear st r ess, and the maximum nor malstr ess induced due to combined T and M.

(b) Name any one theory of fai lure, which maybe used to design this shaft using thesest resses.

13. A company has introduced a new product withfixed cost of Rs. 200 per week and unit var iablecost of Rs 7. The product is sold to a r etai lerwith a quant i ty discount as per the fol lowingschedule:

Quan t i t y Un i t pr i ce

0 - 99 units Rs 10

100 units onwards Rs 8

(a) Deter mine t he pr of i t as a funct ion ofquant ity sold.

(b) I n what range of quant it ies sold does thecompany earn profi t?

14. A furniture manufacturer produces chairs andt ables. The wood wor k ing depar tment i scapable of pr oducing 200 chairs or 100 tablesor any propor t ionate combinat ions of these perweek. The weekly demand for chairs and tablesis l imited to 150 and 80 units respect ively. Theprofi t from a chai r is Rs 100 and that from atable is Rs 300.

(a) Set up the problem as a L inear Progr am.

(b) Determine the optimum mix for maximisingthe profi t .

(c) What is the maximum profi t?

(d) I f the profi t of each table dr ops to Rs 200per unit, what is the opt imal mix and profi t?

15. The precedence relat ions and durat ions of jobsin a project are given below:

Job Predecessor (s) Durat ion (in days)

A – 2

B – 4

C A 6

D A 8

E B, C 6

F B, C 4

G F 2

H F 8

I D.E.G 6

MECHANICAL ENGINEERING GATE : 2002

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(a) Draw the act ivity-on-arc project network.

(b) Deter mine al l cr i t ical path(s) and thei rduration(s)

(c) What is the total float for jobs B and D?

16. A copper tube of 20 mm outer diameter, 1 mmthickness and 20 m long (Thermal conductivity= 400 W/m2– K). is car rying saturated steam at150 C (Convect ive heat t ransfer coefficient =150 W/m2 – K ). The tube is exposed to anambient temperatur e of 27 C. The convect iveheat t r ansfer coefficient of air is 5 W/m2– K .Glass wool is used for insulat ion (Ther malconduct ivity = 0.075 W/m– K). I f the thicknessof the insulat ion used is 5 mm higher than thecr i t ical thickness of insulat ion, calculate therate of heat lost by the steam and the rate ofsteam condensat ion in kg/hr.

(The enthalpy of condensat ion of st eam=2230 kJ/kg.)

17. Air at 327C and 400 kPa wi th a volume flowrate of 5 m3/s flows through a turbine and exitsat 100 kPa and 182 C.

I f the expansion process is polytropic, calculatepower output , rate of heat t ransfer and rate ofchange in entropy (specific heat at constantpressur e of air = 1.0035 kJ/kg-K , and Gasconstant of air = 0.287 kJ/kg-K).

18. An ideal , ai r standard regenerat ive Braytoncycle i s wor k ing bet ween min imum andmaximum temperatures of 300K and 1200Krespect ively.

(a) Find out the value of cr i t ical pressure rat iowhere the degree of regener at ion becomescero.

(b) Calculate the efficiency of the cycle whenthe operat ing pr essur e rat io is 60% of thecr it ical pressure rat io.

19. A centr i fugal pump has an efficiency of 08%.The specificat ions of the pump ar e :

Dischar ge = 70m3/hr, Head = 7 m, speed =1450 rpm and diameter = 200 mm. I f the speedof this pump is increased to 1750 rpm, calculatethe discharge, head developed and power inputrequired without loss in efficiency.

20. The Wil lan’s l ine measured for a four -stroke,four -cyl inder is expressed as:

FC = 0.15 + 0.03 . B.P., where FC is the rate offuel consumption in gm/s and B.P. is the brakepower in k.W. The bore of each cylinder is 75 mmand stroke is 90 mm and the speed is 3000 rpm.

Calculate indicated power, mechanical efficiencyand indicated mean effective pressure, when theengine is developing a brake power of 20 kW.

21. An ice making plant refr igerant R-12 is havingan evapor at or sat ur at i on t emper at ur e of– 25C and t he condenser sat u r at i ontemperature of 35C. The vapour is leaving thecompressor at 65C. Following table shows theproper t ies of the refr igerant .

TempeC Pressur e, kPa Saturation Enthalpy kJ/kg

Liquid Vapour

– 25 123.7 13.3 176.5

35 850.0 69.6 201.5

Enthalpy of superheated refr igerant at 850 kPa and65 C = 225.5 kJ/kg

22. A r ecipr ocat ing compressor is to be designedto compress 4.5 kg/min of air from 100 kPaand 27° C thr ough an overal l pressure rat io of9. The law of compression is pV1.3 = constant .Calculate the savings in power consumpt ionand gain in isothermal efficiency, when a two-stage compressor with complete inter -cool ingis used in place of a single stage compressor.Assume equal pressure equal rat io in both thestages of the two-stage compressor.(Gas Constant = 0.287 kJ/kg-K)