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MECHANICAL ENGINEERING GATE : 2009

1

Q. 1 – Q. 20 Car r y one mar k each .

1. For a matr ix [M] =

35

4535

x

L

N

MMM

O

Q

PPP, the t ranspose of the

mat r ix is equal to the inverse of the matr ix,[M]T = [M]– 1. The value of x is given by

(a) 45

(b) 35

(c)35

(d)45

2. The divergence of the vector field 3xz i + 2xy j – yz2 k

at a point (1, 1, 1) is equal to

(a) 7 (b) 4

(c) 3 (d) 0

3. The inverse Laplace t ransform of 2

1

( )s sis

(a) 1 + et

(b) 1 – et

(c) 1 – e– t

(d) 1 + e– t

4. I f t hr ee coins ar e tossed simul taneously, theprobabil i ty of get t ing at least one head is

(a)18

(b) 38

(c)12

(d) 78

5. I f a closed system is undergoing an ir reversibleprocess, the entropy of the system

(a) must increase

(b) always remains constant

(c) must decrease

(d) can increase, decrease or remain constant

6. A coolant fluid at 30 oC flows over a heated flatplate maintained at a constant temperature of100oC. The boundar y l ayer t emper at u r edistr ibut ion at a given locat ion on the plate maybe approximated as T = 30 + 70 exp (– y) where y(in m) is the distance normal to the plate and T isin oC. I f t her mal conduct ivi ty of the fluid is1.0 W/mK, the local convect ive heat t ransfercoefficient (in W/m2K) at that locat ion wil l be

(a) 0.2 (b) 1

(c) 5 (d) 10

7. A fr ictionless piston-cylinder device contains a gasinit ial ly at 0.8 MPa and 0.015 m3. I t expandsquasi-statical ly at constant temperature to a finalvolume of 0.030 m3. The work output (in kJ)dur ing this process wil l be

(a) 8.32 (b) 12.00

(c) 554.67 (d) 8320.00

8. I n an ideal vapour compression refr iger at ioncycle, the specific enthalpy of refr igerant (in kJ/kg)at the fol lowing states is given as:

Inlet of condenser : 283

Exit of condenser :116

Exit of evaporator : 232

The COP of this cycle is

(a) 2.27 (b) 2.75

(c) 3.27 (d) 3.75

9. A compressor undergoes a reversible, steady flowpr ocess. The gas at i n let and out l et of thecompressor is designated as state 1 and state2 r espect ively. Potent ial and k inet i c ener gychanges ar e t o be i gnor ed. The fol l owi ngnotat ions are used :

v = specific volume and P = pressure of the gas.

The specific work required to be supplied to thecompressor for this gas compression process is

(a)2

1Pdv (b)

2P

1vd

(c) v1 (P2 – P1) (d) – P2 (v1 – v2)

GATE 2009 MECHANICAL ENGINEERING

Time Al l ow ed : 3 Hou r s M ax imum M ar k s : 150

GATE : 2009 MECHANICAL ENGINEERING

2

10. A block weighing 981 N is rest ing on a hor izontalsur face. The coefficient of fr ict ion between theblock and the hor izontal sur face is = 0.2.Avertical cable attached to the block provides partialsuppor t as shown. A man can pull hor izontal lywith a force of 100 N. What wil be the tension, T(in N) in the cable i f the man is just able to movethe block to the r ight ?

(a) 176.2

(b) 196.0

(c) 481.0

(d) 981.0

11. I f the pr incipal st resses in a plane stress problemare 1 = 100 MPa, 2 = 40 MPa, the magnitude ofthe maximum shear stress (in MPa) wil l be(a) 176.2

(b) 196.0

(c) 481.0

(d) 981.0

12. A simple quick return mechanism is shown in thefigure. The forward to return rat io of the quickreturn mechanism is 2 : 1. I f the radius of thecrank O1 P is 125 mm, then the distance 'd' (in mm)between the crank centre to lever pivot centrepoint should be

Q

PO1

d

O2

(a) 144.3

(b) 216.5

(c) 240.0

(d) 250.0

13. The rotor shaft of a large electr ic motor suppor tedbetween shor t bear ings at both the ends shows adeflect ion of 1.8 mm in the middle of the rotor.Assuming the rotor to be per fect ly balanced andsuppor ted at knife edges at both the ends, thel ikely cr i t ical speed (in rpm) of the shaft is

(a) 350 (b) 705

(c) 2810 (d) 4430

14. A solid circular shaft of diameter d is subjected toa combined bending moment M and torque, T. Themater ial property to be used for designing the shaft

using the relat ion 16 2 2M T3d

is

(a) ult imate tensile st rength (Su)

(b) tensi le yield strength (Sy)

(c) tor isonal yield strength (Ssy)

(d) endurance strength (Se)

15. The effect ive number of lat t ice points in the unitcel l of simple cubic, body centered cubic, and facecentered cubic space lat t ices, respect ively, are

(a) 1, 2, 2 (b) 1, 2, 4

(c) 2, 3, 4 (d) 2, 4, 4

16. Fr ict ion at the tool-chip inter face can be reducedby

(a) decreasing the rake angle

(b) increasing the depth of cut

(c) decreasing the cut t ing speed

(d) increasing the cut t ing speed

17. Two streams of l iquid metal which are not hotenough to fuse properly result into a cast ing defectknown as

(a) cold shut

(b) swell

(c) sand wash

(d) scab

18. The expected t ime (te) of a PERT act ivity in termsof opt imist ic t ime (to) , pessimist ic t ime (tp) andmost l ikely t ime (t l) is given by

(a)4

6o l p

e

t t tt

(b)

4

6o p l

e

t t tt

(c)4

3o l p

e

t t tt

(d)

4

3o p l

e

t t tt


3

19. Which of t he fol l owing i s t he cor r ect dat astructure for soi ld models ?

(a) sol id par t faces edges ver t ices

(b) sol id par t edges faces ver t ices

(c) ver t ices edges faces solid par ts

(d) ver t ices faces edges solid par ts

20. Which of the fol lowing forecasting methods takesa fract ion of forecast er ror into account for thenext per iod forecast ?

(a) simple average method

(b) moving average method

(c) weighted moving average method

(d) exponent ial smoothening method

Q. 21 t o Q. 60 car r y t w o mar k s each .

21. An analytic function of a complex var iable z = x + iyis expresed as f (z) = u, (x, y) + i v (x, y) where

i = 1. I f u = xy, the expression of v should be

(a)2( )

2x y

k

(b) 2 2

2x y

k

(c)2 2

2y x

k

(d) 2( )

2x y

k

22. The solut ion of 4dyx y x

dx with the condit ion

y (1) = 65

is

(a)4 1

5x

yx

(b)44 4

5 5x

yx

(c)4

15x

y

(d)5

15x

y

23. A path AB in the form of one quar ter of a circle ofunit radius is shown in the figure. Integrat ion of(x + y)2 on path AB traversed in a counter-clockwisesense is

Y

B

A X

(a) 12 (b) 1

2

(c)2

(d) 1

24. The distance between the or igin and the pointnearest to i t on the sur face z2 = 1 + xy is

(a) 1

(b) 3

2

(c) 3

(d) 2

25. The area enclosed between the curves y2 = 4x andx2 = 4y is

(a)163

(b) 8

(c)323

(d) 16

26. The standard deviation of a uniformly dist r ibutedrandom var iable between 0 and 1 is

(a)1

12

(b)1

3

(c)5

12

(d)7

12


4

27. Consider steady, incompressible and ir rotat ionalflow through a reducer in a horizontal pipe wherethe diameter is reduced from 20 cm to 10 cm. Thepressure in the 20 cm pipe just upstream of thereducer is 150kPa. The fluid has a vapour pressureof 50 kPa and a speci fic weight of 5 kN/m3.Neglect i ng fr i ct i onal ef fect s, t he maximumdischarge (in m3/s) that can pass through thereducer withoud causing cavitat ion is

(a) 0.05

(b) 0.16

(c) 0.27

(d) 0.38

28. I n a paral lel flow heat exchanger operating understeady state, the heat capacity rates (product ofspecific heat at constant pressure and mass flowrate) of the hot and cold fluid are equal. The hotfluid, flowing at 1 kg/s with Cp = 4 kJ/kgK, entersthe heat exchanger at 102oC while the cold fluidhas an inlet temperature of 15oC. The overal l heatt ransfer coefficient for the heat exchanger isest imated to be 1 kW/m2K and the cor respondingheat t ransfer sur face area is 5 m2. Neglect heatt r ansfer between the heat exchanger and theambient . The heat exchanger is character ized bythe fol lowing relat ion:

2= – exp (– 2NTU)

The exit temperature (in oC) for the cold fluid is

(a) 45

(b) 55

(c) 65

(d) 75

29. I n an air -standard Otto cycle, the compressionrat io is 10. The condit ion at the beginning of thecompression process is 100 kPa and 27oC. Heatadded at constant volume is 1500 kJ/kg, while700 kJ/kg of heat is rejected dur ing the otherconstant volume process in the cycle. Specific gasconst ant for ai r = 0.287 k J/kgK . The meaneffect ive pressure (in kPa) of the cycle is

(a) 103

(b) 310

(c) 515

(d) 1032

30. An ir reversible heat engine extracts heat from ahigh temperature source at a rate of 100 kW and

rejects heat to a sink at a rate of 50 kW. Theent ire work output of the heat engine is used to

dr ive a reversible heat pump operat ing betweena set of independent isothermal heat reservoirs

at 17oC and 75oC. The rate (in kW) at which theheat pump delivers heat to i ts high temperature

sink is

(a) 50

(b) 250

(c) 300

(d) 360

31. You are asked to evaluate assor ted fluid flows for

their suitabi li ty in a given laboratory applicat ion.The fol lowing three flow choices, expressed in

terms of the two-dimensional velocity fields in thexy-plane, are made available.

P. u = 2y, v = – 3x

Q. u = 3xy, v = 0

R. u = 2x, v = 2y

Which flows (s) should be recommended when theapplication requires the flow to be incompressible

and ir rotat ional ?

(a) P and R

(b) Q

(c) Q and R

(d) R

32. Water at 25 oC is flowing through a 1.0 km long

G.I . pipe of 200 mm diameter at the r ate of0.07 m3/s. I f value of Darcy fr ict ion factor for this

pipe is 0.02 and density of water is 1000 kg/m3,the pumping power (in kW) required to maintain

the flow is

(a) 1.8

(b) 17.4

(c) 20.5

(d) 41.0


5

33. Consider steady-state heat conduct ion across thethinkness in a plane composite wall (as shown inthe figure) exposed to convect ion condit ions onboth sides.

Given: h1 = 20 W/m2K; ho = 50 W/m2K; T, i = 20oC;T,o = 20oC; k 1 = 20 W/mK ; k 2 = 50 W/mK ;L1 = 0.30 m and L2 = 0.15 m. Assuming negligiblecontact resistance between the wall surfaces, theinterface temperature, T (in oC), of the two wallswill be

(a) – 0.50

(b) 2.75

(c) 3.75

(d) 4.50

34. The velocity profi le of a ful ly developed laminarflow in a st raight circular pipe, as shown in thefigure, is given by the expression

2 2

2

R( ) 1

4 R

dp ru r

dx

where dpdx

is a contant .

The average velocity of fluid in the pipe is

(a)2R

8dpdx

(b) 2R

4dpdx

(c)2R

2dpdx

(d) 2R dp

dx

35. A solid shaft of diameter, d and length, L is fixedat both the ends. A torque, T0 is appl ied at a

distance L4

from the left end as shown in the

figure given below.

L/4 3L /4

T0

The maximum shear stress in the shaft is

(a)03

16T

d

(b) 03

12T

d

(c)03

8T

d

(d) 03

4T

d

36. An epicycl ic gear t rain is shown schemat ical ly inthe given figure.

5

3

2

4

The sun gear 2 on the input shaft is a 20 teethexternal gear. The planet gear 3 is a 40 teethexternal gear. The r ing gear 5 is a 100 teethinternal gear. The r ing gear 5 is fixed and thegear 2 is rotat ing at 60 rpm ccw (ccw = counter -clockwise and cw = clockwise). The arm 4 attachedto the output shaft wil l rotate at

(a) 10 rpm ccw

(b) 10 rmp cw

(c) 12 rpm cw

(d) 12 rpm ccw.


6

37. A forged steel l ink wi th uni form diameter of30 mm at the centre is subjected to an axial forcethat var ies from 40 kN in compression to 160 kNin t ension. The t ensi le (Su), y i el d (Sy) andcor rected endurance (Se) st rengths of the steelmater ial are 600 MPa, 420 MPa and 240 MParespect ively. The factor of safety against fat igueendurance as per Soderberg's cr i ter ion is

(a) 1.26 (b) 1.37

(c) 1.45 (d) 2.00

38. An aut omot i ve engine weigh i ng 240 k g i ssuppor t ed on fou r spr i ngs w i t h l i nearcharacter istics. Each of the front two springs havea st i ffness of 16 MN/m while the st i ffness of eachr ear spr i ng i s 32 M N/m. The engine speed(in rpm), at which resonance is likely to to occur, is

(a) 6040 (b) 3020

(c) 1424 (d) 955

39. A vehicle suspension system consists of a spr ingand a damper. The st i ffness of the spr ings is3.6 kN/m and the admping constant of the damperis 400 Ns/m. I f t he mass is 50 kg, then thedamping factor (d) and damped natural frequency(fn) , respect ively, are

(a) 0.471 and 1.19 Hz

(b) 0.471 and 7.48 Hz

(c) 0.666 and 1.35 Hz

(d) 0.666 and 8.50 Hz

40. A frame of two arms of equal length L is shown inthe figure. The flexural r igidity of each arm ofthe frame is EI . The vartical deflection at the pointof applicat ion of load P is

L

L

P

(a)PL3EI

3

(b) 32PL

3EI

(c)32PL

EI(d)

34PL3EI

41. A uniform r igid rod of mass M and length L ishinged at one end as shown in the figure. A forceP is applied at a distance of 2L/3 from the hingeso that the rod swings to the r ight . The react ionat the hinge is

2L3

L

P

(a) – P (b) 0

(c)P3

(d)2P3

42. Match the approaches given below to per formstated kinemat ics/ dynamics analysis of machine.

Analysi s

P. Cont inous relat ive rotat ion

Q. Velocity and accelerat ion

R. Mobility

S. Dynamic-static analysis

Appr oach

1. D' Alembert 's pr inciple

2. Grubler 's cr i ter ion

3. Grashoff 's law

4. Kennedy's theorem

(a) P – 1, Q – 2, R – 3, S – 4

(b) P – 3, Q – 4, R – 2, S – 1

(c) P – 2, Q – 3, R – 4, S – 1

(d) P – 4, Q – 2, R – 1, S – 3

43. A company uses 2555 units of an i tem annually.Delivery lead t ime is 8 days. The reorder point(in number of units) to achieve optimum inventory is

(a) 7

(b) 8

(c) 56

(d) 60


7

44. Consider the fol lowing L inear Pr ogr ammingProblem (LPP):

Maximize z = 3x1 + 2x2

Subject to x1 4

x2 6

3x1 + 2x2 18

x1 0, x2 0

(a) The LPP has a unique opt imal solut ion.

(b) The LPP is infeasible.

(c) The LPP is unbounded.

(d) The LPP has mult iple opt imal solut ions.

45. Six jobs ar r ived in a sequence as given below:

Jobs Processing Time (days)

I 4

II 9

III 5

IV 10

V 6

VI 8

Average flow t ime (in days) for the above jobsusing Shor test Processing t ime rule is

(a) 20.83 (b) 23.16

(c) 125.00 (d) 139.00

46. Minimum shear strain in or thogonal turning witha cut t ing tool of zero rake angle is

(a) 0.0 (b) 0.5

(c) 1.0 (d) 2.0

47. Eelect r ochemical machining is per for med toremove mater ial from an iron surface of 20 mm 20mm under the fol lowing condit ions:

Inter electrode gap = 0.2 mm

Supply voltage (DC) = 12 V

Specific resistance of = 2 cm

electrolyte

Atomic weight of I ron = 55.85

Valency of I ron = 2

Farady's constant = 96540 CoulombsThe matr ial removal rate (in g/s) is

(a) 0.3471 (b) 3.471

(c) 34.71 (d) 347.1

48. Match the fol lowing:

NC code Def i n i t i on

P. M05 1. Absolute coordinate system

Q. G01 2. Dwell

R. G04 3. Spindle stop

S. G09 4. L inear interpolat ion

(a) P – 2, Q – 3, R – 4, S – 1

(b) P – 3, Q – 4, R – 1, S – 2

(c) P – 3, Q – 4, R – 2, S – 1

(d) P – 4, Q – 3, R – 2, S – 1

49. What are the upper and lower l imits of the shaftrepresented by 60 f8 ?

Use the fol lowing data:

Diameter 60 l ies in the diameter step of 50-80mm.

Fundamental tolerance unit , i , in m

= 0.45 D1/3 + 0.001 D

where D is the representat ive size in mm;

Tolerance value for IT8 = 25i,

Fundamental deviat ion for 'f ' shaft = – 5.5Do.41

(a) Lower l imit = 59.924 mm,

Upper l imit = 59.970 mm

(b) Lower l imit = 59.954 mm,

Uppr l imit = 60.000 mm

(c) Lower l imit = 59.970 mm,


(d) Lower l imit = 60.00 mm,


50. Match the i tems in Column I and Column I I .

Column I Column I I

P. Metal lic Chills 1. Suppor t for the core

Q. Metallic Chaplets 2. Reservoir of themolten metal

R. Riser 3. Control cooling ofcr i t ical sect ions

S. Exothermic 4. ProgressivePadding solidification

(a) P – 1, Q – 3, R – 2, S – 4

(b) P – 1, Q – 4, R – 2, S – 3

(c) P – 3, Q – 4, R – 2, S – 1

(d) P – 4, Q – 1, R – 2, S – 3


8

Common Data Quest i ons

Common Dat a for Quest i on 51 and 52 :

The inlet and the out let condit ions of steam foran adiabat ic steam turbine are as indicated in thefigure

The notat ions are as usually fol lowed.

51. I f mass rate of steam through the turbine is 20 kg/s,the power output of the turbine (in MW) is

(a) 12.157

(b) 12.941

(c) 168.001

(d) 168.785

52. Assume the above turbine to be par t of a simpleRankine cycle. The density of water at the inletto the pump is 1000 kg/m3. I gnor ing kinet ic andpotential energy effects, the specific work (in kJ/kg)supplied to the pump is

(a) 0.293

(b) 0.351

(c) 2.930

(d) 3.510

Common Dat a for Quest i ons 53 and 54 :Radiative heat transfer is intended between theinner sufaces of two very large isothermal parallelmetal plates. While the upper plate (designated asplate 1) is a black sur face and is the warmer onebeing maintained at 727 oC, the lower plate(plate 2) is a diffuse and gray surface with anemi8ssivity of 0.7 and is kept at 227 oC.

Assume that the sur faces are sufficiently large toform a two-sur face enclosure and steady-statecondit ions to exist . Stefan-Boltzmann constant isgiven as 5.67 10– 8 W/m2K 4.

53. The ir radiat ion (in kW/m2) for the upper plate(plate 1) is

(a) 2.5 (b) 3.6

(c) 17.0 (d) 19.5

54. I f plate 1 is also a diffuse and gray sur face withan emissivity value of 0.8, the net radiat ion heat

exchange (in kW/m2) between plate 1 and plate 2is

(a) 17.0

(b) 19.5

(c) 23.0

(d) 31.7

Common Dat a for Quest i ons 55 and 56 :

Consider the fol lowing PERT network:

The opt imi st i c t ime, most l i k el y t ime and

pessimist ic t ime of al l the act ivit ies are given inthe table below:

Activity Optimist ic Most l ikely Pessimist ict ime (days) t ime (day) t ime (days)

1-2 1 2 3

1-3 5 6 7

1-4 3 5 7

2-5 5 7 9

3-5 2 4 6

5-6 4 5 6

4-7 4 6 8

6-7 2 3 4

55. The cr it ical path duration of the network (in days)

is

(a) 11

(b) 14

(c) 17

(d) 18


9

56. The standard deviat ion of the cr i t ical path is

(a) 0.33

(b) 0.55

(c) 0.77

(d) 0.66

L ink ed Answ er Quest i ons

St atement for L i nk ed Answer Quest ions 57

and 58:

I n a machining exper iment, tool l i fe was found to

var y with the cut t ing speed in the fol lowing

manner :

Cutt ing speed (m/min) Tool l i fe (minutes)

60 81

90 36

57. The exponent (n) and constant (k) of the Taylor 's

tool l i fe equat ion are

(a) n = 0.5 and k = 540

(b) n = 1 and k = 4860

(c) n = 1 and k = 0.74

(d) n = – 0.5 and k = 1.155

58. What is the percentage increase in tool l i fe whenthe cut t ing speed is halved ?

(a) 50%

(b) 200%

(c) 300%

(d) 400%

St at emen t for L i nk ed Answ er Quest i ons 59and 60

A 20o ful l depth involute spur pinion of 4 mmmodule and 21 teeth is to t r ansmit 15 kW at960 rpm. I ts face width is 25 mm.

59. The tengent ial force t ransmit ted (in N) is

(a) 3552

(b) 2611

(c) 1776

(d) 1305

60. Given that the tooth geometry factor is 0.32 andthe combined effect of dynamic load and al l iedfactors intensifying the stress is 1.5; the minimumallowable stress (in MPa) for the gear mater ial is

(a) 242.0

(b) 166.5

(c) 121.0

(d) 74.0

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