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GATE -1996

ME: Mechanical Engineering

1. Write in your answerbook the correct or the most appropriate answer to the following multiple

choice questions by writing the corresponding letter A, B, C or D against the subquestions number.

(25 xl =25)

(A) 0

Ans. (B)

Explanation.

1t

(C) -2

P eilogi

ei(in/2)

e- 1t/2

1.2 The expression curl (grad f), where f is a scalar function, is

(A) equal to V2 f (B) equal to div (grad t)

(C) a scalar of zero magnitude (0) a vector of zero magnitude

Ans. (0)

1.3 The one dimensional heat condietion parlial differential equation

a T  & T.- = - ISat ax..2'  

(A) parabolic (B) hyperbolic (C) elliptic (D) mixed

Ans. (A)

1.4 III the Gauss elimination method for solving a system of linear algebraic equations, triangularization

leads to

(A) diagonal matrix

(C) upper triangular matrix

Ans. (C)

1.5 If the two principal strains at a point are 1000 x 1 0 -< ' and -600 x 10-<',then the maximum shp-ar strain is

(A) 800 x 10-6 (B) 500 x 10-6

(C) 1600 X 10-6 (0) 200 x 10-6

Ans. (0)

(B) lower triangular matrix

(0) singular matrix

-1 0

(1000 - 600) ~

600 x 10 1E B o 1000 " 10~ E

E ° E•

'600 x 10'"--E- Mulmumlhear

au.in = 200 x 10-6

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1.6 If the ratio of the diameter of rivet hole to the pitch of rivets;s (J.25, then the tearing efficiency of the

  joint is

(A) 0.50 (8) 0.75 (C) 0.25 (D) 0.87

Ans. (8)

0.25

( P ; d ) x 1 0 075%

1. 7 In thick film hydrodynamic journal bearing.~, the coefficient of friction

(A) increases with increases in load

(8) is independent of load

(C) decreases with increase in load(0) may increase or decrease with increase in load

Ans. (C)

1.8 A ball A of mass m falls under gravity from a height h and strikes another ball B of mass m wlrich is

supported at rest on a spring of stiffness k. Assume peifeetly elastic impact. Immediately after the impact 

(A) the velocity of ball A is ~~2gh (8) the velocity o f ball A is zero

(C) the velocity of both balls is ~~2gh (0) none of the above2

Ans. (8)

Explanation. In a perfectly elastic collision between equal masses of two bodies, velocities exchange on

impact.Velocity just before impact = Velocity immediately after impact.

Before impact After impact

V A ~2gh V A = 0

V B 0 V B =~2gh

1.9 The bolts in a rigid flanged coupling connecting two shafts transmitting power are subjected to

(A) shear force and bending moment (8) axial force

(C) torsion (0) torsion and bending moment.

Ans. (A)

1.10 A wheel of mass m and radius r is in accelerated rolling motion without slip under a steady axle torque

T . If the coefficient of killetic friction is p, the friction force from the ground on tire wheel is(A) !! mg (8) T/r

(C) zero (0) none of the above

Ans. (A)

1.11 In spur gears, the circle on which the involute is generated is called the

(A) pitch circle (8) clearance circle

(C) base circle (0) addendum circle

Ans. (A)

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1.12 The specific heats of an ideal gas depend on its

(A) temper~ ,'(B) pressure

(C) volume (D) molecular weight and structure

Ans. (A)

1.13 For reversibile adiabatic compresslonUttl steady flow process, the work transfer per unit mass is

(A)fpdv (B)fvdp

(C) f T ds (D) f s dT

Ans. (B)

1.14 A cycle consisting of two reversible isothermal processes and two reversible isobaric processes is knownas

(A) Atkinson cycle(C) Brayton cycle

Ans. (D)

(B) Stirling cycle(D) Ericsson cycle

1.15 /11 flow through a pipe, the transition from laminar to turbulent flow does not depend on

(A) velocity of the fluid (B) density of the fluid

(C) diameter of the pipe (D) length of the pipe

Ans. (C)

t .16 For a given heat flow and for the same thickness, the temperature drop across the material will bemaximumfor 

(A) copper (B) steel

(C) glass-wool (D) refractory brick  Ans. (C)

I orq = -kA-

2 a x'k' should be least for a T  to be maximum for a given q, A and a x .

1.17 When wet steamflows through a throttle valve and remains wet at exit 

(A) its temperature and quality increases

(B) its temperature decreases but quality increases

(C) its temperature increases but quality decreases

(D) its temperature and quality decreases

Ans. (B)

. • ~ t : ' N l . . The dimenation of surface tension is

  \ \ (A) ML-1

)(C)ML-1TI

Ans. (D)

(B) UT-I

(D) MT-2

1.19 The rake angle in a drill

(A) increases from centre to periphery

(C) remains constant

(B) decreases from centre to periphery

(D) is irrelevant to the drilling operation

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1.20 Light impurities in the molten metal are prevented from reaching the mould cavity by providing a

(A) strainer (B) button well

(C) skim bob (D) all of the above

A D s . (D)

1.21 Prelreating before welding is done to

(A) make the steel softer

(C) prevent cold cracks

A D s . (C)

(B) bum away oil, grease, etc, from the plate surface

(D) prevent plate distortion

1.22 Tirefit on a Irole-shaft system is specified as 117-.'~6.Tire type of fit is

(A) clearance fit (B) running fit (sliding fit)

(C) push fit (transition fit) (D) force fit (interference fit)ADS. (D)

1.23 Tire iron-carbon diagram and the TTT curves are determined under 

(A) equilibrium and non-equilibrium conditions respectively

(B) non-equilibrium and equilibrium conditions respectively

(C) equilibrium conditions for both

(D) non-equilibrium conditions for both.

A D S . (A)

1.24 If at the optimum in a linear programming problem, a dual variable corresponding to a particular 

  primal constraint is zero, tlren it means tlrat 

(A) right hand side of the primal constraint can be altered without affecting the optimum solution(B) changing the right hand side of the primal constraint will disturb the optimum s~ution

(C) the objective function is unbounded

(D) the problem is degenerate

Ans. (C)

1.25 Tire individual human variability in time studies to determine the production standards is taken care of 

b y 

(A) personal allowances (B) work allowances

(C) rating factor (D) none of the above

A D S . (C)

2. Write in your allswerbook the correct or the most appropriate anwer to the following multiple choicequestions by writing the corresponding letter A, B, Cor D against the subquestion number.

2.1 Theeigenvalues of [ :

(A) 0, 0 0

(C) 0,0,3

A D S . (C)

(B) 0, 0 1

(D) 1, 1, 1

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[

1

1 1

Explanation. Given matrix is,

(1- A) 1 I

I (1- A) I

I I (I-A)

(3-A) (3- A) (3- A)

1 (1- A) I

I I (I-A)

1 1

(3- A) (I-A) I

1 (I- A)

0 0

(3- A) -A 0

0 -A

2.2 A ball falls from a height of  Jm, hits the ground and rebounds with half its velocity just before impact.

Then after rising itfalls and hits the ground and again rebounds with halfits velocity just before impact,

and so on. The total distance travelled by the ball till it comes to rest on the ground is

(A) 2 m (B) 5/3 m

(C) 5/3 m (D) 5/4 m

ADS. (B)

e2 x2gh 4h+---x2+e xhx2+ .

2g

h [1+ 2e2 + 2e4 + ...]

h + 2 e 2 h [1 + ~ + _1_ +... J4 16

2.3 The particular solution for the differential equation

d 2

d - - - - . L +3- . 2 : . + 2y = 5cos x isdx 2 dx 

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(A) 0.5 cos x + 1.5 sin x

(C) 1.5 sin x

Ans. (A)

(8) 1.5 cos x + 0.5 sin x

(D) 0.5 cos x

I

02 + 3D + 2 (5 cos x)

5cosx

02+30+2

5(30-I)2 cosx

90 -1

_5_[ -2 sin x - cosx]-10

cosx 3sin x--+--

2 2

0.5 cos x + 1.5 sin x.

2.4 A solid shaft can resist a bending moment of 3.0 kNm and a twisting moment of 4.0 kNm together, then

the maximum torque that can be applied is

(A) 7.0 kNm

(C)4.5 kNm

Ans. (D)

(B) 3.5 kNm

(D) 5.0 kNm

Explanation. Maximum tensile stress due to bending, ~

TR

J

_4_x_I_O_3

_x_d_/_2= 20;7 kPa.

~xd4 d32

When both the stress acts simultaneously, maximum induced shear stress,

~[~(ft)2 +4(f s)2 ]

nd3

-f16 smax.

1 t l ~ 3x~ ( 3 0 ~ 5 3 5 8 r + 4 ( 2 ~ . ~ 7 r5kNm

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2.5 A thick cylinder is subjected to an internal pressure of 60 MPa. If the hoop stress on the outer surface is

150 MPa, then the hoop .••tress on the internal surface is

(A) 105 MPa (B) 180 MPa(C) 210 MPa (D) 135 MPa

Ans. (C)

2Pr2I

(r; -r?)

2x60xrj2

(r; - r?)

r . 2

1

(r; - r?)

( : ~Jpe r ; + rh( r ; - rh

2.6 AB and CD two uniform and identical bars of mass 10 kg each, as shown in Fig. 2.6. The hinges at A

and Bare frictionless. The assembly is released from rest and motion occurs in the vertical plane. At the

instant that the hinge B passes the point B, the angle between the two bars will be

r-1m

I

. I

,I cB 1

(A) 60 degrees

(C) 30 degrees

Ans. (C)

(B) 37.4 degrees

(D) 45 degrees

2.7 A rod of length 1 m is sliding in a corner as shown in Fig. 2.7. At an instant when the rod makes an angle

of 60 degrees with the horizontal plane, the velocity of point A on the rod is 1mls. The angular velocity

of the rod at this instant 

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(A) 2 rad/s

(B) J.5 rad/s 

(C) 0.5 rad/s

(0) 0.75 rad/s

Ans. (A)

Explanation :

v.l!!

•2 . : 2 .

•

2.8 A mass of 1 kg is suspended by means of  3 springs as shown in Fig 2.8. The spring constants X I' X : z and 

 X J  are respectively 1 kN/m, 3kN/m and  2kN/m. The natura/frequency of the system is approximately

(A) 46.90 Hz

(B) 52.44 Hz

(C) 60.55 Hz

(0) 77.46 Hz

I I 4-+-=- kN/m I 3 3

K, +K3

3- +24

! ! .kN 1m 

4

I I x 103

= 52.44 Hz4xl

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2.9 A mercury manometer is used to measure the static pressure at a point in a water pipe as shown in Fig.

2.9. The level difference of mercury in the two limbs is 10mm The gauge presure at that point is(A) 1236 Pa

(B) 1333 Pa

(C) zero

(D) 98 Pa

Ans. (B)

Explanation:

H 2O -- ~ J ... .: 10mm;T:

Hgl--.1lt~...•:::::..:::.- U · ' pgh

. 1013,600 x 9.81 x -- = 1333 Pa

1000

2.10 A solar energy based heat engine which receives 80 kJ of heat at 100 deg C and rejects 70 kJ of /reat to

the ambient at 30 deg C is to be designed. The thermal efficiency of the heat engine is

(A) 70% (B) 1.88%

(C) 12.5% (D) indetermjnate

ADS. (C)

(80-70)

80

1

8

12.5%

2.11For two cycles coupled in series, the topping cycle has an efficiency of 30% and the bottoming cycle hasan efficiency of 20%. The overall combined cycle efficiency is

(A) 50% (B) 44%

(C) 38% (D) 55%

ADS. (B) High Temperature

.2lQ2

From equation (i) and (ii)

h0.7

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  A steam plant has the boiler effICiency of92%, trubine efficiency (mechanical) of94%, generator efficiency

of  95% and cycle efficiency of  44%. If  6% of the generated power is used to run the auxiliaries, the

overall plant efficiency is

(A) 34%

(C) 45%

Ans. (A)

Explanation.

(B) 39%

(D) 30%

Q ,1 l. ~ «11> 1411>

--1Boi••. HStaam cycleHTurbine

Useful output

Energy input

Overall plant efficiency

WOUI = Q inX 0.92 x 0.44 x 0.94 x 0.95 x 0.94

Qin

0.92 x 0.44 x 0.94 x 0.95 x 0.94

= = 0.34 = 34%

2.13 For laminar fiow through a long pipe, the pressure drop per unit length increases

(A) in linear proportion to the cross-sectional area

(B) in proportion to the dimeter of the pipe

(C) in inverse proportion to the cross-sectional area

(D) in inverse proportion to the square of cross-sectional area

Ans. (C)

2.14 For a current wire of 20 mmdiameter exposed to air (h =20 W/m1K), maximum heat dissipation occurs

when thickness of insulation (k  =0.5 W/mK) is

(A) 20 mm (B) 25 mm

(C) 20 mm (D) 10 mm

Ans. (B)

k 0.5 Ir = -=-=-m=25mm

o h 20 40

2.15 Tool life of  10 hours is obtained when cutting wi~h single point tool at  63 m/min. If Taylor's constant  C 

=257.35, tool life on doubling the velocity will be

(A) 5 hours (B) 25.7 min.

(C) 38.3 min (D) unchanged

Ans. (B)

Explanation. VT" Cwhere, V in m/min; and T in minutes.

.. 63 x (10 x 60)n 257.35

or n = 0.22

IWhen velocity is doubled,

~ C = 0.22 257.35

2V 2x63

=25.69 minutes", 25.7 min.

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2.16 A wire of 0.1 mm diameter is drawn from a rod of 15mm diameter. Dies giving reductions of 20%, 40%

and 80% are available. For minimum error in thefinal size, the number of stages and reduction at each

stage respectively would be

(A) 3 stages and 80% reduction for all three stages

(B) 4 stages and 80% reduction for first three stages followed by a finishing stage of20% reduction

(C) 5 stages and reduction of 80%, 80%.40%,40%,20% in a sequence

(D) none of the above

Ans. (D)

2

Reduction required = ~ = 22500

0 .12

Reduction achievable

(A) 5 x 5 x 5 = 125

(B) .5 x 5 x 5 x 1.25 = 156.25

(C) 5 x 5 x 2.5 x 4.5 x 1.25 = 351.5625

(D) None.

2.17 A 50 mm diameter disc is to be punched outfrom a carbon steel sheet 1.0 mm thick. The diameter of the

  punch should be

(A) 49.925 mm

(C) 50.075 mm

Ans. (D)Explanation. Clearance required for punching carbon steel sheet = 7% of thickness

(B) 50.00 mm

(D) none of the above

lx7 --=0.07mm.100

50 -2 x 0.07

49.860 mm.

2.18 In an assembly linefor assembling toys, five workers are assigned tasks which take times of 10,8,6,9

and 10 minutes respectively. The balance delay for the line is

(A) 43.5% (B) 14.8%

(C) 14.0% (D) 16.3%

Ans. (D)

(10-8) (10-6) (10-9) x 100

(10x 5)

16% ~ 16.3%

2.19 Preliminary work sampling studies show that machine was idle 25% of the time based on a sample of 

100 observations. The number of observations needed for a confidence level of95% and an accuracy of 

,± 5% is

(A) 400

(C) 3600

Ans. (D)

(B) 1200

(D) 4800

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where K 2 for 95% confidence level

S 0.05 (accuracy)

P 0.25 (idle time)

Number of observations needed for 95% wnfidence level and ± 5% accuracy is

K2(P) (I-P)N = -----

S2P2

(2)2(0.25) (1-0252 =4800

(0.05)2 (025)2

2.20 The probability of a defective piece being produced in a manufacturing process is 0.01. The probability

that out of 5.\·uccessi"e pieces, only one is defective, is

(A) (0.99)4 (0.01) (B) (0.99) (0.01)4

(C) 5 x (0.99) (0.01)4 (D) 5 x (0.99)4 (0.01)

Ans. (D)

Explanation. Probability 5C,(0.OI)' (0.99)4

5 x (0.99)4 (0.01)

3. Match 4 correct pairs between List  1and List IIfor questions 3.1 through 3.5 below. No credit will be

given for partially correct matching. (5 x 2 = 10)-~"

List I List II

(A) Collision of particles 1. Euler's equation of motion

(B) Stability 2. Minimum kinetic energy

(C) Satellite motion 3. Minimum potential energy(D) Spining top 4. Impulse-momentum principle

5. Conservation of moment of momentum

List I List II

(A) Fourier number 1. Surface tension

(B) Weber number 2. Forced convection

(C) Grashoffnumber 3. Natural convection

(D) Schmidt number 4. Radiation

5. Transient heat conduction

6. Mass diffusion

List I List II

(A) Cetane number 1. Ideal gas

(B) Approach and range 2. VAn der Waals gas

(C) ( ~ ; ) h*0 3. S.l. engine

(D) dh = edT, even when pressure varies 4. C.l. enginep

5. Cooling towers

6. Heat exchangers

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List I List II

(A) Rivets for aircraft body 1. Forging

(B) Carburettor body 2. Cold heading

(C) Crankshafts 3. Aluminium-based alloy

(0) Nails 4. Pressure die casting

5. Investment casting

List I List II

(A) Welding of aluminium aIloy 1. Submerged arc welding

(B) Ship building 2. Election beam welding

(C) Joining of HSS drill bit to carbon steel shank  3. TIG welding

(0) Deep penetration precision welds 4. Gas welding

ADs. (A)-~ (B)-I, (C)-3, (D)-2

4. A thin cylinder of JOOmminternal diameter and 5mm thickness is subjected to an internal pressure of 

10MPa and a torque of 2000 Nm. Calculate the magnitudes of the principal stresses. (5)

= 0.1 m

= 0.005 m

pd

4t

lOx 106

x 0.1 = 50 x 107 N / m 2 = 50 MN / m 2

4 x 0.005

10 x 106 Pa = 107 N/m 2 

pd = 100 MN 1m 2

2t

Torque = 2000 = ס4 ס oo N

Radius 0.05

Force

1tdt

40000 = 25464 x 106 N / m2 = 2546MN 1m 2

1t x OJ x 0.005

PI ;P2 ± (PI ;P2

r+q2

10 + 50 . I ( 100 - 50 \2 (25464)2-2-:LV~2 ) +

75±25.129

= 100.129 MN/m1and 49.871 MN/m 1

5. Two bars AB and BC are connected by afriction less hinge at B. The assembly is supported and loaded 

as shown in Figure 5. Draw the shearforce and bending moment diagramsfor the combined beam AC,

clearly labelling the important values. Also indicate your sign convention. (5)