GATE 2017 Mechanical Engineering

For the vector $$\overrightarrow V = 2yz\widehat i + 3xz\widehat j + 4xy\widehat k,$$ the value of $$\,\nabla .\left( {\nabla \times \overrightarrow \nabla } \right)\,\,$$ is _____...

Saturated steam at $${100^ \circ }C$$ condenses on the outside of a tube. Cold fluid enters the tube at $${20^ \circ }C$$ and exits at $${50^ \circ }C.$$ The value of the Log Mean...

Two cutting tools with tool life equations given below are being compared: $$\eqalign{ & Tool\,\,1:\,\,\,\,\,\,\,V{T^{0.1}} = 150 \cr & Tool\,\,2:\,\,\,\,\,\,\,V{T^{0.3}} = 300 \cr...

The product of eigenvalues of the matrix $$P$$ is $$P = \left[ {\matrix{ 2 & 0 & 1 \cr 4 & { - 3} & 3 \cr 0 & 2 & { - 1} \cr } } \right]$$

The differential equation $$\,{{{d^2}y} \over {d{x^2}}} + 16y = 0$$ for $$y(x)$$ with the two boundary conditions $${\left. {{{dy} \over {dx}}} \right|_{x = 0}} = 1$$ and $${\left....

Two disks A and B with identical mass (m) and radius (R) are initially at rest. They roll down from the top of identical inclined planes without slipping. Disk A has all of its mas...

For a steady flow, the velocity field is $$\overrightarrow V = \left( { - {x^2} + 3y} \right)\widehat i + \left( {2xy} \right)\widehat j.$$ The magnitude of the acceleration of the...

A $$10$$ $$mm$$ deep cylindrical cup with diameter of $$15$$ $$mm$$ is drawn from a circular blank. Neglecting the variation in the sheet thickness, the diameter (upto $$2$$ decima...

For steady flow of a viscous incompressible fluid through a circular pipe of constant diameter, the average velocity in the fully developed region is constant. Which one of the fol...

A Particle of unit mass is moving on a plane. Its trajectory, in polar coordinates, is given by $$r\left( t \right) = {t^2},\theta \left( t \right) = t,$$ where $$t$$ is time. The...

A block of length $$200$$ $$mm$$ is machined by a slab milling cutter $$34$$ $$mm$$ in diameter. The depth of cut and table feed are set at $$2$$ $$mm$$ and $$18$$ $$mm/minute,$$ r...

Consider the two-dimensional velocity field given by $$\overrightarrow V = \left( {5 + {a_1}x + {b_1}y} \right)\widehat i + \left( {4 + {a_2}x + {b_2}y} \right)\widehat j,$$ where...

Metric thread of $$0.8$$ $$mm$$ pitch is to be cut on a lathe. Pitch of the lead screw is $$1.5$$ $$mm.$$ If the spindle rotates at $$1500$$ $$rpm,$$ the speed of rotation of the l...

$$\,\,P\,\,\,\left( {0,3} \right),\,\,Q\,\,\,\left( {0.5,4} \right),\,\,$$ and $$\,\,R\,\,\,\left( {1,5} \right)\,\,\,$$ are three points on the curve defined by $$\,\,f\left( x \r...

A sprue in a sand mould has a top diameter of $$20mm$$ and height of $$200mm.$$ The velocity of the molten metal at the entry of the sprue is $$0.5m/s.$$ Assume acceleration due to...

Consider steady flow of an incompressible fluid through two long and straight pipes of diameters $${d_1}$$ and $${d_2}$$ arranged in series. Both pipes are of equal length and the...

Water (density $$ = 1000$$ $$kg/{m^3}$$) at ambient temperature flows through a horizontal pipe of uniform cross section at the rate of $$1$$ $$kg/s.$$ If the pressure drop across...

The molar specific heat at constant volume of an ideal gas is equal to $$2.5$$ times the universal gas constant ($$8.314$$ $$J/mol.K$$). When the temperature increases by $$100$$ $...

A heat pump absorbs $$10$$ $$kW$$ of heat from outside environment at $$250$$ $$K$$ while absorbing $$15$$ $$kW$$ of work. It delivers the heat to a room that must be kept warm at...

One $$kg$$ of an ideal gas (gas constant, $$R = 400$$ $$J/kg.K;$$ specific heat at constant volume, $${c_v} = 1000\,J/kg.K$$ at $$1$$ bar, and $$300$$ $$K$$ is contained in a seale...

In an arc welding process, welding speed is doubled. Assuming all other process parameters to be constant, the cross sectional area of the weld bead will

The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is

In a metal forming operation when the material has just started yielding, the principal stresses are $${\sigma _1} = \,\, + 180\,\,MPa,\,\,\,$$ $${\sigma _2} = \,\, - 100\,\,MPa,\,...

A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\,\,$$ in the range $$0 \le u \le 1$$ is rotated about the $$...

Circular arc on a part profile is being machined on a vertical $$CNC$$ milling machine. $$CNC$$ part program using metric units with absolute dimensions is listed below: $$\eqalign...

Consider the following partial differential equation for $$u(x,y)$$ with the constant $$c>1:$$ $$\,{{\partial u} \over {\partial y}} + c{{\partial u} \over {dx}} = 0\,\,$$ solution...

Two models, $$P$$ and $$Q,$$ of a product earn profits of Rs. $$100$$ and Rs. $$80$$ per piece, respectively. Production times for $$P$$ and $$Q$$ are $$5$$ hours and $$3$$ hours,...

Heat is generated uniformly in a long solid cylindrical rod ( diameter $$ = 10\,\,mm$$) at the rate of $$4 \times {10^7}\,\,W/{m^3}.$$ The thermal conductivity of the rod material...

A six-face fair dice is rolled a large number of times. The mean value of the outcomes is _________.

The velocity profile inside the boundary layer for flow over a flat plate is given as $${u \over {{U_\infty }}} = \sin \left( {{\pi \over 2}\,{y \over \delta }} \right),$$ where $$...

Cylindrical pins of diameter $${15^{ \pm 0.020}}\,\,mm$$ are being produced on a machine. Statistical quality control tests show a mean of $$14.995$$ $$mm$$ and standard deviation...

Consider a beam with circular cross-section of diameter $$d.$$ The ratio of the second moment of area about the neutral axis to the section modulus of the area is

The damping ratio for a viscously damped spring mass system, governed by the relationship $$\,m{{{d^2}x} \over {d{t^2}}} + c{{dx} \over {dt}} + kx = f\left( t \right),\,\,\,$$ is g...

A motor driving a solid circular steel shaft transmits $$40$$ $$kW$$ of power at $$500$$ $$rpm$$. If the diameter of the shaft is $$40$$ $$mm,$$ the maximum shear stress in the sha...

A rectangular region in a solid is in a state of plane strain. The $$(x,y)$$ coordinates of the corners of the undeformed rectangle are given by $$P(0,0), Q(4,0), R(4,3), S(0,3).$$...

A horizontal bar, fixed at one end (x = 0), has a length of 1 m, and cross-sectional area of 100 mm 2 . Its elastic modulus varies along its length as given by E(x) = 100 e -x GPa,...

The Poisson's ratio for a perfectly incompressible linear elastic material is

In the engineering stress-strain curve for mild steel, the Ultimate Tensile Strength $$(UTS)$$ refers to

Consider a laminar flow at zero over a flat plate . The shear stress at the wall is denoted by $${\tau _w}.$$ The axial positions $${x_1}$$ and $${x_2}$$ on the plate are measured...

A cylindrical pin $$25_{ + 0.010}^{ + 0.020}\,\,mm$$ diameter is electroplated. Plating thickness is $${2.0^{ \pm 0.005}}\,\,mm$$. Neglecting the gauge tolerance, the diameter (in...

The principal stresses at a point in a critical section of a machine component are $${\sigma _1} = 60\,\,MPa,\,\,{\sigma _2} = 5\,\,MPa$$ and $${\sigma _3} = - 40\,\,MPa.$$ For the...

Consider the matrix $$A = \left[ {\matrix{ {50} & {70} \cr {70} & {80} \cr } } \right]$$ whose eigenvectors corresponding to eigen values $${\lambda _1}$$ and $${\lambda _2}$$ are...

In an orthogonal machining with a tool of $${9^ \circ }$$ orthogonal rake angle, the uncut chip thickness is $$0.2$$ $$mm.$$ The chip thickness fluctuates between $$0.25$$ $$mm$$ a...

Maximize $$\,\,\,\,\,\,\,\,\,Z = 5{x_1} + 3{x_2}$$ Subject to $$\,\,\,\,\,\,\,\,\,\,{x_1} + 2{x_2} \le 10,$$ $$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\...

Two coins are tossed simultaneously. The probability (upto two decimal points accuracy) of getting at least one head is _________.

In a counter-flow heat exchanger, water is heated at the rate of $$1.5kg/s$$ from $$\,40{}^ \circ C\,$$ to $$\,80{}^ \circ C\,$$ by an oil entering at $$\,120{}^ \circ C\,$$ and le...

It is desired to make a product having T-shaped cross - section from a rectangular aluminium block. Which one of the following processes is expected to provide the highest strength...

The heat loss from a fin is $$6W.$$ The effectiveness of the fin are $$3$$ and $$0.75,$$ respectively. The heat loss (in $$W$$) from the fin, keeping the entire fin surface at base...

A simple of $$15$$ data is as follows; $$17,18,17,17,13,18,5,5,6,7,8,9,20,17,3.$$ The mode of the data is

The surface integral $$\int {\int\limits_s {F.ndS} } $$ over the surface $$S$$ of the sphere $${x^2} + {y^2} + {z^2} = 9,$$ where $$\,F = \left( {x + y} \right){\rm I} + \left( {x...

One $$kg$$ of an ideal gas (gas constant $$R=287$$ $$J/kg$$-$$K$$) undergoes an irreversible process from state- $${\rm I}$$ ( $$1$$ bar, $$300$$ $$K$$ ) to state-$$2$$ ( $$2$$ bar...

The volume and temperature of air (assumed to be an ideal gas) in a closed vessel is $$2.87{m^3}$$ and $$300$$ $$K,$$ respectively. The gauge pressure indicated by a manometer fitt...

A metal ball of diameter $$60mm$$ is initially at $${220^ \circ }C.$$ The ball is suddenly cooled by an air jet of $${20^ \circ }C.$$ The heat transfer coefficient is $$200\,W/{m^2...

In the Rankine cycle for a system power plant the turbine entry and exit enthalpies are $$2803$$ $$kJ/kg$$ and $$1800$$ $$kJ/kg$$, respectively. The enthalpies of water at pump ent...

A mass $$m$$ of a perfect gas at pressure $${p_1}$$ and volume $${V_1}$$ undergoes an isothermal process. The final pressure is $${p_2}$$ and volume is $${V_2}.$$ The work done on...

A calorically perfect gas (specific heat at constant pressure $$1000$$$$J/kg.K$$) enters and leaves a gas turbine with the same velocity. The temperatures of the gas at turbine ent...

For the stability of a floating body the

During the turning of a $$20$$ $$mm-$$diameter steel bar at a spindle speed of $$400$$ $$rpm,$$ a tool life of $$20$$ minute is obtained. When the same bar is turned at $$200$$ $$r...

A rod of length $$20$$ $$mm$$ is stretched to make a rod of length $$40$$ $$mm.$$ Subsequently, it is compressed to make a rod of final length $$10$$ $$mm.$$ Consider the longitudi...

A single – plate clutch has a friction disc with inner and outer radii of $$20mm$$ and $$40$$ $$mm,$$ respectively. The friction lining in the disc is made in such a way that the c...

A strip of $$120mm$$ width and $$8mm$$ thickness is rolled between two $$300$$ $$mm$$-diameter rolls to get a strip of $$120mm$$ width and $$7.2mm$$ thickness. The speed of the str...

Which one of the following statement is TRUE?

A machine component made of a ductile material is subjected to a variable loading with $${\sigma _{\min }} = - 50MPa$$ and $${\sigma _{\max }} = 50MPa.$$ If the corrected endurance...

If $$f\left( z \right) = \left( {{x^2} + a{y^2}} \right) + ibxy$$ is a complex analytic function of $$z=x+iy,$$ where $${\rm I} = \sqrt { - 1} ,$$ then

For a single server with Poisson arrival and exponential service time, the arrival rate is $$12$$ per hour. Which one of the following service rates will provide a steady state fin...

The standard deviation of linear dimensions $$P$$ and $$Q$$ are $$3\mu m$$ and $$4\mu m$$, respectively. When assembled, the standard deviation (in $$\mu m$$) of the following line...

The determinant of a $$2 \times 2$$ matrix is $$50.$$ If one eigenvalue of the matrix is $$10,$$ the other eigenvalue is __________.

Consider the differential equation $$\,\,3y''\left( x \right) + 27y\left( x \right) = 0\,\,$$ with initial conditions $$y\left( 0 \right) = 0$$ and $$y'\left( 0 \right) = 2000.\,\,...

The emissive power of a blackbody is $$P.$$ If its absolute temperature is doubled, the emissive power becomes

The radius of gyration of a compound pendulum about the point of suspension is $$100$$ mm. The distance between the point of suspension and the centre of mass is $$250$$ mm. Consid...

A cantilever beam of length $$L$$ and flexural modulus $$EI$$ is subjected to a point load $$P$$ at the free end. The elastic strain energy stored in the beam due to bending (negle...

A helical compression spring made of wire of circular cross-section is subjected to a compressive load. The maximum shear stress induced in the cross-section of the wire is 24 MPa....

For a loaded cantilever beam of uniform cross-section, the bending moment (in N-mm) along the length is M(x) = 5x 2 +10x, where x is the distance (in mm) measured from the free end...

The state of stress at a point is $${\sigma _x} = {\sigma _y} = {\sigma _z} = {\tau _{xz}} = {\tau _{zx}} = {\tau _{yz}} = {\tau _{zy}} = 0$$ and $${\tau _{xy}} = {\tau _{yx}} = 50...