GATE 2008 Mechanical Engineering
46 questions across 1 session
A journal bearing has a shaft diameter of $$40mm$$ and a length of $$40mm.$$ The shaft is rotating at $$20$$ $$rad/s$$ and viscocity of the lubricant is $$20mPa.s.$$ The clearance...
The directional derivative of the scalar function $$f(x, y, z)$$$$ = {x^2} + 2{y^2} + z\,\,$$ at the point $$P = \left( {1,1,2} \right)$$ in the direction of the vector $$\,\overri...
A journal bearing has a shaft diameter of $$40$$ $$mm$$ and a length of $$40mm.$$ The shaft is rotating at $$20$$ $$rad/s$$ and the viscosity of the lubricant is $$20$$ $$mPa$$-$$s...
In a single pass rolling operation, a $$20mm$$ thick plate with plate width of $$100mm$$, is reduced to $$18mm$$ thick. The roller radius is $$250mm$$ and rotational speed is $$10$...
In an $$M/M/1$$ queuing system, the number of arrivals in an interval of length $$T$$ is a Poisson random variable (i.e., the probability of there being $$n$$ arrivals in an interv...
While cooling, a cubical casting of side $$40mm$$ undergoes $$3\% ,\,\,4\% $$ and $$5\% $$ volume shrinkage during the liquid state, phase transition and solid state respectively....
For a continuity equation given $$\nabla .\overrightarrow V = 0$$ to be valid, $$\overrightarrow V $$ where is the velocity vector, which one of the folllowing is a necessary condi...
Let $$\,\,f = {y^x}.$$ What is $$\,\,{{{\partial ^2}f} \over {\partial x\partial y}}\,\,$$ at $$x=2,$$ $$y=1$$?
A hollow enclosure is formed between two infinitely concentric cylinders of radii $$1m$$ and $$2m$$ respectively. Radiative heat exchange takes place between the inner surface of t...
A two dimensional fluid element rotates like a rigid body. At a point with in the element, the pressure is $$1$$ unit. Radius of the Mohr's circle, charactering the state of stress...
A spur gate has a module of $$3mm,$$ number of teeth $$16,$$ a face width of $$36mm$$ and a pressure angle of $${20^ \circ }$$. It is transmitting a power of $$3KW$$ at $$20rev/s.$...
The integral $$\oint {f(z)dz} $$ evaluated around the unit circle on the complex plane for $$f(z) = {{\cos z} \over z}$$ is
The value of $$\,\,\mathop {Lim}\limits_{x \to 8} {{{x^{1/3}} - 2} \over {x - 8}}\,\,$$ is
Consider the Linear programme $$(LP)$$ Max $$4x$$ + $$6y$$ Subject to $$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,3x + 2y \le 6 \cr & \,\,\,\,\,\,\,\,\,\,\,2x + 3y \le 6 \cr & \,\,\,\,\,\,...
In the deep drawing of cups, blanks show a tendency to wrinkle up around the periphery (flange). The most likely cause and remedy of the phenomenon are respectively,
Orthogonal turning is performed on a cylindrical work piece with shear strength of $$250Mpa.$$ The following conditions are used: cutting velocity is $$180m/min,$$ feed is $$0.2mm/...
Which of the following integrals is unbounded?
The length of the curve $$\,y = {2 \over 3}{x^{3/2}}$$ between $$x=0$$ & $$x=1$$ is
Internal gear cutting operation can be performed by
The divergence of the vector field $$\left( {x - y} \right)\widehat i + \left( {y - x} \right)\widehat j + \left( {x + y + z} \right)\widehat k$$ is
$$A$$ gas expands in a frictionless piston-cylinder arrangement. The expansion process is very slow, and is resisted by an ambient pressure of $$100kPa.$$ During the expansion proc...
A balloon containing an ideal gas is initially kept in an evacuated and insulated room. The balloon ruptures and the gas fills up the entire room. Which one of the following statem...
In a steady state steady flow process taking place in a device with a single inlet and a single outlet, the work done per unit mass flow rate is given by $$W = - \int_{inlet}^{outl...
$$2$$ moles of oxygen are mixed adiabatically with another $$2$$ moles of oxygen in mixing chamber, so that the final total pressure and temperature of the mixture become same as t...
Match the type of gears with their most appropriate description Type of gear (P)$$\,\,\,\,\,$$ Helical (Q)$$\,\,\,\,\,$$ Spiral Bevel (R)$$\,\,\,\,\,$$ Hypoid (S)$$\,\,\,\,\,$$ Rac...
In the Taylor series expansion of $${e^x}$$ about $$x=2,$$ the coefficient of $$\,\,{\left( {x - 2} \right)^4}\,\,$$ is
In arc welding of a butt joint, the welding speed is to be selected such that highest cooling rate is achieved. Melting efficiency and heat transfer efficiency are $$0.5$$ and $$0....
For generating coon's surface we require
For what values of 'a' if any will the following system of equations in $$x, y$$ and $$z$$ have a solution? $$$2x+3y=4,$$$ $$$x+y+z=4,$$$ $$$x+2y-z=a$$$
A moving average system is used for forecasting weekly demand. $${F_1}\left( t \right)$$ and $${F_2}\left( t \right)$$ are sequences of forecasts with parameters $${m_1}$$ and $${m...
Orthogonal turning is performed on a cylindrical work piece with shear strength of $$250Mpa.$$ The following conditions are used: cutting velocity is $$180m/min,$$ feed is $$0.2mm/...
The matrix $$\left[ {\matrix{ 1 & 2 & 4 \cr 3 & 0 & 6 \cr 1 & 1 & P \cr } } \right]$$ has one eigen value to $$3.$$ The sum of the other two eigen values is
For flow of fluid over a heated plate, the following fluid properties are known: Viscosity $$=0.001Pa.s;$$ Specific heat at constant pressure $$=1$$ $$kJ/kgK;$$ Thermal conductivit...
A clutch has outer and inner diameters $$100mm$$ and $$40mm$$ respectively. Assuming a uniform pressure of $$2MPa$$ and coefficient of friction of inner material $$0.4,$$ the torqu...
A planar mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mechanism, using Gruebler's criterion, is
The $$LMTD$$ of a counter flow heat exchanger is $${20^ \circ }C$$. The cold fluid enters at $${20^ \circ }C$$ and the hot fluid enters at $${100^ \circ }C$$. Mass flow rate of the...
It is given that $$y'' + 2y' + y = 0,\,\,\,\,y\left( 0 \right) = 0,y\left( 1 \right) = 0.$$ What is $$y(0.5)$$?
The eigen vectors of the matrix $$\left[ {\matrix{ 1 & 2 \cr 0 & 2 \cr } } \right]$$ are written in the form $$\left[ {\matrix{ 1 \cr a \cr } } \right]\,\,\& \,\,\left[ {\matrix{ 1...
A straight rod length $$L(t),$$ hinged at one end freely extensible at the other end, roates through an angle $$\theta \left( t \right)$$ about the hinge. At time $$t,$$ $$L(t)=1m,...
Given that $$\mathop x\limits^{ \bullet \bullet } + 3x = 0$$ and $$x\left( 0 \right) = 1,\,\,\mathop x\limits^ \bullet \left( 0 \right) = 1,$$ What is $$x(1)$$ ________.
In a single point turning tool, the side rake angle and orthogonal rake angle are equal. $$\varphi $$ is the principle cutting edge angle and its range is $$0$$ to $$90.$$ The chip...
A coin is tossed $$4$$ times. What is the probability of getting heads exactly $$3$$ times?
A compression spring is made of music wire of $$20$$ mm diameter having a shear strength and shear modulus of $$800$$ MPa and $$80$$ GPa respectively. The mean coil diameter is $$2...
A solid circular shaft of diameter 100 mm is subjected to an axial stress of 50 MPa. It further subjected to a torque of 10 KN-m. The maximum principal stress experienced on the sh...
A two dimensional fluid element rotates like a rigid body. At a point within the element, the pressure is $$1$$ unit. Radius of the Mohr's circle, characterizing the state at that...
A uniform rigid rod of mass $$m =1kg$$ and length $$L=1$$ m is hinged at its centre & laterally supported at one end by a spring of spring constant $$k=300N/m.$$ The natural freque...