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GATE Mechanical 2023 | Question: 16
The principal stresses at a point $\mathrm{P}$ in a solid are $70 ~ \mathrm{MPa},-70 ~ \mathrm{MPa}$ and $0$. The yield stress of the material is $100 ~ \mathrm{MPa}$. Which prediction(s) about material ... material fails Maximum normal stress theory predicts that the material does not fail Maximum shear stress theory predicts that the material does not fail
The principal stresses at a point $\mathrm{P}$ in a solid are $70 ~ \mathrm{MPa},-70 ~ \mathrm{MPa}$ and $0$. The yield stress of the material is $100 ~ \mathrm{MPa}$. Wh...
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GATE Mechanical 2023 | Question: 17
Which of the plot(s) shown is/are valid Mohr's circle representations of a plane stress state in a material? (The center of each circle is indicated by $0$.) $\text{M1}$ $\text{M2}$ $\text{M3}$ $\text{M4}$
Which of the plot(s) shown is/are valid Mohr's circle representations of a plane stress state in a material? (The center of each circle is indicated by $0$.)$\text{M1}$$\...
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GATE Mechanical 2023 | Question: 18
Consider a laterally insulated rod of length $\mathrm{L}$ and constant thermal conductivity. Assuming one-dimensional heat conduction in the rod, which of the following steady-state temperature profile(s) can occur without internal heat generation?
Consider a laterally insulated rod of length $\mathrm{L}$ and constant thermal conductivity. Assuming one-dimensional heat conduction in the rod, which of the following s...
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GATE Mechanical 2023 | Question: 19
Two meshing spur gears $1$ and $2$ with diametral pitch of $8$ teeth per mm and an angular velocity ratio $\left|\omega_{2}\right| /\left|\omega_{1}\right|=1 / 4$, have their centers $30 \mathrm{~mm}$ apart. The number of teeth on the driver (gear $1$) is___________. (Answer in integer)
Two meshing spur gears $1$ and $2$ with diametral pitch of $8$ teeth per mm and an angular velocity ratio $\left|\omega_{2}\right| /\left|\omega_{1}\right|=1 / 4$, have t...
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GATE Mechanical 2023 | Question: 20
The figure shows a block of mass $m=20 \mathrm{~kg}$ attached to a pair of identical linear springs, each having a spring constant $k=1000 \mathrm{~N} / \mathrm{m}$. The block oscillates on a frictionless horizontal surface. ... the time taken by the block to complete ten oscillations is_____________seconds. (Rounded off to two decimal places) Take $\pi=3.14$
The figure shows a block of mass $m=20 \mathrm{~kg}$ attached to a pair of identical linear springs, each having a spring constant $k=1000 \mathrm{~N} / \mathrm{m}$. The ...
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GATE Mechanical 2023 | Question: 21
A vector field $ \mathbf{B}(x, y, z)=x \; \hat{\imath}+y \; \hat{\jmath}-2 z \; \hat{\mathrm{k}} $ is defined over a conical region having height $h=2$, base radius $r=3$ and axis along $z$, as shown in the figure. The ... $S$ of the cone, the value of the integral $ \int_S \mathbf{B} \cdot \boldsymbol{n} \; \mathrm{dS} $ equals____________. (Answer in integer)
A vector field$$\mathbf{B}(x, y, z)=x \; \hat{\imath}+y \; \hat{\jmath}-2 z \; \hat{\mathrm{k}}$$is defined over a conical region having height $h=2$, base radius $r=3$ a...
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GATE Mechanical 2023 | Question: 22
A linear transformation maps a point $(x, y)$ in the plane to the point $(\hat{x}, \hat{y})$ according to the rule $\hat{x}=3 y, \quad \hat{y}=2 x .$ Then, the disc $x^2+y^2 \leq 1$ gets transformed to a region with an area equal to ________. (Rounded off to two decimals) Use $\pi=3.14$
A linear transformation maps a point $(x, y)$ in the plane to the point $(\hat{x}, \hat{y})$ according to the rule$$\hat{x}=3 y, \quad \hat{y}=2 x .$$Then, the disc $x^2+...
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GATE Mechanical 2023 | Question: 23
The value of $k$ that makes the complex-valued function $ f(z)=e^{-k x}(\cos 2 y-i \sin 2 y) $ analytic, where $z=x+i y$, is __________. (Answer in integer)
The value of $k$ that makes the complex-valued function$$f(z)=e^{-k x}(\cos 2 y-i \sin 2 y)$$analytic, where $z=x+i y$, is __________. (Answer in integer)
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GATE Mechanical 2023 | Question: 24
The braking system shown in the figure uses a belt to slow down a pulley rotating in the clockwise direction by the application of a force $\mathbf{P}$. The belt wraps around the pulley over an angle $\alpha=270$ degrees. The coefficient of friction ... $T_1$ to $T_2$ in the belt is equal to_____________. (Rounded off to two decimal places) Take $\pi=3.14$
The braking system shown in the figure uses a belt to slow down a pulley rotating in the clockwise direction by the application of a force $\mathbf{P}$. The belt wraps ar...
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GATE Mechanical 2023 | Question: 25
Consider a counter-flow heat exchanger with the inlet temperatures of two fluids $(1$ and $2)$ being $T_1$, in $=300 \mathrm{~K}$ and $T_2$, in $=350 \mathrm{~K}$ ... and the effectiveness of the heat exchanger is $0.5.$ The actual heat transfer rate is _________ $\mathrm{kW}$. (Answer in integer)
Consider a counter-flow heat exchanger with the inlet temperatures of two fluids $(1$ and $2)$ being $T_1$, in $=300 \mathrm{~K}$ and $T_2$, in $=350 \mathrm{~K}$. The he...
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GATE Mechanical 2023 | Question: 26
Which one of the options given is the inverse Laplace transform of $\frac{1}{s^3-S} ?$ $u(t)$ denotes the unit-step function. $\left(-1+\frac{1}{2} e^{-t}+\frac{1}{2} e^t\right) u(t)$ $\left(\frac{1}{3} e^{-t}-e^t\right) u(t)$ ... $\left(-1-\frac{1}{2} e^{-(t-1)}-\frac{1}{2} e^{(t-1)}\right) u(t-1)$
Which one of the options given is the inverse Laplace transform of $\frac{1}{s^3-S} ?$$u(t)$ denotes the unit-step function.$\left(-1+\frac{1}{2} e^{-t}+\frac{1}{2} e^t\r...
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GATE Mechanical 2023 | Question: 27
A spherical ball weighing $2 \mathrm{~kg}$ is dropped from a height of $4.9 \mathrm{~m}$ onto an immovable rigid block as shown in the figure. If the collision is perfectly elastic, what is the momentum vector of the ball (in $\mathrm{kg} \; \mathrm{m} / \mathrm{s}$ ... $19.6 \hat{\jmath}$ $17.0 \hat{\imath}+9.8 \hat{\jmath}$ $9.8 \hat{\imath}+17.0 \hat{\jmath}$
A spherical ball weighing $2 \mathrm{~kg}$ is dropped from a height of $4.9 \mathrm{~m}$ onto an immovable rigid block as shown in the figure. If the collision is perfect...
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GATE Mechanical 2023 | Question: 28
The figure shows a wheel rolling without slipping on a horizontal plane with angular velocity $\omega_1$. A rigid bar $\text{PQ}$ is pinned to the wheel at $\text{P}$ while the end $\text{Q}$ slides on the floor. What is the angular velocity $\omega_2$ ... bar $\text{PQ}$? $\omega_2=2 \omega_1$ $\omega_2=\omega_1$ $\omega_2=0.5 \omega_1$ $\omega_2=0.25 \omega_1$
The figure shows a wheel rolling without slipping on a horizontal plane with angular velocity $\omega_1$. A rigid bar $\text{PQ}$ is pinned to the wheel at $\text{P}$ whi...
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GATE Mechanical 2023 | Question: 29
A beam of length $L$ is loaded in the $x y-$ plane by a uniformly distributed load, and by a concentrated tip load parallel to the $z$-axis, as shown in the figure. The resulting bending moment distributions about the $y$ ... Which one of the options given depicts qualitatively $\text{CORRECT}$ variations of $M_y$ and $M_z$ along the length of the beam?
A beam of length $L$ is loaded in the $x y-$ plane by a uniformly distributed load, and by a concentrated tip load parallel to the $z$-axis, as shown in the figure. The r...
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GATE Mechanical 2023 | Question: 30
The figure shows a thin-walled open-top cylindrical vessel of radius $r$ and wall thickness $t$. The vessel is held along the brim and contains a constant-density liquid to height $h$ from the base. Neglect atmospheric pressure, the weight ... magnitudes of axial wall stress $\left(\sigma_1\right)$ and circumferential wall stress $\left(\sigma_2\right)$ on $y$?
The figure shows a thin-walled open-top cylindrical vessel of radius $r$ and wall thickness $t$. The vessel is held along the brim and contains a constant-density liquid ...
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GATE Mechanical 2023 | Question: 31
Which one of the following statements is $\text{FALSE?}$ For an ideal gas, the enthalpy is independent of pressure. For a real gas going through an adiabatic reversible process, the process equation is given by $P V^\gamma=$ constant, ... is the mass of the gas. Any real gas behaves as an ideal gas at sufficiently low pressure or sufficiently high temperature.
Which one of the following statements is $\text{FALSE?}$For an ideal gas, the enthalpy is independent of pressure.For a real gas going through an adiabatic reversible pro...
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GATE Mechanical 2023 | Question: 32
Consider a fully adiabatic piston-cylinder arrangement as shown in the figure. The piston is massless and cross-sectional area of the cylinder is $A$. The fluid inside the cylinder is air (considered as a perfect gas), with $\gamma$ being the ratio of the specific heat at ... $P_1 A L_1 \ln \frac{L_1}{L_2}$ $\frac{\left(P_2 L_2-P_1 L_1\right) A}{(1-\gamma)}$
Consider a fully adiabatic piston-cylinder arrangement as shown in the figure. The piston is massless and cross-sectional area of the cylinder is $A$. The fluid inside th...
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GATE Mechanical 2023 | Question: 33
A cylindrical rod of length $h$ and diameter $d$ is placed inside a cubic enclosure of side length $L. S$ denotes the inner surface of the cube. The view-factor $F_{S-S}$ is $0$ $1$ $\frac{\left(\pi d h+\pi d^2 / 2\right)}{6 L^2}$ $1-\frac{\left(\pi d h+\pi d^2 / 2\right)}{6 L^2}$
A cylindrical rod of length $h$ and diameter $d$ is placed inside a cubic enclosure of side length $L. S$ denotes the inner surface of the cube. The view-factor $F_{S-S}$...
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GATE Mechanical 2023 | Question: 34
In an ideal orthogonal cutting experiment (see figure), the cutting speed $V$ is $1 \mathrm{~m} / \mathrm{s}$, the rake angle of the tool $\alpha=5^{\circ}$, and the shear angle, $\phi$, is known to be $45^{\circ}$. Applying the ideal orthogonal cutting model, ... at the shear zone? $1.84 \times 10^4$ $5.20 \times 10^4$ $0.71 \times 10^4$ $1.30 \times 10^4$
In an ideal orthogonal cutting experiment (see figure), the cutting speed $V$ is $1 \mathrm{~m} / \mathrm{s}$, the rake angle of the tool $\alpha=5^{\circ}$, and the shea...
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GATE Mechanical 2023 | Question: 35
A $\text{CNC}$ machine has one of its linear positioning axes as shown in the figure, consisting of a motor rotating a lead screw, which in turn moves a nut horizontally on which a table is mounted. The motor moves in discrete rotational steps ... total number of controllable locations at which the table can be positioned on this axis? $5000$ $2$ $1000$ $500$
A $\text{CNC}$ machine has one of its linear positioning axes as shown in the figure, consisting of a motor rotating a lead screw, which in turn moves a nut horizontally ...
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GATE Mechanical 2023 | Question: 36
Cylindrical bars $\mathrm{P}$ and $\mathrm{Q}$ have identical lengths and radii, but are composed of different linear elastic materials. The Young's modulus and coefficient of thermal expansion of $\mathrm{Q}$ ... $\mathrm{P}$ and $\mathrm{Q}$ moves to the right after heating $\sigma_1<\sigma_2$ $\sigma_1=\sigma_2$
Cylindrical bars $\mathrm{P}$ and $\mathrm{Q}$ have identical lengths and radii, but are composed of different linear elastic materials. The Young's modulus and coefficie...
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GATE Mechanical 2023 | Question: 37
A very large metal plate of thickness $d$ and thermal conductivity $k$ is cooled by a stream of air at temperature $T_{\infty}=300 \mathrm{~K}$ with a heat transfer coefficient $h$, as shown in the figure. The centerline temperature of the plate is $T_{\mathrm{P}}$. In ...
A very large metal plate of thickness $d$ and thermal conductivity $k$ is cooled by a stream of air at temperature $T_{\infty}=300 \mathrm{~K}$ with a heat transfer coeff...
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GATE Mechanical 2023 | Question: 38
The smallest perimeter that a rectangle with area of $4$ square units can have is __________ units. (Answer in integer)
The smallest perimeter that a rectangle with area of $4$ square units can have is __________ units. (Answer in integer)
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GATE Mechanical 2023 | Question: 39
Consider the second-order linear ordinary differential equation $ x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y=0, \quad x \geq 1 $ with the initial conditions $ y(x=1)=6,\left.\quad \frac{d y}{d x}\right|_{x=1}=2 . $ The value of $y$ at $x=2$ equals __________. (Answer in integer)
Consider the second-order linear ordinary differential equation$$x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}-y=0, \quad x \geq 1$$with the initial conditions$$y(x=1)=6,\lef...
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GATE Mechanical 2023 | Question: 40
The initial value problem $ \frac{d y}{d t}+2 y=0, \quad y(0)=1 $ is solved numerically using the forward Euler's method with a constant and positive time step of $\Delta t$. Let $y_n$ represent the numerical solution obtained after $n$ ... $\Delta t$ does not exceed _________. (Answer in integer)
The initial value problem$$\frac{d y}{d t}+2 y=0, \quad y(0)=1$$is solved numerically using the forward Euler's method with a constant and positive time step of $\Delta t...
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GATE Mechanical 2023 | Question: 41
The atomic radius of a hypothetical face-centered cubic (FCC) metal is $(\sqrt{2} / 10) \; \mathrm{nm}$. The atomic weight of the metal is $24.092 \mathrm{~g} /$ mol. Taking Avogadro's number to be $6.023 \times 10^{23} \; \mathrm{atom/mol}$, the density of the metal is _________ $\mathrm{kg} / \mathrm{m}^3$. (Answer in integer)
The atomic radius of a hypothetical face-centered cubic (FCC) metal is $(\sqrt{2} / 10) \; \mathrm{nm}$. The atomic weight of the metal is $24.092 \mathrm{~g} /$ mol. Tak...
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GATE Mechanical 2023 | Question: 42
A steel sample with $1.5 \; \mathrm{wt}. \%$ carbon (no other alloying elements present) is slowly cooled from $1100^{\circ} \mathrm{C}$ ... the total cementite content in the microstructure that develops just below the eutectoid temperature is ___________. (Rounded off to two decimal places)
A steel sample with $1.5 \; \mathrm{wt}. \%$ carbon (no other alloying elements present) is slowly cooled from $1100^{\circ} \mathrm{C}$ to just below the eutectoid tempe...
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GATE Mechanical 2023 | Question: 43
A part, produced in high volumes, is dimensioned as shown. The machining process making this part is known to be statistically in control based on sampling data. The sampling data shows that $\text{D1}$ follows a normal distribution with a mean of ... upper limit of dimension $\mathrm{C}$ should be _____________ $\mathrm{mm}$. (Rounded off to one decimal place)
A part, produced in high volumes, is dimensioned as shown. The machining process making this part is known to be statistically in control based on sampling data. The samp...
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GATE Mechanical 2023 | Question: 44
A coordinate measuring machine (CMM) is used to determine the distance between Surface $\text{SP}$ and Surface $\text{SQ}$ of an approximately cuboidal shaped part. Surface $\text{SP}$ is declared as the datum as per the engineering ... to fit the necessary planes. The distance between the two fitted planes is ____________ $\mathrm{mm}$. (Answer in integer)
A coordinate measuring machine (CMM) is used to determine the distance between Surface $\text{SP}$ and Surface $\text{SQ}$ of an approximately cuboidal shaped part. Surfa...
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GATE Mechanical 2023 | Question: 45
A solid part (see figure) of polymer material is to be fabricated by additive manufacturing $\text{(AM)}$ in square-shaped layers starting from the bottom of the part working upwards. The nozzle diameter of the $\text{(AM)}$ ... delays between layers or the serpentine scan lines. The time taken to fabricate this part is ____________ minutes. (Answer in integer)
A solid part (see figure) of polymer material is to be fabricated by additive manufacturing $\text{(AM)}$ in square-shaped layers starting from the bottom of the part wor...
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GATE Mechanical 2023 | Question: 46
An optical flat is used to measure the height difference between a reference slip gauge $\text{A}$ and a slip gauge $\text{B}$. Upon viewing via the optical flat using a monochromatic light of wavelength $0.5 \mu \mathrm{m}, 12$ fringes were ... $45 \mathrm{~mm}$ apart, the height difference of the gauges is ____________ $\mu \mathrm{m}$. (Answer in integer)
An optical flat is used to measure the height difference between a reference slip gauge $\text{A}$ and a slip gauge $\text{B}$. Upon viewing via the optical flat using a ...
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GATE Mechanical 2023 | Question: 47
Ignoring the small elastic region, the true stress $(\sigma)-\operatorname{true}$ strain $(\varepsilon)$ variation of a material beyond yielding follows the equation $\sigma=400 \varepsilon^{0.3} \mathrm{MPa}$. The engineering ultimate tensile strength value of this material is ____________ $\text{MPa}$. (Rounded off to one decimal place)
Ignoring the small elastic region, the true stress $(\sigma)-\operatorname{true}$ strain $(\varepsilon)$ variation of a material beyond yielding follows the equation $\si...
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GATE Mechanical 2023 | Question: 48
The area moment of inertia about the $y$-axis of a linearly tapered section shown in the figure is __________ $\mathrm{m}^4$. (Answer in integer)
The area moment of inertia about the $y$-axis of a linearly tapered section shown in the figure is __________ $\mathrm{m}^4$.(Answer in integer)
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GATE Mechanical 2023 | Question: 49
A cylindrical bar has a length $L=5 \mathrm{~m}$ and cross section area $S=10 \mathrm{~m}^{2}$. The bar is made of a linear elastic material with a density $\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}$ ... J. (Rounded off to two decimal places) Take the acceleration due to gravity as $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$.
A cylindrical bar has a length $L=5 \mathrm{~m}$ and cross section area $S=10 \mathrm{~m}^{2}$. The bar is made of a linear elastic material with a density $\rho=2700 \ma...
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GATE Mechanical 2023 | Question: 50
A cylindrical transmission shaft of length $1.5 \mathrm{~m}$ and diameter $100 \mathrm{~mm}$ is made of a linear elastic material with a shear modulus of $80 \mathrm{GPa}$. While operating at $500 \mathrm{rpm}$, the angle of twist across ... transmitted by the shaft at this speed is ____________ $\mathrm{kW}$. (Rounded off to two decimal places) Take $\pi=3.14$
A cylindrical transmission shaft of length $1.5 \mathrm{~m}$ and diameter $100 \mathrm{~mm}$ is made of a linear elastic material with a shear modulus of $80 \mathrm{GPa}...
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GATE Mechanical 2023 | Question: 51
Consider a mixture of two ideal gases, $\mathrm{X}$ and $\mathrm{Y}$, with molar masses $\bar{M}_{\mathrm{X}}=10 \mathrm{~kg} / \mathrm{kmol}$ and $\bar{M}_{\mathrm{Y}}=20 \mathrm{~kg} / \mathrm{kmol}$, respectively, in a container. The ... off to one decimal place) Assume that the universal gas constant is $8314 \mathrm{~J} \mathrm{kmol}^{-1} \mathrm{~K}^{-1}$.
Consider a mixture of two ideal gases, $\mathrm{X}$ and $\mathrm{Y}$, with molar masses $\bar{M}_{\mathrm{X}}=10 \mathrm{~kg} / \mathrm{kmol}$ and $\bar{M}_{\mathrm{Y}}=2...
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GATE Mechanical 2023 | Question: 52
The velocity field of a certain two-dimensional flow is given by $\boldsymbol{V}(x, y)=k(x \hat{\imath}-y \hat{\jmath})$ where $k=2 \mathrm{~s}^{-1}$. The coordinates $x$ and $y$ are in meters. Assume gravitational effects to be negligible. If ... $(2 \mathrm{~m}, 2 \mathrm{~m})$ is __________ $\mathrm{kPa}$. (Answer in integer)
The velocity field of a certain two-dimensional flow is given by$\boldsymbol{V}(x, y)=k(x \hat{\imath}-y \hat{\jmath})$where $k=2 \mathrm{~s}^{-1}$. The coordinates $x$ a...
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GATE Mechanical 2023 | Question: 53
Consider a unidirectional fluid flow with the velocity field given by $ V(x, y, z, t)=u(x, t) \hat{\imath} $ where $u(0, t)=1$. If the spatially homogeneous density field varies with time $t$ as $ \rho(t)=1+0.2 e^{-t} $ the value of $u(2,1)$ is ______________. (Rounded off to two decimal places) Assume all quantities to be dimensionless.
Consider a unidirectional fluid flow with the velocity field given by$$V(x, y, z, t)=u(x, t) \hat{\imath}$$where $u(0, t)=1$. If the spatially homogeneous density field v...
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GATE Mechanical 2023 | Question: 54
The figure shows two fluids held by a hinged gate. The atmospheric pressure is $P_a=100 \mathrm{kPa}$. The moment per unit width about the base of the hinge is ____________ $\mathrm{kNm} / \mathrm{m}$. (Rounded off to one decimal place) Take the acceleration due to gravity to be $\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2$.
The figure shows two fluids held by a hinged gate. The atmospheric pressure is $P_a=100 \mathrm{kPa}$. The moment per unit width about the base of the hinge is __________...
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GATE Mechanical 2023 | Question: 55
An explosion at time $t=0$ releases energy $\text{E}$ at the origin in a space filled with a gas of density $\rho$. Subsequently, a hemispherical blast wave propagates radially outwards as shown in the figure. Let $\text{R}$ denote the ... where $k$ is a dimensionless constant. The value of exponent $\text{a}$ is _____________. (Rounded off to one decimal place)
An explosion at time $t=0$ releases energy $\text{E}$ at the origin in a space filled with a gas of density $\rho$. Subsequently, a hemispherical blast wave propagates ra...
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