GATE Overflow for GATE XE - Questions without answers in Solid Mechanics

GATE XE 2026 | Question: 52

Tue, 24 Feb 2026 15:45:43 +0000

At the peak (also denoted as UTS) of an engineering stress vs. engineering strain curve for ductile metal, the engineering strain is $0.2$. The corresponding true stress $(\sigma)$ vs. true strain ( $\epsilon$ ) relationship follows the equation: $\sigma=K \epsilon^{n}$, where $K$ and $n$ are constants.

The engineering stress at the peak in $\mathrm{MPa}$ is $\_\_\_\_$ (rounded off to one decimal place).

Given: $K=200 \mathrm{MPa}$

GATE XE 2026 | Question: 56

Tue, 24 Feb 2026 15:45:38 +0000

Two axial members, namely $\text{OP}$ and $\text{OQ}$, are pin joined at $\text{O}$ as shown in the figure. A force $F$ acts at point $\text{P}$ along the positive $x$ direction and a force $\sqrt{3} F$ acts at point $\text{Q}$ along the positive $y$ direction.

The resultant of the applied forces makes an angle $\theta$ (anticlockwise from the positive $x$ -axis).

The value of $\theta$ is $\_\_\_\_$.

$30^{\circ}$
$45^{\circ}$
$60^{\circ}$
$75^{\circ}$

GATE XE 2026 | Question: 57

Tue, 24 Feb 2026 15:45:34 +0000

Two smooth drums each weighing $W$ and radius, $r$ are connected by a stiff rope of length, $h$ as shown in the figure. Force $F$ is applied using a massless lever (RS) of length, $l$. The friction between the drums and the lever (RS) is negligible. The system is in static equilibrium.

Which one of the following represents the CORRECT free body diagram of the lever (RS)?

Figures are not to scale. $N_{1}, N_{2}$, and $N_{3}$ in the options are reaction forces and $F_{\mathrm{f}}$ is the frictional force.

GATE XE 2026 | Question: 59

Tue, 24 Feb 2026 15:45:12 +0000

A rigid block of mass $m$ on a horizontal surface is connected to three springs each having spring constant $k$ as shown in the figure.

Which one of the following is the CORRECT natural frequency of the system?

$\sqrt{\frac{3 k}{4 m}}$
$\sqrt{\frac{2 k}{3 m}}$
$\sqrt{\frac{3 k}{2 m}}$
$\sqrt{\frac{3 k}{m}}$

GATE XE 2026 | Question: 60

Tue, 24 Feb 2026 15:44:55 +0000

Which among the following options is/are CORRECT unit(s) of stress?

$\mathrm{Nm}^{2}$
$\text{Pa}$
$\text{Nm}$
$\mathrm{N} / \mathrm{m}^{2}$

GATE XE 2026 | Question: 61

Tue, 24 Feb 2026 15:44:35 +0000

A car is moving on a horizontal surface in a straight line with a constant velocity of $3 \mathrm{~m} / \mathrm{s}$. A ball is thrown vertically upwards at time $t=0$ from the top of the moving car with a velocity of $20 \mathrm{~m} / \mathrm{s}$. The acceleration due to gravity is $10 \mathrm{~m} / \mathrm{s}^{2}$.

At what value(s) of time $t$ in second(s), the ball is at a height of $15$ $\mathrm{m}$ from the top of the moving car?

GATE XE 2026 | Question: 62

Tue, 24 Feb 2026 15:44:30 +0000

A cart of weight $(W) 5 \mathrm{kN}$, stands on an inclined smooth surface at an angle $(\theta)$ of $45^{\circ}$ as shown in the figure. The cart is maintained in equilibrium by a horizontal force $F$ acting at point $\text{S}$.

The value of $F$ is $\_\_\_\_$ kN (in integer).

GATE XE 2026 | Question: 63

Tue, 24 Feb 2026 15:44:28 +0000

A $\text{2D}$ state of stress at a point in a body is given by $\sigma_{x x}=-40 \mathrm{~MPa}$, $\sigma_{y y}=100 \mathrm{~MPa}$ and $\tau_{x y}=50 \mathrm{~MPa}$.

The radius of the Mohr's circle for the given state of stress is $\_\_\_\_$ $\mathrm{MPa}$ (rounded off to two decimal places).

GATE XE 2026 | Question: 64

Tue, 24 Feb 2026 15:44:26 +0000

Sum of the principal stresses of a $\text{2D}$ stress state $\left[\begin{array}{cc}11 & 4 \\ 4 & 5\end{array}\right]$ is $\_\_\_\_$ (in integer).

GATE XE 2026 | Question: 65

Tue, 24 Feb 2026 15:44:25 +0000

The force $(F)$ acting on the block, as shown in the figure is $2\text{N}$. The stiffness of the spring ( $k$ ) is $100 \mathrm{~N} / \mathrm{m}$.

Assume that the block is in static equilibrium and its mass is negligible.

If the static deflection of the spring $(\Delta)$ is $0.01$ m, which one of the following is the corresponding angle $(\theta)$ in radians?

$\dfrac{\pi}{3}$
$\dfrac{\pi}{4}$
$\dfrac{\pi}{5}$
$\dfrac{\pi}{6}$

GATE XE 2026 | Question: 66

Tue, 24 Feb 2026 15:44:21 +0000

Which one of the following options is the CORRECT absolute value of the bending moment at $\text{R}$ in the frame as shown in the figure?

$0$
$\mathrm{Fl}$
$2 \text{Fl}$
$3 \text{Fl}$

GATE XE 2026 | Question: 67

Tue, 24 Feb 2026 15:43:44 +0000

One column with square cross section of side $r$ and another column with rectangular cross section of breadth $p$ and width $q(<p)$ are made from the same material. Both the columns have one end fixed, and the other end is free. They are subjected to axial loads along the centroidal axis.

Consider the area of cross sections of both the columns to be the same. The minimum critical Euler buckling loads of the columns with rectangular and square cross-sections are $F_{\text {rect }}$ and $F_{\mathrm{sq}}$, respectively.

Then $\dfrac{F_{\text {rect }}}{F_{\text {sq }}}$ is $\_\_\_\_$ .

$\dfrac{p}{q}$
$\dfrac{q}{p}$
$\dfrac{r}{q}$
$\dfrac{r^{2}}{p q}$

GATE XE 2026 | Question: 68

Tue, 24 Feb 2026 15:43:40 +0000

Two prismatic rods of identical lengths are designed for the same strain energy density when subjected to the same axial load. One of the rods is made of steel (Young's modulus $=210 \mathrm{GPa}$ ) and another is made of aluminum (Young's modulus $=70 \mathrm{GPa}$).

If the diameter of the aluminum rod is $70$ $\mathrm{mm}$, then which one of the following options corresponds to the diameter of the steel rod in mm?

$53.19$
$70$
$210$
$29.13$

GATE XE 2026 | Question: 69

Tue, 24 Feb 2026 15:43:19 +0000

A spring of stiffness $k$ is connected to the center of a homogeneous right circular cylinder of radius $r$ placed on a horizontal surface as shown in the figure. The cylinder is assumed to be rolling without slipping.

Considering small amplitude of oscillation, which one of the following options is the CORRECT natural frequency of the system?

$\sqrt{\dfrac{k}{2 m}}$
$\sqrt{\dfrac{3 k}{2 m}}$
$\sqrt{\dfrac{k}{m}}$
$\sqrt{\dfrac{2 k}{3 m}}$

GATE XE 2026 | Question: 70

Tue, 24 Feb 2026 15:43:02 +0000

A $\text{2D}$ stress state of pure shear is shown in the figure.

Which one of the following options is equivalent to the given stress state?

Figures are not to scale.

GATE XE 2026 | Question: 71

Tue, 24 Feb 2026 15:42:58 +0000

The point $\text{Q}$ of a thin rigid equilateral triangular plate $\text{PQR}$ is constrained to move in a horizontal channel. Point $\text{P}$ of the same plate is constrained to move in the vertical channel as shown in the figure. The length of the side $\text{PQ}$ is $4$ $\mathrm{m}$.

At the instant when the angle $\theta=\dfrac{\pi}{3} \mathrm{rad}$ and the velocity of point $\text{Q}$ in the positive $x$ direction is $20 \mathrm{~m} / \mathrm{s}$, which one of the following options is the magnitude of the angular velocity vector of the line $\text{SR}$ on the plate in $\mathrm{rad/s}$?

$5$
$10$
$20$
$25$

GATE XE 2026 | Question: 73

Tue, 24 Feb 2026 15:42:39 +0000

The bending moment diagram for a simply supported beam is piecewise linear as shown in the figure. The bending moment $M(\mathrm{x})$ at $\mathrm{x}=0.5 \mathrm{~m}$ is $5$ $\mathrm{Nm}$. The beam has a rectangular cross-section with an area of $1 \mathrm{~m}^{2}$.

The absolute value of the maximum shear stress on the cross-section at $x=0.75 \mathrm{~m}$ is $\_\_\_\_$ $\mathrm{N} / \mathrm{m}^{2}$ (in integer).

GATE XE 2026 | Question: 74

Tue, 24 Feb 2026 15:42:26 +0000

A thin walled cylindrical container of diameter $2$ $\mathrm{m}$ and wall thickness of $2.5$ $\mathrm{cm}$ is made of steel whose Young's modulus is $200$ $\mathrm{GPa}$ and yield stress is $450$ $\mathrm{MPa}$.

Using von Mises criteria, the maximum permissible pressure is $\_\_\_\_$ $\mathrm{MPa}$ (rounded off to the nearest integer).

GATE XE 2026 | Question: 75

Tue, 24 Feb 2026 15:42:25 +0000

A solid axial bar made of steel with Young's modulus, $200$ $\mathrm{GPa}$ and Poisson's ratio, $0.3$, is subjected to uniaxial stress of $50$ $\mathrm{MPa}$.

The absolute value of the maximum shear strain on the outer surface of the bar is $\_\_\_\_$ $\times 10^{-4}$ (rounded off to two decimal places).

GATE XE 2026 | Question: 76

Tue, 24 Feb 2026 15:42:24 +0000

$\text{EFGH}$ (solid lines) is the initial configuration and $\mathrm{E}^{\prime} \mathrm{F}^{\prime} \mathrm{G}^{\prime} \mathrm{H}^{\prime}$ (dashed lines) is the deformed configuration of an object as shown in the figure. $\mathrm{E}^{\prime}$ coincides with $\text{E}$ and $\mathrm{F}^{\prime}$ coincides with $\text{F}$.

The average normal strain along the line segment $\text{OP}$ is $\_\_\_\_$ (rounded off to three decimal places).

Figure is not to scale.
All units are in $\mathrm{mm}$.

GATE XE 2026 | Question: 77

Tue, 24 Feb 2026 15:42:21 +0000

A simply supported beam of length $l=2 \mathrm{~m}$ is subjected to a concentrated moment $M=150 \mathrm{~kNm}$ at a distance $l / 2$ from the left end as shown in the figure. The elastic-strain energy $U$ of the beam is given by the following expression:

$$U=\frac{M^{2} l}{48 E I}$$

The section modulus of the beam is $E I=25 \times 10^{5} \mathrm{Nm}^{2}$.

The absolute value of the slope of the beam at a distance $l / 2$ from the left end is $\_\_\_\_$ (rounded off to three decimal places).

GATE XE 2026 | Question: 118

Tue, 24 Feb 2026 15:32:58 +0000

A rectangular plastic specimen with a cross-sectional area of $240 \mathrm{~mm}^{2}$ is subjected to tension with a force of $15000$ $\mathrm{N}$ resulting in elastic deformation. If the Young's modulus of the specimen is $34.55$ $\mathrm{MPa}$, the value of strain is $\_\_\_\_\_\_$ (rounded off to two decimal places).

GATE XE 2025 | Question: 66

Sun, 04 May 2025 19:06:53 +0000

Consider a spring-mass system with mass $m$ and spring stiffness $k$ as shown in the illustration. At time $t=0$, the mass is displaced by $P$ units and the velocity of the mass is zero. The displacement of the mass, $x(t)$, is measured from the equilibrium position.

Which one of the following functions represent $x(t)$?

$P \cos \left(\sqrt{\frac{k}{m}} t\right)+P \sin \left(\sqrt{\frac{k}{m}} t\right)$
$P \sin \left(\sqrt{\frac{k}{m}} t\right)$
$\frac{P}{2} \cos \left(\sqrt{\frac{k}{m}} t\right)+\frac{P}{2} \sin \left(\sqrt{\frac{k}{m}} t\right)$
$P \cos \left(\sqrt{\frac{k}{m}} t\right)$

GATE XE 2025 | Question: 67

Sun, 04 May 2025 19:06:51 +0000

A ball of mass $5 \mathrm{m}$ approaches a stationary ball of mass $\mathrm{m}$ with a horizontal velocity of $2 \mathrm{~m} / \mathrm{s}$ from left to right. After a perfectly elastic central collision, the horizontal velocity of the heavier ball is $1 \mathrm{~m} / \mathrm{s}$ from left to right.

Which one of the following statements, regarding the velocity (in $\mathrm{m} / \mathrm{s}$ ) of the lighter ball after impact, is TRUE?

Comes to rest
Moves from left to right at $5 \mathrm{~m} / \mathrm{s}$
Moves from right to left at $5 \mathrm{~m} / \mathrm{s}$
Moves from left to right at $1 \mathrm{~m} / \mathrm{s}$

GATE XE 2025 | Question: 68

Sun, 04 May 2025 19:06:48 +0000

The Mohr's circle corresponding to an infinitesimal element is shown in the figure. The plane $\text{PQ}$ in the infinitesimal element, at an angle of $\theta$ from the $x$-axis, is in a state of pure shear.

Which one of the following values of $\theta$ (in degrees) is CORRECT?

$90$
$60$
$45$
$120$

GATE XE 2025 | Question: 69

Sun, 04 May 2025 19:06:46 +0000

The two-dimensional state of stress, in an infinitesimal element, is given by

$\sigma_{x x}=800 \mathrm{MPa}, \sigma_{x y}=300 \mathrm{MPa}$ and $\sigma_{y y}=0 \mathrm{MPa}$.

Which one of the following options is the maximum shear stress (in $\mathrm{MPa}$) in the element?

$500$
$400$
$800$
$300$

GATE XE 2025 | Question: 70

Sun, 04 May 2025 19:06:44 +0000

Two cars $\text{P}$ and $\text{Q}$ are travelling on a straight path and are $60 \: \mathrm{m}$ apart as shown in the figure; Car $\text{P}$ is moving with a constant velocity of $36 \: \mathrm{kmph}$ , while car $\text{Q}$ is moving at a constant velocity of $18\: \mathrm{kmph}$. At this instant, the driver in car $\text{P}$ applies the brake and collision occurs with car $\text{Q}$ after $30$ seconds.

Assuming uniform deceleration due to braking, which one of the following is the $\textsf{CORRECT}$ velocity (in $\mathrm{m} / \mathrm{s}$ ) of the car $\text{P}$ just before the collision?

$1$
$16$
$5$
$4$

GATE XE 2025 | Question: 71

Sun, 04 May 2025 19:06:41 +0000

The natural frequency of a spring-mass system is $10 \: \mathrm{rad} / \mathrm{s}$.

Which of the following statements is/are $\textsf{CORRECT}$?

The mass is $100 \:\mathrm{kg}$ and the stiffness is $1 \mathrm{~N} / \mathrm{m}$.
The mass is $1.25 \: \mathrm{kg}$ and the stiffness is $125 \mathrm{~N} / \mathrm{m}$.
The stiffness is $620 \mathrm{~N} / \mathrm{m}$ and the mass is $6.2 \:\mathrm{kg}$.
The stiffness is $62 \mathrm{~N} / \mathrm{m}$ and the mass is $620 \mathrm{kg}$ .

GATE XE 2025 | Question: 72

Sun, 04 May 2025 19:06:39 +0000

Consider a beam with a square box cross-section as shown in the figure. The outer square has a length of $10 \: \mathrm{mm}$. The thickness of the section is $1 \: \mathrm{mm}$.

The area moment of inertia about the $x$-axis is $\_\_\_\_\_\_ \: \mathrm{mm}^{4}$ (in integer).

GATE XE 2025 | Question: 73

Sun, 04 May 2025 19:06:38 +0000

For a certain linear elastic isotropic material, the Young's modulus is $140 \: \mathrm{GPa}$ and the shear modulus is $50 \: \mathrm{GPa}$.

The Poisson's ratio for the material is $\_\_\_\_\_\_$ (rounded off up to two decimal places).

GATE XE 2025 | Question: 75

Sun, 04 May 2025 19:06:35 +0000

Consider two blocks, $\mathbf{P}$ of mass $100 \: \mathrm{kg}$ and $\mathbf{Q}$ of mass $150 \: \mathrm{ kg}$, resting as shown in the figure. The angle $\theta=30^{\circ}$. The coefficient of friction between the two blocks is $0.2$. Assume no friction exists at all other interfaces. The minimum force required to move the block $\mathbf{P}$ upward is $\mathrm{W}$.

Which one of the following options is closest to the $\textsf{CORRECT}$ magnitude of $\mathrm{W}$ (in $\text{N}$)?

$862.2$
$1116.6$
$2900.0$
$406.2$

GATE XE 2025 | Question: 76

Sun, 04 May 2025 19:06:33 +0000

Which one of the following vertical columns, of circular cross-section, sustains the highest load without buckling?

Cantilever column with a length $\text{L}$ and cross-section diameter $\text{d}$
Column with hinge at one end and roller at the other end with a length $2 \:\text{L}$ and cross-section diameter $\text{d}$.
Cantilever column with a length $\text{L}$ and cross-section diameter $2\:\text{d}$
Column with hinge at one end and roller at the other end with a length $\text{L}$ and cross-section diameter $\text{d}$.

GATE XE 2025 | Question: 77

Sun, 04 May 2025 19:06:30 +0000

The figure shows a rod $\text{PQ}$, hinged at $\text{P}$, rotating counter-clockwise with a uniform angular speed of $15 \: \mathrm{rad} / \mathrm{s}$. A block $R$ translates along a slot cut out in rod $\text{PQ}$. At the instant shown the distance $\text{PR}=0.5 \mathrm{~m}$ and $\theta=60^{\circ}$. The relative velocity of $\text{R}$ with respect to the rod $\text{PQ}$ is $5 \mathrm{~m} / \mathrm{s}$ at the instant shown. The relative acceleration of $\text{R}$ with respect to the rod $\text{PQ}$ is zero at the instant shown.

Which one of the following is the $\textsf{CORRECT}$ magnitude of the absolute acceleration (in $\mathrm{m} / \mathrm{s}^{2}$) of block $\text{R}$?

$135.2$
$187.5$
$112.5$
$150.0$

GATE XE 2025 | Question: 78

Sun, 04 May 2025 19:06:28 +0000

The frame shown in the figure is loaded at $\text{S}$ with a force of $2000 \: \text{N}$. The reactions at $\text{T}$ are denoted by $\text{T}_{x}$ and $\text{T}_{y}$, while the reaction at $\text{W}$ is $\text{W}_{y}$. Neglect the weight of the members.

Which one of the following options for the magnitudes of the forces (in $N$), $\mathrm{T}_{\mathrm{x}}, \mathrm{T}_{\mathrm{y}}$ and $W_{y}$, is $\textsf{CORRECT}$?

$\mathrm{T}_{\mathrm{x}}=0, \mathrm{~T}_{\mathrm{y}}=1000$ and $\mathrm{W}_{\mathrm{y}}=1000$
$\mathrm{T}_{\mathrm{x}}=0, \mathrm{~T}_{\mathrm{y}}=1500$ and $\mathrm{W}_{\mathrm{y}}=500$
$\mathrm{T}_{\mathrm{x}}=0, \mathrm{~T}_{\mathrm{y}}=800$ and $\mathrm{W}_{\mathrm{y}}=1200$
$\mathrm{T}_{\mathrm{x}}=0, \mathrm{~T}_{\mathrm{y}}=500$ and $\mathrm{W}_{\mathrm{y}}=1500$

GATE XE 2025 | Question: 79

Sun, 04 May 2025 19:06:25 +0000

A closed thin cylindrical tank with a mean diameter $\mathrm{d}=300 \mathrm{~mm}$ and thickness $\mathrm{t}=2 \mathrm{~mm}$, is subjected to a uniform internal gas pressure $p$. The allowable shear stress on the curved wall of the tank is $70 \: \mathrm{MPa}$.

Based on the Tresca criteria, which one of the following options for the maximum safe value of $p$ (in $\mathrm{MPa}$) is CORRECT?

$3.73$
$7.46$
$1.87$
$5.60$

GATE XE 2025 | Question: 80

Sun, 04 May 2025 19:06:22 +0000

An infinitesimal square element $\text{PQRS}$ is shown in the figure. The $x$ and $y$ axes are also marked in the figure. The strains on the element are given by

$\varepsilon_{x x}=500 \times 10^{-6}, \: \varepsilon_{y y}=100 \times 10^{-6} \text { and } \varepsilon_{x y}=0$

Which of the following statements is/are $\textsf{CORRECT}$?

Percentage change in length of the diagonal $\text{PR}$ is $0.03$.
Change in angle between $\text{PR}$ and $\text{QS}$ is $4 \times 10^{-4} \: \mathrm{rad}$.
Change in angle between $\text{PR}$ and $\text{QS}$ is $2 \times 10^{-4} \: \mathrm{rad}$.
Percentage change in length of the diagonal $\text{QS}$ is $0.03$.

GATE XE 2025 | Question: 81

Sun, 04 May 2025 19:06:19 +0000

The figure shows the stress distribution across an internal surface of a rectangular beam of height $30 \: \mathrm{mm}$ and depth $10 \: \mathrm{mm}$. The normal stress distribution is given by the expression $\sigma_{x x}=200 y+500 \mathrm{~N} / \mathrm{mm}^{2} ; y$ is the distance in $\mathrm{mm}$ from the centroidal axis of the beam. Assume that there is no variation in the stress distribution along the $z$ -direction.

Which of the following statements is/are $\textsf{CORRECT}$?

The net force in the $x$ direction is $150 \: \mathrm{kN}$.
The net force in the $x$ direction is $75 \: \mathrm{kN}$.
The net moment about the $z$ axis is $4500\: \mathrm{Nm}$.
The net moment about the $z$ axis is $2250 \: \mathrm{Nm}$.

GATE XE 2025 | Question: 82

Sun, 04 May 2025 19:06:16 +0000

A vertical column fixed at one end is subjected to a compressive axial load at the free end. The column's section modulus, $\text{EI}$, is $9.82 \times 10^{5} \: \mathrm{Nm}^{2}$ and the cross-section area is $7.85 \times 10^{-3} \mathrm{~m}^{2}$. The length of the column is $2 \: \mathrm{m}$. The yield stress of the material is $145 \: \mathrm{MPa}$.

If the column can fail either in buckling or by Tresca's criterion, the maximum load that the structure can safely sustain is $\_\_\_\_\_\_ \: \mathrm{kN}$ (rounded off to one decimal place).

GATE XE 2025 | Question: 83

Sun, 04 May 2025 19:06:15 +0000

A simply-supported beam, with a point load $\text{P}=150 \: \mathrm{kN}$ at a distance of $\text{L} / 3$ from the left end, is shown in the figure. The elastic-strain energy $\text{(U)}$ of the beam is given by the following expression:

$$\mathrm{U}=\frac{2}{243} \frac{\mathrm{P}^{2} \mathrm{~L}^{3}}{\mathrm{EI}}$$

where the section modulus, $\mathrm{EI}$, is $16.66 \times 10^{5} \: \mathrm{Nm}^{2}$ and the length of the beam $\mathrm{L}$ is $1 \: \mathrm{m}$.

The deflection at the loading point is $\_\_\_\_\_\_\_$ mm (rounded off to two decimal places).

GATE XE 2025 | Question: 84

Sun, 04 May 2025 19:06:13 +0000

A simply-supported beam has a circular cross-section with a diameter of $20 \: \mathrm{mm}$ , area of $314.2 \mathrm{~mm}^{2}$, area moment of inertia of $7854 \mathrm{~mm}^{4}$ and a length $\text{L}$ of $4 \: \mathrm{m}$ . A point load $\mathrm{P}=100 \mathrm{~N}$ acts at the center and an axial load $\mathrm{Q}=20 \mathrm{kN}$ acts through the centroidal axis as shown in the figure.

The magnitude of the offset between the neutral axis and the centroidal axis, at $\mathrm{L} / 2$ from the left, is $\_\_\_\_\_\_$ mm (rounded off to one decimal place).

GATE XE 2025 | Question: 85

Sun, 04 May 2025 19:06:12 +0000

A massless cantilever beam, with a tip mass m of $10 \: \mathrm{kg}$, is modeled as an equivalent spring-mass system as shown in the figure. The beam is of length $\mathrm{L}=1 \mathrm{~m}$, with a circular cross-section of diameter $\mathrm{d}=20 \mathrm{~mm}$. The Young's modulus of the beam material is $200 \: \mathrm{GPa}$.

The natural frequency of the spring-mass system is $\_\_\_\_\_\_\_ \: \mathrm{Hz}$ (rounded off to two decimal places).

GATE XE 2025 | Question: 86

Sun, 04 May 2025 19:06:10 +0000

An electric motor's rotor is spinning at $1500 \: \mathrm{rpm}$ when its load and power are cutoff. The rotor which has a mass of $50 \: \mathrm{ kg}$ and a radius of gyration of $200 \: \mathrm{mm}$, then coasts down to rest. Due to kinetic friction, a constant torque of $10 \: \mathrm{Nm}$ acts on the rotor as it coasts down.

The number of revolutions executed by the rotor before it comes to rest is $\_\_\_\_\_\_$ (in integer).

GATE XE 2025 | Question: 87

Sun, 04 May 2025 19:06:09 +0000

A bar of length $\text{L}=1 \mathrm{~m}$ is fixed at one end. Before heating its free end has a gap of $\delta=0.1 \mathrm{~mm}$ from a rigid wall as shown in the figure. Now the bar is heated resulting in a uniform temperature rise of $10^{\circ} \mathrm{C}$. The coefficient of linear thermal expansion of the material is $20 \times 10^{-6} /{ }^{\circ} \mathrm{C}$ and the Young's modulus of elasticity is $100 \: \mathrm{GPa}$. Assume that the material properties do not change with temperature.

The magnitude of the resulting axial stress on the bar is $\_\_\_\_\_\_ \: \mathrm{ MPa}$ (in integer).

GATE XE 2025 | Question: 122

Sun, 04 May 2025 19:04:54 +0000

During material testing, stress is applied from time $t_{i}$ to $t_{f}$ as shown below:

The corresponding strain responses for three different materials are shown in plots $\mathbf{P}, \mathbf{Q}$ and $\mathbf{R}$.

Choose the option(s) where the strain response is correctly mapped to its material class.

$\text{P}$- purely elastic; $\text{Q}$- purely viscous; $\text{R}$- viscoelastic
$\text{P}$- purely elastic; $\text{Q}$- viscoelastic; $\text{R}$- purely viscous
$\text{P}$- purely viscous; $\text{Q}$- purely elastic; $\text{R}$- viscoelastic
$\text{P}$- purely viscous; $\text{Q}$- viscoelastic; $\text{R}$- purely elastic

GATE XE 2024 | Question: 35

Sun, 21 Jul 2024 16:42:18 +0000

Mechanical behaviour of a crystalline ceramic material is best described as

ductile
brittle
viscoelastic
Viscoplastic

GATE XE 2024 | Question: 56

Sun, 21 Jul 2024 16:42:01 +0000

The engineering stress $(\sigma)$ vs. engineering strain $(\varepsilon)$ curve obtained by conducting uniaxial tension test on a steel specimen is shown in the figure $\text{(the sketched curve is not to the scale)}$. The specimen exhibits cup-and-cone failure within its gage length. Which point on the curve corresponds to the beginning of necking in the test specimen?

GATE XE 2024 | Question: 57

Sun, 21 Jul 2024 16:42:00 +0000

An $\text{L}$-shaped rigid member is fixed at the midpoint of a simply-supported beam, as shown in figure $\text{(i)}$. The member is subjected to a vertically downward force $P$ at its free end. In an equivalent system, the member along with the applied load is replaced with a force $Q=P$ and a moment $M$ $\text{(see figure (ii))}$. Which of the following statements is correct?
$\text{(Neglect the mass of the beam and of the rigid member)}$

$M=P a$
$M=P b$
$M=P(a+b)$
$M=0$

GATE XE 2024 | Question: 58

Sun, 21 Jul 2024 16:41:59 +0000

A block of weight $W$, placed on a surface, is subjected to a horizontal force $P$ as shown in the figure. The line of action of force $P$ passes through the center-ofgravity of the block. The magnitude of $P$ is such that the block remains at rest. If $N$ is the resultant normal reaction exerted by the surface, and $F$ is the frictional force acting on the bottom surface of the block, then which of the following represents the correct free body diagram of the block?

GATE XE 2024 | Question: 59

Sun, 21 Jul 2024 16:41:58 +0000

A mass $M$ is hung from a frictionless, massless pulley. The pulley is suspended by using an inextensible, massless rope of which one end is directly fixed to a support, and the other end is connected to the support through a linear spring of stiffness constant $k$ $\text{(see figure)}$. The natural frequency of this system is

$\sqrt{\frac{4 k}{M}}$
$\sqrt{\frac{2 k}{M}}$
$\sqrt{\frac{k}{M}}$
$\sqrt{\frac{k}{2M}}$

GATE XE 2024 | Question: 60

Sun, 21 Jul 2024 16:41:57 +0000

A simply-supported beam of rectangular cross-section $\text{(width $w$ and height $h$)}$ is subjected to the loads as shown in the figure $\text{(i)}$. The enlarged view of the beam cross-section is shown in figure $\text{(ii)}$. The coordinate system is indicated in the figures. Assuming Euler-Bernoulli beam approximation, the shear stress $\tau_{x z}$ and normal stress $\sigma_{x x}$ at the origin, $\text{O}$ are respectively given by

$\frac{3 P}{2 w h}, \frac{3 P L}{2 w h^{2}}$
$0, \frac{3 P L}{2 w h^{2}}$
$\frac{3 P}{2 w h}, 0$
$0,0$