GATE Overflow for GATE XE - Questions without answers in Mechanics of rigid bodies

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: 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: 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: 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: 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 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: 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: 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: 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: 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 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: 61

Sun, 21 Jul 2024 16:41:56 +0000

An ice-skater starts spinning during her performance. As she retracts her arms and legs closer to her body, her angular velocity __________.

increases
decreases
remains the same
goes to zero

GATE XE 2024 | Question: 64

Sun, 21 Jul 2024 16:41:54 +0000

A particle of mass $1$ kg is attached to one end of a spring having stiffness of $125$ $\mathrm{N} / \mathrm{m}$. The free length of the spring is $100$ mm . The system is rotated about the other end of the spring at a uniform angular velocity of $5 \mathrm{rad} / \mathrm{s}$. Ignore gravity and consider that the elongation of the spring may be comparable to the free length of the spring. The elongation of the spring $\text{(in mm , rounded off to the nearest integer)}$ is ____________.

GATE XE 2024 | Question: 66

Sun, 21 Jul 2024 16:41:51 +0000

A rigid bar PQR is hinged at its end P . As shown in the figure, the bar is pinconnected through two identical links QS and RT at points Q and R , respectively. The other ends of the links, S and T , are fixed. Both links are made of the same material. If the temperature of the links is uniformly increased by $\wedge T$, then which one of the following statements is correct? $\text{(Neglect the weight of the rigid bar)}$

Both QS and RT will be stress free.
Both QS and RT will be under tension.
Both QS and RT will be under compression.
QS will be under compression and RT will be under tension.

GATE XE 2024 | Question: 67

Sun, 21 Jul 2024 16:41:50 +0000

A pin-jointed truss has a pin support at the point E and a roller support at the point F. A horizontal force P is applied at pin H as shown in the figure. Which one of the members is in compression?

GATE XE 2024 | Question: 68

Sun, 21 Jul 2024 16:41:49 +0000

Two identical rigid slender bars of length $2$ L and mass $m$ are acted upon by a transverse force F at one of the ends as shown in the figure. In the first case, the bar is pinned at the other end. In the second case, the bar is pinned at its mid-point. What should be the magnitude of the force F such that the resulting angular accelerations of the two bars are equal?

$m g$
$\frac{m g}{2}$
$\frac{m g}{4}$
$\frac{m g}{6}$

GATE XE 2024 | Question: 71

Sun, 21 Jul 2024 16:41:47 +0000

A rigid massless bar $\text{PQR}$ is hinged at its end $\text{P}$ and supported through a spring of stiffness $k$ at point $\text{Q}$. A vertically downward force $W=560 \mathrm{~N}$ is applied at the free end $\text{R}$ of the bar. If the vertical component of the displacement at $\text{R}$ is $30$ mm , then the stiffness of the spring $\text{(in $\mathrm{kN} / \mathrm{m}$, rounded off to one decimal place)}$ is _____________.

GATE XE 2024 | Question: 75

Sun, 21 Jul 2024 16:41:44 +0000

A turn-table is rotating about its center with an angular velocity of $2 \mathrm{rad} / \mathrm{s}$ in the counter-clockwise direction. There is a groove in the turn-table within which a marble moves with a constant speed $v \mathrm{~m} / \mathrm{s}$ relative to the turn-table. At a given instant, the marble is at a radial distance of $1$ m from the center and the line joining the center of the turn-table with the marble makes an angle of $30^{\circ}$ with the groove. The value of $v$ $\text{(in $\mathrm{m} / \mathrm{s}$, rounded off to one decimal place)}$ for which there is no radial acceleration for the marble at this instant is ____________.

GATE XE 2024 | Question: 76

Sun, 21 Jul 2024 16:41:43 +0000

The system shown in the figure is in static equilibrium. The spring is massless and has a spring constant of $\mathrm{K}=1 \mathrm{kN} / \mathrm{m}$. The free length of the spring is $1$ m . All bodies in the system except the spring are rigid and have mass of $\mathrm{M}=1 \mathrm{~kg}$. All surfaces are frictionless and pin joints are ideal. The elongation of the spring $\text{(in mm, rounded off to the nearest integer)}$ in this configuration is __________. Take the acceleration due to gravity $g=10 \mathrm{~m} / \mathrm{s}^{2}$.

GATE XE 2024 | Question: 77

Sun, 21 Jul 2024 16:41:42 +0000

A square block of side $1$ m and mass $10$ kg is resting on a horizontal surface. The coefficient of static friction between the block and the surface is $0.75$. A horizontal force P acts on the block as shown in the figure. The force P is gradually increased from zero until the block either slides or topples. The maximum value of $h$ $\text{(in m, rounded off to two decimal places)}$ for which the block slides without toppling is ____________. Take the acceleration due to gravity $g=10 \mathrm{~m} / \mathrm{s}^{2}$.

GATE XE 2023 | Question: 56

Wed, 14 Feb 2024 18:09:30 +0000

\begin{tabular}{l}
Q. 66 \\
A plane truss is simply supported at $P$ and $R$ as shown. A downward force $F$ is \\
applied at hinge $Q$. The axial force developed in member $P S$ is \\

$\frac{\sqrt{5}}{2} F$ Tensile \\
$\frac{\sqrt{5}}{2} F$ Compressive \\
$\sqrt{5} F$ Tensile \\
\hline
$\sqrt{5} F$ Compressive
\end{tabular}

GATE XE 2023 | Question: 57

Wed, 14 Feb 2024 18:09:29 +0000

\begin{tabular}{l}
Q.67 A massless rigid rod $O P$ of length $L$ is hinged frictionlessly at $O$. A concentrated \\
mass $m$ is attached to end $P$ of the rod. Initially, the rod $O P$ is horizontal. Then, it \\
is released from rest. There is gravity as shown. The rod acquires an angular \\
velocity as it swings. The clockwise angular velocity of the rod, when it first \\
reaches the vertical position as shown, is \\

\\
(D) \\
\\
\\
\\
$\frac{1}{\frac{g}{L}}$ \\
\hline$\frac{2}{L}$
\end{tabular}

GATE XE 2023 | Question: 62

Wed, 14 Feb 2024 18:09:22 +0000

Q. 72 A rod is subjected to three forces as shown in the figure on the left. An equivalent
force system with forces $F_{1}, F_{2}$ and moment $M$ is shown in the figure on the right.
The value of $M($ in $\mathrm{N}-\mathrm{m})$ is

GATE XE 2023 | Question: 65

Wed, 14 Feb 2024 18:09:20 +0000

\begin{tabular}{|c|c|}
\hline Q. 75 & \begin{tabular}{l}
A solid uniform rigid disk of mass $m$ and radius $R$ rolls without slipping along a \\
horizontal surface $\mathrm{PQ}$. The speed of the center of the disk is $v$. The disk then \\
strikes a hurdle of height $\frac{3 R}{20}$ at point $\mathrm{S}$. During the impact, there is no rebound or \\
slip at $\mathrm{S}$ and no impulse from the surface $\mathrm{PQ}$. The magnitude of the velocity of the \\
center of the disk immediately after the impact is
\end{tabular} \\
\hline & $\underbrace{R}$ \\
\hline

& $0.1 v$ \\
\hline
& $0.3 v$ \\
\hline
& $0.7 v$ \\
\hline
& $0.9 v$ \\
\hline
\end{tabular}

GATE XE 2023 | Question: 71

Wed, 14 Feb 2024 18:09:13 +0000

Q. 81 A steel ball of mass $m=10 \mathrm{~kg}$ is suspended from the ceiling of a moving carriage by two inextensible strings making $60^{\circ}$ with the horizontal as shown. The carriage has an acceleration $a$ such that the tension in the string on the right is double the tension in the string on the left. Take the acceleration due to gravity $(g)$ as $10 \mathrm{~m} / \mathrm{s}^{2}$. The acceleration $a$ (in $\mathrm{m} / \mathrm{s}^{2}$ ) is (rounded off to one decimal place).

GATE XE 2023 | Question: 72

Wed, 14 Feb 2024 18:09:12 +0000

Q. 82 A block of mass $m=10 \mathrm{~kg}$ is lying on an inclined plane $\mathrm{PQ}$. The mass is
restrained from sliding down the inclined plane by a force $F$. The coefficient of
friction between the block and the inclined plane is 0.3 . Take the acceleration due
to gravity as $10 \mathrm{~m} / \mathrm{s}^{2}$. The smallest force $F$ (in $\mathrm{N}$ ) required to prevent the block
from sliding down is__rounded off to one decimal place).

GATE XE 2023 | Question: 73

Wed, 14 Feb 2024 18:09:11 +0000

Q. 83
A spherical rigid ball of mass $10 \mathrm{~kg}$ is moving with a speed of $2 \mathrm{~m} / \mathrm{s}$ in the
direction shown. The ball collides with a rigid frictionless wall and rebounds at an
angle $\alpha$ with a speed of $v$, as shown. The coefficient of restitution is 0.9 . The
angle $\alpha$ (in degrees) is__rounded off to one decimal place).

GATE XE 2023 | Question: 160

Wed, 14 Feb 2024 18:07:48 +0000

Q. 170 Two balls each of $5 \mathrm{~cm}$ in diameter are placed $200 \mathrm{~m}$ apart on a horizontal frictionless plane at $45^{\circ} \mathrm{N}$. The balls are impulsively propelled directly at each other with equal speeds. What must be the speed in $\mathrm{m} \mathrm{s}^{-1}$ so that the two balls just miss each other?
(consider the value of $\Omega$ as $7.29 \times 10^{-5} \mathrm{rad} \mathrm{s}^{-1}$, rounded off to two decimal places).