GATE 2021 Question Paper XE D Engineering Sciences - Solid Mechanics

Q.1 – Q.8 Multiple Choice Question (MCQ), carry ONE mark each (for each wrong
answer: – 1/3).

Q.1 A force F = 40 kN is applied on the hook as shown. The equivalent force-
couple system at B is

y

x

B

A

100 mm
40 kN

(A) 40 kN in +y direction and M = 0

(B) 40 kN in –y direction and M = 0

(C) 40 kN in +y direction and M =4000 Nm counter clockwise

(D) 40 kN in -y direction and M = 4000 Nm clockwise

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Q.2 A rigid rod OA rotates clockwise at an angular velocity of 10 rad/s. A bead
B (OB = 1 m) translates outward on the rod at a speed of 5 m/s and
acceleration 2.5 m/s2 (both quantities with respect to the rod). The Coriolis
component of acceleration is

y

A

x
B

10 rad/s
O

(A) 2.5 m/s2 in +x direction

(B) 100 m/s2 in +x direction

(C) 100 m/s2 in –y direction

(D) 25 m/s2 in + y direction

Q.3 A two force member in equilibrium is one in which

(A) Forces act at two points and forces are collinear

(B) Forces act at two points and member is always straight

(C) Forces act at two points but the member is free to carry moment at any point

(D) Force acts at one point and moment acts at second point

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Q.4 If the yield point shear stress obtained from the torsion test of a cylindrical
specimen is τy, then what is the maximum value of principal strain at
yielding? (µ is Poisson’s ratio and E is Young’s modulus)

(A) 𝜏𝑦
𝐸
(B) (1 + µ)𝜏𝑦
𝐸
(C) 𝜏𝑦
2𝐸
(D) (1 − µ)𝜏𝑦
𝐸

Q.5 If the ratio of Young’s modulus to bulk modulus of a material is 3/2, then
the ratio of shear modulus to the Young’s modulus of the material is

(A) 1

(B) 2/5

(C) 1/3

(D) 3/5

Q.6 With respect to the plane of maximum shear stress, which of the following
statements is INCORRECT?

(A) The normal stress on this plane is zero.

(B) The maximum shear stress is equal to the largest of the one half the difference of
principal stresses

(C) The plane of maximum shear stress occurs at 45o to the principal planes.

(D) The magnitude of the maximum shear stress is equal to the largest of the radius
of the Mohr’s circles.

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Q.7 A simply supported beam of length L is loaded by two symmetrically
applied point loads P at L/3 from each support. Both the loads are then
shifted to new points which are at a distance L/4 from each support. The
bending moments at the mid-section of the beam in both the cases are
same. The magnitude of P1 in terms of P is
P P
1
A B C D
L/3 L/3 L/3

P1 P1

A E F D
L/2 L/4
L/4

(A) P/4

(B) 8P/3

(C) 4P/3

(D) P/3

Q.8 A beam having rectangular cross section is subjected to transverse loading.
The ratio of maximum shear stress developed in the beam to the average
shear stress is

(A) 1.50

(B) 1.25

(C) 1.33

(D) 1.66

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Q.9 Numerical Answer Type (NAT), carry ONE mark each (no negative marks).

Q.9 During an earthquake, a structure vibrates and the vibration can be
assumed to be in simple harmonic motion at 5 Hz. At a measurement
point, the RMS value of acceleration is 10 m/s2. The approximate
amplitude of motion (in mm) at this point (rounded off to two decimal
places) is ___________

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Q. 10–Q. 15 Multiple Choice Question (MCQ), carry TWO marks each (for each wrong
answer: – 2/3).

Q.10 For the state of plane stress shown, the components of normal and shear
stresses are given in terms of stress σ and unknown constants m and n. If
the normal and shear components of stress on a 45o plane are 2σ and zero,
the values of m and n would be:

2σ

σ

45o
nσ

mσ

(A) m = 1, n = 2

(B) m = 2, n = 1

(C) m = 1, n = 1

(D) m = 2, n = 2

Q.11 For a state of plane strain, the normal strains are given by
εxx = 1000 × 10-6, εyy = 200 × 10-6 and the maximum shear strain is
γmax = 1000 × 10-6. The value of shear strain γxy for this strain state is

(A) 600 × 10-6

(B) 183 × 10-6

(C) 1000 × 10-6

(D) 800 × 10-6

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Q.12 A thin cylinder (closed at its ends) of radius r and thickness t ( r ≫ t ) is
subjected to internal pressure p. The maximum shear stress in the wall of
the cylinder is

(A) 𝑝𝑟
𝑡
(B) 𝑝𝑟
2𝑡
(C) 𝑝𝑟
4𝑡
(D) 3𝑝𝑟
2𝑡

Q.13 The truss shown is subjected to a force P. All members of the truss have the
same length L. The reaction at A and force in member AB are

B P

L

A

(A) 𝑃√3 𝑃
and
4 2
(B) 𝑃√3 𝑃√3
and
8 4
(C) 𝑃√3 𝑃
and
4 4
(D) 𝑃
𝑃 and
4

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Q.14

(A) 100kN

(B) 100kN

(C) 100kN

(D) 100kN

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Q.15 A hammer of mass 1 kg is used to break an almond shell. The velocity time
graph of the hammer during the impact duration is shown in the figure.
The shape of force time graph is also given, which can be approximated as
a triangle. A force of 300 N is required for breaking the shell, while a force
of 200 N will not be able to break it, but just introduce a crack. Which one
of the following events will happen?
0.3 m/s
Velocity
(m/s)
t (milliseconds)
-0.03 m/s

Force
(N) t (milliseconds)

3 ms

(A) The almond shell will crack but not break

(B) The almond shell will not crack.

(C) The almond shell will break

(D) Cannot be determined from the given data

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Q. 16 – Q. 22 Numerical Answer Type (NAT), carry TWO marks each (no negative
marks).

Q.16 A rigid circular disc of radius 0.2 m and mass 10 kg rolls without slip on
the ground at A. The coefficient of static friction µ between ground and
disc is 0.7. A torque T of 9 Nm acts on the disc as shown. Given
acceleration due to gravity g = 10 m/s2 . The friction force (in N) acting on
the disc (in integer) is _________

y

x

T

A

Q.17 A prismatic solid circular rod of diameter d is bent to introduce an offset
s = d as shown. The rod is further subjected to an axial load P. If the
maximum longitudinal stress at a section A-B in the rod (with offset) is n
times the longitudinal stress in the straight rod, the value of n (in integer)
would be ___________________

P
d P
’

P
d s=d P
’ A

B

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Q.18 A naturally curved steel beam AB having Young’s modulus 208 GPa, area
moment of inertia I = 26.7 cm4 and radius R = 2 m is subjected to a vertical
load P = 1000 N at B. The end A at θ = 900 is rigidly fixed. The bending
strain energy of the beam
(in Nm, rounded off to two decimal places) is _______________
B

P

R
θ
A
O

Q.19 At room temperature of 25oC, a gap of 1 mm exists between the ends of the
rods 1 and 2 as shown. Given the cross section area A of the rods is
1500 mm2, Young’s modulus E = 75 GPa and the coefficient of thermal
expansion α = 23 × 10-6 /oC. When the temperature has reached 150oC, the
magnitude of normal stress in each of the rods (in MPa, rounded off to two
decimal places) is ______________
1 mm
300 mm 300 mm

1 2

Q.20 A tube of inner radius 4 cm and outer radius 5 cm can carry a maximum
torque of T. This tube is now replaced by a solid circular shaft of the same
material. The minimum radius of the solid circular shaft (in cm, rounded
off to two decimal places) to carry the same amount of torque T is
_____________________

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Q.21 In System A, a rectangular block of mass M is centrally supported on a
spring of stiffness K as shown. In the System B, the mass is hinged at one of
its ends and is supported centrally by the spring. The ratio of natural
frequency of System B to that of System A (rounded off to two decimal
places) is _____________________

M M

K K

System A System B

Q.22 A coronavirus droplet of mass 1 microgram ejects from the mouth of a
patient with a velocity of 0.7 m/s and travels through air. The gravitational
force experienced by it can be neglected due to the buoyancy effect.
However, the droplet experiences air drag force proportional to its velocity
and the drag coefficient is given as 1.0 μN-s/m. The distance travelled by
the droplet before its velocity drops to 10% of its initial velocity (in m,
rounded off to two decimal places) is __________________.

END OF THE QUESTION PAPER

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