Solid Mechanics (XE-D) - GATE XE (Engineering Sciences) Subject-Wise PYQ Practice

Question 1

Year: 2025 | Marks: 1

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)?

• A. 𝑃 𝑐𝑜𝑠(√𝑘 𝑚𝑡)+ 𝑃 𝑠𝑖𝑛(√𝑘 𝑚𝑡)
• B. 𝑃 𝑠𝑖𝑛(√𝑘 𝑚𝑡)
• C. 𝑃 2 𝑐𝑜𝑠(√𝑘 𝑚𝑡)+ 𝑃 2 𝑠𝑖𝑛(√𝑘 𝑚𝑡)
• D. 𝑃𝑐𝑜𝑠(√𝑘 𝑚𝑡)

Answer: D. 𝑃𝑐𝑜𝑠(√𝑘 𝑚𝑡)

Question 2

Year: 2025 | Marks: 1

A ball of mass 5m approaches a stationary ball of mass m with a horizontal velocity of 2 m/s from left to right. After a perfectly elastic central collision, the horizontal velocity of the heavier ball is 1 m/s from left to right. Which one of the following statements, regarding the velocity (in m/s) of the lighter ball after impact, is TRUE?

• A. Comes to rest
• B. Moves from left to right at 5 m/s
• C. Moves from right to left at 5 m/s
• D. Moves from left to right at 1m/s

Answer: B. Moves from left to right at 5 m/s

Question 3

Year: 2025 | Marks: 1

The Mohr’s circle corresponding to an infinitesimal element is shown in the figure. The plane PQ in the infinitesimal element, at an angle of 𝜃 from the x-axis, is in a state of pure shear. Which one of the following values of 𝜃 (in degrees) is CORRECT?

• A. 90
• B. 60
• C. 45
• D. 120

Answer: C. 45

Question 4

Year: 2025 | Marks: 1

The two-dimensional state of stress, in an infinitesimal element, is given by 𝜎𝑥𝑥= 800 MPa, 𝜎𝑥𝑦= 300 MPa and 𝜎𝑦𝑦= 0 MPa. Which one of the following options is the maximum shear stress (in MPa) in the element?

• A. 500
• B. 400
• C. 800
• D. 300

Answer: A. 500

Question 5

Year: 2025 | Marks: 1

Two cars P and Q are travelling on a straight path and are 60 m apart as shown in the figure; Car P is moving with a constant velocity of 36 kmph, while car Q is moving at a constant velocity of 18 kmph. At this instant, the driver in car P applies the brake and collision occurs with car Q after 30 seconds. Assuming uniform deceleration due to braking, which one of the following is the CORRECT velocity (in m/s) of the car P just before the collision?

• A. 1
• B. 16
• C. 5
• D. 4

Answer: D. 4

Question 6

Year: 2025 | Type: MSQ | Marks: 1

The natural frequency of a spring-mass system is 10 rad/s. Which of the following statements is/are CORRECT?

• A. The mass is 100 kg and the stiffness is 1 N/m.
• B. The mass is 1.25 kg and the stiffness is 125 N/m.
• C. The stiffness is 620 N/m and the mass is 6.2 kg.
• D. The stiffness is 62 N/m and the mass is 620 kg.

Answer: B. The mass is 1.25 kg and the stiffness is 125 N/m.; C. The stiffness is 620 N/m and the mass is 6.2 kg.

Question 7

Year: 2025 | Type: NAT | Marks: 1

Consider a beam with a square box cross-section as shown in the figure. The outer square has a length of 10 mm. The thickness of the section is 1 mm. The area moment of inertia about the x-axis is _____ mm4 (in integer).

Answer: 490 to 494

Question 8

Year: 2025 | Type: NAT | Marks: 1

For a certain linear elastic isotropic material, the Young’s modulus is 140 GPa and the shear modulus is 50 GPa. The Poisson’s ratio for the material is _____ (rounded off up to two decimal places).

Answer: 0.40

Question 9

Year: 2025 | Type: NAT | Marks: 1

A force of P = 100 N is applied at the ends of the pliers as shown in the figure. Neglecting friction, the force exerted by the upper jaw on the workpiece is _____ N (in integer).

Answer: 400

Question 10

Year: 2025 | Marks: 2

Consider two blocks, P of mass 100 kg and Q of mass 150 kg, resting as shown in the figure. The angle 𝜃= 30°. 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 P upward is W. Which one of the following options is closest to the CORRECT magnitude of W (in N)?

• A. 862.2
• B. 1116.6
• C. 2900.0
• D. 406.2

Answer: A. 862.2

Question 11

Year: 2025 | Marks: 2

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

• A. Cantilever column with a length L and cross-section diameter d
• B. Column with hinge at one end and roller at the other end with a length 2L and cross-section diameter d
• C. Cantilever column with a length L and cross-section diameter 2d
• D. Column with hinge at one end and roller at the other end with a length L and cross-section diameter d

Answer: C. Cantilever column with a length L and cross-section diameter 2d

Question 12

Year: 2025 | Marks: 2

The figure shows a rod PQ, hinged at P, rotating counter-clockwise with a uniform angular speed of 15 rad/s. A block R translates along a slot cut out in rod PQ. At the instant shown the distance PR = 0.5 m and θ = 60o. The relative velocity of R with respect to the rod PQ is 5 m/s at the instant shown. The relative acceleration of R with respect to the rod PQ is zero at the instant shown. Which one of the following is the CORRECT magnitude of the absolute acceleration (in m/s2) of block R?

• A. 135.2
• B. 187.5
• C. 112.5
• D. 150.0

Answer: B. 187.5

Question 13

Year: 2025 | Marks: 2

The frame shown in the figure is loaded at S with a force of 2000 N. The reactions at T are denoted by Tx and Ty, while the reaction at W is Wy. Neglect the weight of the members. Which one of the following options for the magnitudes of the forces (in N), Tx, Ty and Wy, is CORRECT?

• A. Tx = 0, Ty = 1000 and Wy = 1000
• B. Tx= 0, Ty =1500 and Wy = 500
• C. Tx = 0, Ty = 800 and Wy = 1200
• D. Tx = 0, Ty = 500 and Wy = 1500

Answer: D. Tx = 0, Ty = 500 and Wy = 1500

Question 14

Year: 2025 | Marks: 2

A closed thin cylindrical tank with a mean diameter d = 300 mm and thickness t = 2 mm, is subjected to a uniform internal gas pressure p. The allowable shear stress on the curved wall of the tank is 70 MPa. Based on the Tresca criteria, which one of the following options for the maximum safe value of p (in MPa) is CORRECT?

• A. 3.73
• B. 7.46
• C. 1.87
• D. 5.60

Answer: C. 1.87

Question 15

Year: 2025 | Type: MSQ | Marks: 2

An infinitesimal square element 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 𝜀𝑥𝑥= 500 × 10−6, 𝜀𝑦𝑦= 100 × 10−6 and 𝜀𝑥𝑦= 0. Which of the following statements is/are CORRECT?

• A. Percentage change in length of the diagonal PR is 0.03.
• B. Change in angle between PR and QS is 4 × 10−4 rad.
• C. Change in angle between PR and QS is 2 × 10−4 rad.
• D. Percentage change in length of the diagonal QS is 0.03.

Answer: A. Percentage change in length of the diagonal PR is 0.03.; B. Change in angle between PR and QS is 4 × 10−4 rad.; D. Percentage change in length of the diagonal QS is 0.03.

Question 16

Year: 2025 | Type: MSQ | Marks: 2

The figure shows the stress distribution across an internal surface of a rectangular beam of height 30 mm and depth 10 mm. The normal stress distribution is given by the expression 𝜎𝑥𝑥= 200𝑦+ 500 N/mm2; y is the distance in 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 CORRECT?

• A. The net force in the x direction is 150 kN.
• B. The net force in the x direction is 75 kN.
• C. The net moment about the z axis is 4500 Nm.
• D. The net moment about the z axis is 2250 Nm.

Answer: A. The net force in the x direction is 150 kN.; C. The net moment about the z axis is 4500 Nm.

Question 17

Year: 2025 | Type: NAT | Marks: 2

A vertical column fixed at one end is subjected to a compressive axial load at the free end. The column’s section modulus, EI, is 9.82 × 105 Nm2 and the cross-section area is 7.85 × 10−3 m2. The length of the column is 2 m. The yield stress of the material is 145 MPa. If the column can fail either in buckling or by Tresca’s criterion, the maximum load that the structure can safely sustain is _____ kN (rounded off to one decimal place).

Answer: 603.5 to 607.5

Question 18

Year: 2025 | Type: NAT | Marks: 2

A simply-supported beam, with a point load P = 150 kN at a distance of L/3 from the left end, is shown in the figure. The elastic-strain energy (U) of the beam is given by the following expression: U = 2 243 P2L3 EI where the section modulus, EI, is 16.66 × 105 Nm2 and the length of the beam L is 1 m. The deflection at the loading point is ____ mm (rounded off to two decimal places).

Answer: 1.46 to 1.50

Question 19

Year: 2025 | Type: NAT | Marks: 2

A simply-supported beam has a circular cross-section with a diameter of 20 mm, area of 314.2 mm2, area moment of inertia of 7854 mm4 and a length L of 4 m. A point load P =100 N acts at the center and an axial load Q = 20 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 L/2 from the left, is __________ mm (rounded off to one decimal place).

Answer: 4.9 to 5.1

Question 20

Year: 2025 | Type: NAT | Marks: 2

A massless cantilever beam, with a tip mass m of 10 kg, is modeled as an equivalent spring-mass system as shown in the figure. The beam is of length L = 1 m, with a circular cross-section of diameter d = 20 mm. The Young’s modulus of the beam material is 200 GPa. The natural frequency of the spring-mass system is _____ Hz (rounded off to two decimal places).

Answer: 3.43 to 3.47

Question 21

Year: 2025 | Type: NAT | Marks: 2

An electric motor’s rotor is spinning at 1500 rpm when its load and power are cut- off. The rotor which has a mass of 50 kg and a radius of gyration of 200 mm, then coasts down to rest. Due to kinetic friction, a constant torque of 10 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).

Answer: 392 to 394

Question 22

Year: 2025 | Type: NAT | Marks: 2

A bar of length L = 1 m is fixed at one end. Before heating its free end has a gap of δ = 0.1 mm from a rigid wall as shown in the figure. Now the bar is heated resulting in a uniform temperature rise of 10 ℃. The coefficient of linear thermal expansion of the material is 20 × 10−6/ ℃ and the Young’s modulus of elasticity is 100 GPa. Assume that the material properties do not change with temperature. The magnitude of the resulting axial stress on the bar is ______ MPa (in integer).

Answer: 10

Question 23

Year: 2024 | Marks: 1

The engineering stress (𝜎) vs. engineering strain (𝜀) curve obtained by conducting uniaxial tension test on a steel specimen is shown in the figure (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?

• A. P
• B. Q
• C. R
• D. S

Answer: D. S

Question 24

Year: 2024 | Marks: 1

An L-shaped rigid member is fixed at the midpoint of a simply-supported beam, as shown in figure (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 (see figure (ii)). Which of the following statements is correct? (Neglect the mass of the beam and of the rigid member)

• A. M = P a
• B. M = P b
• C. M = P (a + b)
• D. M = 0

Answer: A. M = P a

Question 25

Year: 2024 | Marks: 1

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-of- gravity 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?

• A.
• B.
• C.
• D.

Answer: B.

Question 26

Year: 2024 | Marks: 1

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 (see figure). The natural frequency of this system is

• A. √4𝑘 𝑀
• B. √2𝑘 𝑀
• C. √𝑘 𝑀
• D. √𝑘 2𝑀

Answer: A. √4𝑘 𝑀

Question 27

Year: 2024 | Marks: 1

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

• A. 3𝑃 2𝑤ℎ⁡⁡, ⁡ 3𝑃𝐿 2𝑤ℎ2
• B. 0 , ⁡ 3𝑃𝐿 2𝑤ℎ2
• C. 3𝑃 2𝑤ℎ , 0
• D. 0 , 0

Answer: D. 0 , 0

Question 28

Year: 2024 | Marks: 1

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

• A. increases
• B. decreases
• C. remains the same
• D. goes to zero

Answer: A. increases

Question 29

Year: 2024 | Type: MSQ | Marks: 1

A solid circular shaft of diameter 100 mm is subjected to a torque 3𝜋 kNm. Which of the following statements about the state of stress in the shaft is/are correct?

• A. The maximum shear stress is 48 MPa
• B. The maximum tensile stress is 48 MPa
• C. The magnitude of maximum compressive stress is 48 MPa
• D. The magnitude of shear stress is 48 MPa at all points in the shaft

Answer: A. The maximum shear stress is 48 MPa; B. The maximum tensile stress is 48 MPa; C. The magnitude of maximum compressive stress is 48 MPa

Question 30

Year: 2024 | Type: NAT | Marks: 1

A spring is connected to an elastic bar as shown in the figure. The spring has a stiffness constant of 107 N/m. The bar is 70 mm long, and has an area of cross- section 10 mm2. The Young’s modulus of the bar material is 70,000 MPa. A force F = 5000 N is applied at point O along the axis of the bar and the spring. The resulting deflection of Point O in mm (rounded to one decimal place) is ______

Answer: 0.9 to 1.1

Question 31

Year: 2024 | Type: NAT | Marks: 1

A particle of mass 1 kg is attached to one end of a spring having stiffness of 125 N/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 rad/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 (in mm, rounded off to the nearest integer) is ______ .

Answer: 24 to 26

Question 32

Year: 2024 | Marks: 2

A 0-45-90 strain gauge rosette is mounted on an aircraft wing. The co-ordinate system is placed such that the strain gauges p, q and r are oriented at angles 0o, 45o and 90o, respectively, from the x axis (see figure). The strain readings associated to p, q and r are denoted by 𝜀𝑝, 𝜀𝑞, and 𝜀𝑟 , respectively. While conducting a test the strain gauges show the following readings. 𝜀𝑝= 150 × 10−6, 𝜀𝑞= 180 × 10−6, 𝜀𝑟= −90 × 10−6 The developed engineering shear strain 𝛾𝑥𝑦 associated to the strain gauge data is

• A. 120 × 10−6
• B. 180 × 10−6
• C. 300 × 10−6
• D. 240 × 10−6

Answer: C. 300 × 10−6

Question 33

Year: 2024 | Marks: 2

A rigid bar PQR is hinged at its end P. As shown in the figure, the bar is pin- connected 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 ΔT, then which one of the following statements is correct? (Neglect the weight of the rigid bar)

• A. Both QS and RT will be stress free.
• B. Both QS and RT will be under tension.
• C. Both QS and RT will be under compression.
• D. QS will be under compression and RT will be under tension.

Answer: D. QS will be under compression and RT will be under tension.

Question 34

Year: 2024 | Marks: 2

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 ?

• A. EF
• B. FG
• C. EG
• D. EH

Answer: C. EG

Question 35

Year: 2024 | Marks: 2

Two identical rigid slender bars of length 2L 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?

• A. 𝑚⁡𝑔
• B. 𝑚⁡𝑔 2
• C. 𝑚⁡𝑔 4
• D. 𝑚⁡𝑔 6

Answer: B. 𝑚⁡𝑔 2

Question 36

Year: 2024 | Type: MSQ | Marks: 2

A critical point on a component is subjected to the state of stress [𝜎] as given in the following. The yield strength of the material is 400 MPa. By considering maximum shear stress (Tresca) theory, the possible value(s) of 𝜎0 at the onset of yielding is/are, [𝜎] =⁡[ 280 0 0 0 𝜎𝑜 0 0 0 −60 ] MPa

• A. 340
• B. 680
• C. ̶ 120
• D. ̶ 460

Answer: A. 340; C. ̶ 120

Question 37

Year: 2024 | Type: NAT | Marks: 2

A plane passing through a point Q inside a body is shown. The unit normal of the plane is 𝑛̂ = 0.6⁡𝑖̂ + 0.8⁡𝑗̂ , as shown in the figure. The traction (stress) vector on the plane at point Q is given by 𝑡 = (50⁡𝑖̂ + 20⁡𝑗̂) MPa. Given that at point Q, 𝜎𝑥𝑥= 𝜎𝑦𝑦 , the shear stress component 𝜏𝑥𝑦 (in MPa, rounded off to two decimal places) is ___________ .

Answer: 99.00 to 101.00

Question 38

Year: 2024 | Type: NAT | Marks: 2

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

Answer: 41.0 to 43.0

Question 39

Year: 2024 | Type: NAT | Marks: 2

A beam of rectangular cross-section, as shown in the figure, is made of two perfectly bonded layers of different materials and equal thickness. The Young’s moduli of the two materials are 𝐸1 and 𝐸2, where 𝐸1 = 2𝐸2. The beam is subjected to pure bending. If t = 1 mm, the distance of the neutral plane from the top surface of the beam is ______ (in mm, rounded off to two decimal places).

Answer: 0.80 to 0.85

Question 40

Year: 2024 | Type: NAT | Marks: 2

A stepped beam is made of a material whose Young’s modulus is E. The dimensions of the two stepped sections are such that the sectional moments of inertia, 𝐼1⁡and 𝐼2, are related as 𝐼1 = 8𝐼2. The beam is fixed at one end and a load of F is applied at the free end as shown in the figure. Under this loading condition, if the strain energy of the stepped beam is written as 𝑈= 𝛽 𝐹2𝐿3 𝐸𝐼1 , then the value of 𝛽 is ________ (rounded off to two decimal places).

Answer: 0.29 to 0.33

Question 41

Year: 2024 | Type: NAT | Marks: 2

A cylindrical pressure vessel is constructed by bolting two symmetric halves of flanged semi-cylindrical shells. A cross-sectional view of the vessel is shown in the figure. The inner diameter of the vessel is 2 m and the length is 10 m. Each row comprises 100 bolts along the length of the vessel. If the vessel is pressurized to a net pressure of 6 x 105 N/m2, and assuming the end caps of the vessel do not take any load in the radial direction, then the load borne by each bolt is __________ (in kN, rounded off to one decimal place)

Answer: 59.5 to 60.5

Question 42

Year: 2024 | Type: NAT | Marks: 2

A turn-table is rotating about its center with an angular velocity of 2 rad/s in the counter-clockwise direction. There is a groove in the turn-table within which a marble moves with a constant speed v m/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° with the groove. The value of v (in m/s, rounded off to one decimal place) for which there is no radial acceleration for the marble at this instant is _____.

Answer: 1.9 to 2.1

Question 43

Year: 2024 | Type: NAT | Marks: 2

The system shown in the figure is in static equilibrium. The spring is massless and has a spring constant of K=1 kN/m. The free length of the spring is 1 m. All bodies in the system except the spring are rigid and have mass of M=1 kg. All surfaces are frictionless and pin joints are ideal. The elongation of the spring (in mm, rounded off to the nearest integer) in this configuration is ______. Take the acceleration due to gravity 𝑔= 10 m/s2.

Answer: 14 to 16

Question 44

Year: 2024 | Type: NAT | Marks: 2

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 (in m, rounded off to two decimal places) for which the block slides without toppling is _____. Take the acceleration due to gravity 𝑔= 10 m/s2.

Answer: 0.66 to 0.68

Question 45

Year: 2023 | Marks: 1

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 PS is

• A. √5 2 𝐹 Tensile
• B. √5 2 𝐹 Compressive
• C. √5 𝐹 Tensile
• D. √5 𝐹 Compressive

Answer: B. √5 2 𝐹 Compressive

Question 46

Year: 2023 | Marks: 1

A massless rigid rod OP of length L is hinged frictionlessly at O. A concentrated mass m is attached to end P of the rod. Initially, the rod OP 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

• A. 2√𝑔 𝐿
• B. √ 2 𝑔 𝐿
• C. √𝑔 𝐿
• D. 1 2 √𝑔 𝐿

Answer: B. √ 2 𝑔 𝐿

Question 47

Year: 2023 | Marks: 1

Two equivalent descriptions of the state of stress at a point are shown in the figure. The normal stresses 𝜎1 and 𝜎2 as shown on the right must be, respectively,

• A. 𝜏o and −𝜏o
• B. −𝜏o and 𝜏o
• C. 𝜏o √2 and − 𝜏o √2
• D. − 𝜏o √2 and 𝜏o √2

Answer: A. 𝜏o and −𝜏o

Question 48

Year: 2023 | Marks: 1

The state of strain at a point in a machine component is given as 𝜀𝑥𝑥= 2.5 × 10−4, 𝜀𝑦𝑦= 2.0 × 10−4, 𝜀𝑧𝑧= −1.5 × 10−4, 𝜀𝑥𝑦= 2.5 × 10−4, 𝜀𝑦𝑧= −0.5 × 10−4, 𝜀𝑧𝑥= −1.0 × 10−4. The volumetric strain at this point is

• A. 4 × 10−4
• B. 3 × 10−4
• C. −5 × 10−4
• D. −3 × 10−4

Answer: B. 3 × 10−4

Question 49

Year: 2023 | Marks: 1

A thin walled, closed cylindrical vessel of inside diameter 𝑑 and wall thickness 𝑡 contains a fluid under pressure 𝑝. The figure below shows a part of the cylindrical vessel; end caps are not shown. Consider the small element shown with sides parallel and perpendicular to the axis of the cylinder. The stresses 𝜎1 and 𝜎2 are

• A. 𝜎1 = 𝑝𝑑 2𝑡; 𝜎2 = 𝑝𝑑 4𝑡
• B. 𝜎1 = 𝑝𝑑 𝑡; 𝜎2 = 𝑝𝑑 2𝑡
• C. 𝜎1 = 𝑝𝑑 4𝑡; 𝜎2 = 𝑝𝑑 2𝑡
• D. 𝜎1 = 𝑝𝑑 2𝑡; 𝜎2 = 0

Answer: A. 𝜎1 = 𝑝𝑑 2𝑡; 𝜎2 = 𝑝𝑑 4𝑡

Question 50

Year: 2023 | Type: MSQ | Marks: 1

A spring mass system is shown in the figure below. Take the acceleration due to gravity as 𝑔 = 9.81 m/s2. The static deflection due to weight and the time period of oscillations, respectively, are

• A. 0.392 m and 1.26 s
• B. 0.392 m and 3.52 s
• C. 0.626 m and 1.26 s
• D. 0.626 m and 3.52 s

Answer: A. 0.392 m and 1.26 s

Question 51

Year: 2023 | Type: NAT | Marks: 1

A rod is subjected to three forces as shown in the figure on the left. An equivalent force system with forces F1, F2 and moment M is shown in the figure on the right. The value of M (in N-m) is______(rounded off to one decimal place).

Answer: 117.0 to 118.0

Question 52

Year: 2023 | Type: NAT | Marks: 1

A simply supported beam of length 3 m is loaded as shown in the figure. The magnitude of the shear force (in kN) at the mid-point of the beam is______(rounded off to one decimal place).

Answer: 4.9 to 5.1

Question 53

Year: 2023 | Type: NAT | Marks: 1

For a plane stress problem, the principal stresses are 100 MPa and 50 MPa. The magnitude of maximum shear stress (in MPa) in the material is______(rounded off to one decimal place).

Answer: 49.5 to 50.5

Question 54

Year: 2023 | Marks: 2

A solid uniform rigid disk of mass 𝑚 and radius 𝑅 rolls without slipping along a horizontal surface PQ. The speed of the center of the disk is 𝑣. The disk then strikes a hurdle of height 3𝑅 20 at point S. During the impact, there is no rebound or slip at S and no impulse from the surface PQ. The magnitude of the velocity of the center of the disk immediately after the impact is

• A. 0.1𝑣
• B. 0.3𝑣
• C. 0.7𝑣
• D. 0.9𝑣

Answer: D. 0.9𝑣

Question 55

Year: 2023 | Marks: 2

A cylinder made of rubber (length = L and diameter = d) is inserted in a rigid container as shown in the figure. The rubber cylinder fits snugly in the rigid container. There is no wall friction. The modulus of elasticity of the rubber is E and its Poisson’s ratio is ν. The cylinder is subjected to a small uniform pressure 𝑝 as shown in the figure. The resulting axial strain (𝜀𝑧𝑧) is

• A. − 𝑝 𝐸
• B. − 𝑝 𝐸(1 −2ν)
• C. − 𝑝 𝐸[(1 + ν)(1 −2ν) (1 −ν) ]
• D. − 𝑝 𝐸[(1 −ν)(1 −2ν) (1 + ν) ]

Answer: C. − 𝑝 𝐸[(1 + ν)(1 −2ν) (1 −ν) ]

Question 56

Year: 2023 | Marks: 2

The state of stress at the critical location in a structure is 𝜎𝑥𝑥 = 420 MPa, 𝜎𝑦𝑦 = 100 MPa, 𝜎𝑧𝑧= 𝜎𝑥𝑦 = 𝜎𝑦𝑧= 𝜎𝑧𝑥= 0. The yield stress of the material in uniaxial tension is 400 MPa. Select the correct statement among the following.

• A. The structure is safe by both Tresca (maximum shear stress) theory and von-Mises (distortion energy) theory.
• B. The structure is safe by Tresca (maximum shear stress) theory and unsafe by von-Mises (distortion energy) theory.
• C. The structure is unsafe by Tresca (maximum shear stress) theory and safe by von-Mises (distortion energy) theory.
• D. The structure is unsafe by both Tresca (maximum shear stress) theory and von-Mises (distortion energy) theory.

Answer: C. The structure is unsafe by Tresca (maximum shear stress) theory and safe by von-Mises (distortion energy) theory.

Question 57

Year: 2023 | Marks: 2

The figure shows a column of rectangular cross section 100 mm × 80 mm. It carries a load of 60 kN at a point 30 mm from the edge 𝑃𝑄. The values of stress component 𝜎𝑧𝑧 on surfaces 𝑃𝑄𝑄′𝑃′ and 𝑆𝑅𝑅′𝑆′, at points far away from both ends of the column, are respectively

• A. 18.75 N/mm2 (Compressive) and 3.75 N/mm2 (Tensile)
• B. 18.75 N/mm2 (Compressive) and 3.75 N/mm2 (Compressive)
• C. 13.13 N/mm2 (Compressive) and 1.88 N/mm2 (Tensile)
• D. 13.13 N/mm2 (Compressive) and 1.88 N/mm2 (Compressive)

Answer: D. 13.13 N/mm2 (Compressive) and 1.88 N/mm2 (Compressive)

Question 58

Year: 2023 | Marks: 2

Consider an electric pole with dimensions as shown in the figure. Let the end R be subjected to a vertical force F. The flexural rigidity of both vertical and horizontal bars is EI. Neglect the axial deflection of the vertical bar, and all effects of self-weight. The vertical deflection at end R is

• A. 7𝐹𝐿3 3𝐸𝐼
• B. 10𝐹𝐿3 3𝐸𝐼
• C. 5𝐹𝐿3 3𝐸𝐼
• D. 8𝐹𝐿3 3𝐸𝐼

Answer: A. 7𝐹𝐿3 3𝐸𝐼

Question 59

Year: 2023 | Type: MSQ | Marks: 2

A uniform cantilever beam has flexural rigidity 𝐸𝐼 and length 𝐿. It is subjected to a concentrated force 𝐹 and moment 𝑀= 2𝐹𝐿 at the free end as shown. The deflection (𝛿) at the free end is

• A. 11𝐹𝐿3 12𝐸𝐼
• B. 8𝐹𝐿3 9𝐸𝐼
• C. 4𝐹𝐿3 3𝐸𝐼
• D. 7𝐹𝐿3 6𝐸𝐼

Answer: C. 4𝐹𝐿3 3𝐸𝐼

Question 60

Year: 2023 | Type: NAT | Marks: 2

A steel ball of mass 𝑚 = 10 kg is suspended from the ceiling of a moving carriage by two inextensible strings making 60° 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 m/s2. The acceleration 𝑎 (in m/s2) is______(rounded off to one decimal place).

Answer: 1.8 to 2.0

Question 61

Year: 2023 | Type: NAT | Marks: 2

A block of mass 𝑚= 10 kg is lying on an inclined plane 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 m/s2. The smallest force 𝐹 (in N) required to prevent the block from sliding down is______(rounded off to one decimal place).

Answer: 23.0 to 25.0

Question 62

Year: 2023 | Type: NAT | Marks: 2

A spherical rigid ball of mass 10 kg is moving with a speed of 2 m/s in the direction shown. The ball collides with a rigid frictionless wall and rebounds at an angle 𝛼 with a speed of 𝑣, as shown. The coefficient of restitution is 0.9. The angle 𝛼 (in degrees) is______(rounded off to one decimal place).

Answer: 57.0 to 58.0

Question 63

Year: 2023 | Type: NAT | Marks: 2

A thin steel plate is loaded in the x-y plane as shown in the figure. Take the Poisson’s ratio of steel to be 0.3 and the modulus of elasticity of steel to be 200 GPa. The strain along the 𝑧-direction is 𝜀𝑧𝑧= −3 × 10−4. The value of 𝜎𝑦𝑦 (in MPa) is______(rounded off to one decimal place).

Answer: 79.5 to 80.5

Question 64

Year: 2023 | Type: NAT | Marks: 2

A composite rod made of steel and copper is fixed immovably at its ends as shown in the figure. The length of each portion of the rod is 1 m as shown. The cross-sections of both portions are the same. The moduli of elasticity of steel and copper are 200 GPa and 100 GPa, respectively. The coefficients of thermal expansion of steel and copper are 12 × 10−6 /℃ and 18 × 10−6 /℃, respectively. The composite rod is initially stress free. Then, the temperature of the composite rod is increased by 100 ℃. The magnitude of axial stress (in MPa) developed in the steel rod is______(rounded off to one decimal place).

Answer: 199.0 to 201.0

Question 65

Year: 2023 | Type: NAT | Marks: 2

A slender uniform elastic rod of length 1 m and of solid circular cross-section of diameter 50 mm is originally straight. It is then loaded by equal and opposite end moments as indicated in the figure. The resulting lateral displacement of the mid-point of the rod is 10 mm (displacements are exaggerated in the figure). The maximum longitudinal strain in the rod is 𝑝 × 10−3, where 𝑝 is______(rounded off to one decimal place).

Answer: 1.9 to 2.1

Question 66

Year: 2023 | Type: NAT | Marks: 2

Consider a solid cylindrical shaft and a hollow cylindrical shaft. Both shafts are axisymmetric and elastic, and have the same cross-sectional area. The hollow shaft has an outside diameter of 150 mm and an inside diameter of 120 mm. When both the shafts are twisted by the same twisting moment, the ratio of maximum shear stress developed in the hollow shaft (𝜏ℎ) to maximum shear stress developed in the solid shaft (𝜏𝑠) will be______(rounded off to three decimal places).

Answer: 0.350 to 0.380

Question 67

Year: 2022 | Marks: 1

A force 𝐹 is applied at an angle 𝜃ൌ30௢ on an elastic column as shown in the figure. 𝐸 and 𝐼 are respectively the Young’s modulus and area moment of inertia. The smallest magnitude of 𝐹 needed to cause buckling is

• A. ଶ √ଷ గమாூ ௅మ
• B. √ଷ ଶ గమாூ ௅మ
• C. గమாூ ଶ௅మ
• D. ଶగమாூ ௅మ

Answer: A. ଶ √ଷ గమாூ ௅మ

Question 68

Year: 2022 | Marks: 1

The shear stress due to a transverse shear force in a linear elastic isotropic beam of rectangular cross-section

• A. varies linearly along the depth in the transverse direction of the beam
• B. is zero at the neutral axis
• C. is maximum at the neutral axis
• D. remains constant along the depth in the transverse direction of the beam

Answer: C. is maximum at the neutral axis

Question 69

Year: 2022 | Marks: 1

A massless semicircular rod held fixed at end A is in the xy-plane, as shown in the figure. A force P along the negative z direction is acting at point B on the rod. The unit vectors along x, y and z directions are denoted respectively as i, j and k. Due to the applied force P, the cross-section of the rod at point D will be subjected to

• A. a twisting moment 𝑃𝑅ሺ1 െcos 𝜃ሻ𝐢, a bending moment 𝑃𝑅sin 𝜃 𝐣, and a shear force െ𝑃 𝐤
• B. a twisting moment 𝑃𝑅ሺ1 െsin 𝜃ሻ𝐢, a bending moment 𝑃𝑅cos 𝜃 𝐣, and a shear force 𝑃 𝐤
• C. a twisting moment 𝑃𝑅ሺcos 𝜃െ1ሻ𝐢, a bending moment െ𝑃𝑅sin 𝜃 𝐣, and a shear force െ𝑃 𝐤
• D. a twisting moment 𝑃𝑅sin 𝜃𝐢, a bending moment 𝑃𝑅ሺ1 െcosሻ𝜃 𝐣, and a shear force 𝑃 𝐤

Answer: A. a twisting moment 𝑃𝑅ሺ1 െcos 𝜃ሻ𝐢, a bending moment 𝑃𝑅sin 𝜃 𝐣, and a shear force െ𝑃 𝐤

Question 70

Year: 2022 | Marks: 1

In the truss shown in the figure, all the members are pin jointed to each other. The members AB, BD, DE and DC have the same length. For the given loading, which of the following is the correct statement?

• A. BD is a zero-force member, and AB and ED are in compression
• B. AB is in tension, ED is in compression, and BD is a zero-force member
• C. AB and DC are in tension, and BC is in compression
• D. ED is in tension, and DC and BC are in compression

Answer: B. AB is in tension, ED is in compression, and BD is a zero-force member

Question 71

Year: 2022 | Marks: 1

End B of the 2 m long rigid rod AB is constrained to move horizontally in the slot as shown in the figure and has a velocity of 1.0 i m/s. The angular velocity of the rod at the instant shown is 2 rad/s. The unit vectors along x and y directions are denoted respectively as i and j. The velocity of point A in m/s is then given by

• A. ൫1 െ2√3൯𝐢൅2𝐣
• B. ൫1 ൅2√3൯𝐢െ2𝐣
• C. െ2√3𝐢൅2𝐣
• D. 2√3𝐢െ2𝐣

Answer: A. ൫1 െ2√3൯𝐢൅2𝐣

Question 72

Year: 2022 | Marks: 1

The assembly of four masses connected by rigid mass-less rods is kept on a smooth horizontal floor as shown in the figure. Under the applied force 2F, the magnitude of angular acceleration of the assembly at the instant shown is

• A. 𝐹 𝑚𝑐
• B. 𝐹 2𝑚𝑐
• C. 2𝐹 3𝑚𝑐
• D. 𝐹 3𝑚𝑐

Answer: B. 𝐹 2𝑚𝑐

Question 73

Year: 2022 | Marks: 1

A particle is constrained to move at a constant speed on an inclined plane (ABCD) along the curved path shown in the figure. Edges AD and BC are parallel to the y axis. The inclined plane makes an angle 𝜃 with the 𝑥𝑦-plane. The velocity vector of the particle makes an angle 𝜙 with the dotted line which is parallel to edge AB. If the speed of the particle is 2 m/s, 𝜙ൌ30°, and 𝜃ൌ40°, then the 𝑧-component of the velocity of the particle in m/s is __________

• A. െ1.32
• B. െ1.00
• C. െ1.11
• D. െ1.50

Answer: C. െ1.11

Question 74

Year: 2022 | Type: NAT | Marks: 1

A uniform elastic rod of constant cross-section is fixed at its left end as shown in the figure. An axial force 𝑃 is acting as shown. Assume that plane sections remain plane during deformation. The ratio of axial displacements at point A ሺ𝑥ൌ4𝐿ሻ to that at point B ሺ𝑥ൌ𝐿ሻ is ______ (rounded off to one decimal place)

Answer: 1.9 to 2.1

Question 75

Year: 2022 | Type: NAT | Marks: 1

A thin-walled spherical pressure vessel has a mean radius of 500 mm and wall thickness of 10 mm. The yield strength of the material is 500 MPa. The internal pressure in MPa at which the spherical pressure vessel will yield according to the Tresca criterion is ____________ (rounded off to one decimal place)

Answer: 19 to 21

Question 76

Year: 2022 | Marks: 2

The beam in the figure is subjected to a moment M0 at mid span as shown. Which of the following is the vertical reaction at B?

• A. 0 9 8 M L
• B. 0 15 8 M L
• C. 0 3 4 M L
• D. 0 9 4 M L

Answer: A. 0 9 8 M L

Question 77

Year: 2022 | Marks: 2

A spring-mass system having a mass m and spring constant k, placed horizontally on a foundation, is connected to a vertically hanging mass m with the help of an inextensible string. Ignore the friction in the pulleys and also the inertia of pulleys, string and spring. Gravity is acting vertically downward as shown. The natural frequency of the system in rad/s is

• A. ඨ4𝑘 3𝑚
• B. ඨ𝑘 2𝑚
• C. ඨ𝑘 3𝑚
• D. ඨ4𝑘 5𝑚

Answer: D. ඨ4𝑘 5𝑚

Question 78

Year: 2022 | Marks: 2

One end of a uniform rigid rod OA of length L and mass m is attached to a frictionless hinge at O. The other end of the rod is connected to the roof at B with a mass-less inextensible thread AB. Initially the rod is horizontal and at rest. The gravity is acting vertically downward as shown. Immediately after the thread AB is cut, the reaction on the rod at O is

• A. ௠௚ ସ in the positive y-direction
• B. ௠௚ ଶ in the negative y-direction
• C. ଷ௠௚ ସ in the negative y-direction
• D. 𝑚𝑔 in the positive y-direction

Answer: A. ௠௚ ସ in the positive y-direction

Question 79

Year: 2022 | Marks: 2

A circular shaft is rigidly connected to a wall at one end. The shaft has a solid portion and a hollow portion as shown in the figure. The length of each portion is 𝐿 and the shear modulus of the material is 𝐺. The polar moment of inertia of the hollow portion is 𝐽 and that of the solid portion is 50𝐽. A torque 𝑇 is applied at the right most end as shown. The rotation of the section PQ is

• A. ଶ଻்௅ ଵ଴଴௃ீ
• B. ௅ ସ଴௃ீ
• C. ହ்௅ ସ௃ீ
• D. ଷ்௅ ସ௃ீ

Answer: A. ଶ଻்௅ ଵ଴଴௃ீ

Question 80

Year: 2022 | Marks: 2

A rectangular plate of uniform thickness having initial length a and width b is placed between two rigid immovable walls. The temperature of the plate is increased by ΔT. The plate is free to expand along the y and z directions. The mid-surface of the plate remains in the xy-plane. The Poisson’s ratio is ν and the coefficient of thermal expansion is α. Assuming that the plate is initially free of stresses, the change in length of the plate after the increase in temperature is given by

• A. (1 ) a T να − Δ
• B. (1 ) a T να + Δ
• C. a T αΔ
• D. 2a T αΔ

Answer: B. (1 ) a T να + Δ

Question 81

Year: 2022 | Marks: 2

A mass m = 10 kg is attached to a spring as shown in the figure. The coefficient of friction between the mass and the inclined plane is 0.25. Assume that the acceleration due to gravity is 10 m/s2 and that static and kinematic friction coefficients are the same. Equilibrium of the mass is impossible if the spring force is

• A. 30 N
• B. 45 N
• C. 60 N
• D. 75 N

Answer: A. 30 N

Question 82

Year: 2022 | Type: NAT | Marks: 2

The frame shown in the figure is subjected to a uniformly distributed load of 1000 N/m over a distance of 0.6 m. Neglecting the weight of the frame, the maximum shear force (in N) in the region between the supports A and B of the frame is ________.

Answer: 295 to 305

Question 83

Year: 2022 | Type: NAT | Marks: 2

A slider moving in a frictionless slot is connected with a linear spring OA as shown in the figure. The following is known: stiffness of the spring = 2 kN/m, mass of the slider = 10 kg, and the unstretched length of the spring = 1 m. If the slider is released from rest at A the magnitude of its velocity (in m/s) when it passes through point B is ____________(rounded off to the nearest integer)

Answer: 39 to 41

Question 84

Year: 2022 | Type: NAT | Marks: 2

A sphere A of mass 𝑚 is thrown into the air at 50 m/s along a direction tanିଵሺ3/4ሻ up from the horizontal. At the topmost point of its trajectory, it has a central (non- oblique) collision with another sphere B which is at rest on top of a vertical pole. Sphere B has a mass of 3𝑚. Acceleration due to gravity is 10 m/s2. Neglect contact friction and air-resistance. If the coefficient of restitution is 0.3, then the speed (in m/s) of sphere A immediately after the collision is_________ (rounded off to one decimal place)

Answer: 0.9 to 1.1

Question 85

Year: 2022 | Type: NAT | Marks: 2

The truck shown in the figure is moving on a horizontal road at a speed of 20 m/s. It is carrying a box of mass 1000 kg, which is simply placed on the truck platform. The coefficient of friction between the truck platform and the box is 0.25. Take acceleration due to gravity as 10 m/s2. Assume uniform deceleration during braking, and the coefficients of static and kinetic friction to be the same. The shortest distance in meters in which the truck can be brought to rest without the box slipping is _______(round off to the nearest integer)

Answer: 79 to 81

Question 86

Year: 2022 | Type: NAT | Marks: 2

The stepped rod of length 2 m, shown in the figure, is fixed at both ends (A and C). The area of cross-section of portion AB is 200 mm2 and that of portion BC is 100 mm2. Force F is applied at section B such that the section is displaced by 0.1 mm in the direction of the force. Young’s modulus of the rod is 200 GPa. The applied force F in N is _______ (round off to the nearest integer)

Answer: 5990 to 6010

Question 87

Year: 2022 | Type: NAT | Marks: 2

A cantilever beam has a span of 1 m and carries a uniformly distributed load of q = 1250 N/m over a portion as shown. A force F =1000 N acts at a distance L from the fixed end. The distance L is such that the bending moment at the fixed end is zero. The beam has a rectangular cross-section of depth 20 mm and width 24 mm. For this loading, the magnitude of the maximum bending stress in the beam in MPa is________________(round off to the nearest integer)

Answer: 123 to 127

Question 88

Year: 2022 | Type: NAT | Marks: 2

The figure shows two identical mass-less beams AB and CD, each clamped at one of their ends. The left end of beam CD rests on the right end of beam AB such that the ends of the beams are just in contact. The beams are unstressed before the application of load. Assume no friction at the contact. Now, if a uniformly distributed load of 800 N/m is applied on beam CD, the bending moment at the end B of beam AB in N.m is ___________ (rounded off to the nearest integer)

Answer: 148 to 152

Question 89

Year: 2018 | Marks: 1

A thin cylinder (thickness, ݐ and diameter, ) with closed ends is subjected to an internal pressure, ݌. The ratio of hoop stress to longitudinal stress developed in the wall of the cylinder is

• A. 1:4
• B. 1:2
• C. 1:1
• D. 2:1

Answer: D. 2:1

Question 90

Year: 2018 | Marks: 1

A steel rod is fixed at one end and free at the other end. The coefficient of thermal expansion of the steel is α, and modulus of elasticity is E. If the temperature of the rod is increased by ∆ܶ then the stress and strain developed in the rod are respectively

• A. zero, α∆ܶ
• B. Eα∆ܶ, α∆ܶ
• C. Eα∆ܶ, zero
• D. zero, zero

Answer: A. zero, α∆ܶ

Question 91

Year: 2018 | Marks: 1

The effective lengths of the columns with ideal boundary conditions shown in Case-I, Case-II, and Case-III are respectively

• A. ܮ, 4ܮ, 2ܮ
• B. ܮ, 2ܮ, ܮ/4
• C. ܮ, ܮ, ܮ
• D. ܮ, 2ܮ, ܮ/2

Answer: D. ܮ, 2ܮ, ܮ/2

Question 92

Year: 2018 | Marks: 1

At a point in a stressed body, sum of the normal stresses acting on perpendicular faces of an arbitrarily oriented plane stress element is always

• A. dependent on the angle of orientation of the element
• B. constant and independent of angle of orientation of the element
• C. one half of the sum of the principal stresses
• D. zero Case-I Case-II Case-III L L L P P P Hinged-Hinged Fixed-Free Fixed-Fixed

Answer: B. constant and independent of angle of orientation of the element

Question 93

Year: 2018 | Marks: 1

The principal stresses on a plane stress element are shown in the figure. The maximum shear stress (in MPa) is

• A. 200
• B. 100
• C. 50
• D. 0

Answer: B. 100

Question 94

Year: 2018 | Type: NAT | Marks: 1

A rigid and thin L-shaped bracket is fixed to the wall at point B, and a force F is applied at point A as shown. For a given force F, the point B experiences the maximum clockwise moment when the inclination θ (in degrees) with the x-axis is……….[up to two decimal places].

Answer: 52.50 to 53.50

Question 95

Year: 2018 | Type: NAT | Marks: 1

A cantilever beam with length, L = 1 m, modulus of elasticity, E = 210 GPa, and area moment of inertia, I = 1.2 ൈ10ି଻m4 carries a concentrated mass m = 100 kg at its free- end. By idealizing it as a single degree-of-freedom system and neglecting the mass of the cantilever beam, the natural frequency (in rad/s) of small transverse oscillations of the mass m is …….. [up to two decimal places]

Answer: 27.00 to 28.00

Question 96

Year: 2018 | Type: NAT | Marks: 1

For a typical grade of steel, the value of modulus of elasticity (E) and Poisson’s ratio (ν) are 208 GPa and 0.3 respectively. The value of shear modulus (G) of the steel (in GPa) is …….. 8 mm P = 20 kN P = 20 kN 50 mm

Answer: 80

Question 97

Year: 2018 | Type: NAT | Marks: 1

A tensile test is performed on a metallic specimen of diameter 8 mm and gauge length 50 mm. When the tensile load P reaches a value of 20 kN, the distance between the gauge marks increases by 0.09 mm. If the sample remains within the elastic limit, the modulus of elasticity (in GPa) of the test metal is………..[up to two decimal places]

Answer: 220.00 to 222.00

Question 98

Year: 2018 | Marks: 2

A beam ABC is subjected to load P at its free end C as shown in the figure. The flexural rigidity of the beam is ܧܫ. The vertical support reaction at point B is

• A. ହ௉ ସ
• B. ହ௉ ଶ
• C. ସ௉ ହ
• D. ଶ௉ ହ ܮ/2 ܮܲ ܣ ܤ ܥ

Answer: B. ହ௉ ଶ

Question 99

Year: 2018 | Marks: 2

A horizontal effort P is applied to raise a block of weight W on a rough surface inclined at an angle θ with the horizontal. If ߤ௦ is the coefficient of static friction between the block and the surface, the minimum effort P required to impend the upward motion of the block along the surface is

• A. ቀ ఓೞି௧௔௡ఏ ଵାఓೞ௧௔௡ఏቁ
• B. ቀ ఓೞା௧௔௡ఏ ଵିఓೞ௧௔௡ఏቁ
• C. ቀ ఓೞି௧௔௡ఏ ଵିఓೞ௧௔௡ఏቁ
• D. ቀ ఓೞା௧௔௡ఏ ଵାఓೞ௧௔௡ఏቁ

Answer: B. ቀ ఓೞା௧௔௡ఏ ଵିఓೞ௧௔௡ఏቁ

Question 100

Year: 2018 | Marks: 2

Three round bars of same material, equal lengths, and different cross-sectional dimensions are shown in the figures as Case-I, Case-II and Case-III. All the bars are clamped at the upper end, and a concentrated load P is applied at the lower end of each bar. If the elastic strain energy stored in the bar shown in Case-I is ଵ then the elastic strain energy stored in Case-II and Case-III respectively is

• A. ቀ8 5 ൗቁܷ ଵ, ቀ5 8 ൗቁܷ ଵ
• B. ቀ5 16 ൗ ቁܷ ଵ, ቀ5 16 ൗ ቁܷ ଵ
• C. ቀ5 8 ൗቁܷ ଵ, ቀ8 5 ൗቁܷ ଵ
• D. ቀ5 8 ൗቁܷ ଵ, ቀ5 8 ൗቁܷ ଵ Case-I Case-II Case-III 2d L d P 2d d L/4 L/2 L/4 d 2d d P P L/4 L/2 L/4

Answer: D. ቀ5 8 ൗቁܷ ଵ, ቀ5 8 ൗቁܷ ଵ Case-I Case-II Case-III 2d L d P 2d d L/4 L/2 L/4 d 2d d P P L/4 L/2 L/4

Question 101

Year: 2018 | Marks: 2

A rigid uniform rod with mass m, length L and center of gravity G is freely suspended from a hinge as shown in the figure. The rod is given a small angular displacement θ in the counter-clockwise direction from the position in which it hangs vertically (θ = 0). If ݃ is the acceleration due to gravity, the natural frequency of oscillations (in rad/s) is

• A. ට ଺௚ ௅
• B. ට ଶ௚ ௅
• C. ට ଷ௚ ଶ௅
• D. ට ௚ ௅

Answer: C. ට ଷ௚ ଶ௅

Question 102

Year: 2018 | Marks: 2

A cylindrical member, made up of ductile material, is subjected to pure torsion as shown in the figure. The failure plane (from Option-I to Option-IV) for ductile material as per maximum shear stress theory is represented by

• A. Option I
• B. Option II
• C. Option III
• D. Option IV 45° 45° Option-I 30° T T Option-II Option-III Option-IV 90°

Answer: C. Option III

Question 103

Year: 2018 | Marks: 2

A simply supported beam of span L is subjected to a couple Mo at a distance a from support A. Among the four options (Option-I to Option-IV) shown, the correct shear force diagram of the beam is

• A. Option-I
• B. Option-II
• C. Option-III
• D. Option-IV

Answer: A. Option-I

Question 104

Year: 2018 | Marks: 2

A circular shaft ABC of diameter, ݀ and length, ܮ is fixed at end A. It is subjected to the torsional moments at point B and point C as shown in the figure. The ratio of angle of twists at point B to point C is

• A. 1.5 : 1
• B. 1 : 3
• C. 1 : 2
• D. 1 : 1.5 Mo/L Option-IV Mo/L Mo/L Mo a /L Mo (a/L-1) Option-I Option-II Mo L a A B Mo/L Option-III ܮ ܮ/2 ܣ ܤ ܥ + Sign convention of shear force for all options (+) (+) (-) (+) (-) (+)

Answer: D. 1 : 1.5 Mo/L Option-IV Mo/L Mo/L Mo a /L Mo (a/L-1) Option-I Option-II Mo L a A B Mo/L Option-III ܮ ܮ/2 ܣ ܤ ܥ + Sign convention of shear force for all options (+) (+) (-) (+) (-) (+)

Question 105

Year: 2018 | Type: NAT | Marks: 2

A circular steel bar of diameter 10 mm is bent into the shape as shown in figure, and lies in x-z plane. A horizontal force P is applied along the positive z-direction as shown. The yield strength of the steel is 200 MPa. Neglecting the effect of transverse shear, the load P (in Newton) required to initiate yielding as per maximum shear stress theory of failure is…………[up to two decimal places]

Answer: 195.00 to 197.00

Question 106

Year: 2018 | Type: NAT | Marks: 2

Two sliders A and B, connected by a rigid link of length L, slide in two mutually perpendicular and frictionless guide-ways. At a particular instance, the slider A is moving in the downward direction with a speed of 0.05 m/s. At this instance, the magnitude of the velocity of slider B (in m/s) is ……[up to two decimal places]

Answer: 0.01 to 0.02

Question 107

Year: 2018 | Type: NAT | Marks: 2

A simply supported beam ABC is subjected to load as shown in the figure. The 20 kN load is applied at point B with the help of a welded bracket as shown. The beam has a rectangular cross-section of 15 mm width and 100 mm depth as shown. The maximum transverse shear stress developed in the beam (in MPa) is ……..[up to one decimal place] ݖ ݕ 100 50 ݔ Enlarged view of the cross-section 1 m 20݇N ܣ ܤ ܥ 1 m 0.25 m 5݇N/m 0.5 m 0.5 m 100 mm 15 mm

Answer: 14.5 to 15.5

Question 108

Year: 2018 | Type: NAT | Marks: 2

A rigid rod ABC, in the form of a quarter-circular arc of radius R, is hinged at C and supported by a roller at B. A vertical force P is applied at the end A of the bar. For the reactions at B and C to be equal in magnitude, the value of the angle θ (in degrees) is ………. [up to two decimal places].

Answer: 26.00 to 27.00

Question 109

Year: 2018 | Type: NAT | Marks: 2

A marble of mass m slides along a frictionless linear slide AB kept in a vertical plane. The marble is released from point A with zero initial velocity, and it reaches point B under the action of gravity. Assuming the acceleration due to gravity to be 9.81 m/s2, the speed (in m/s) of the marble when it just reaches point B is………[up to two decimal places].

Answer: 3.10 to 3.20

Question 110

Year: 2018 | Type: NAT | Marks: 2

A horizontal force 10 N is applied at support B on the frame as shown in figure. Considering only bending deformation, the vertical reaction (in N) at support B is ………. [up to one decimal place] ܴ ܣ ܤ 10 N

Answer: 13.5 to 14.5

Question 111

Year: 2016 | Marks: 1

A single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 N-s/m. To make the system critically damped

• A. only the mass is to be increased by 1.2 times.
• B. only the stiffness is to be reduced to half.
• C. only the damping coefficient is to be doubled.
• D. no change in any of the system parameters is required.

Answer: D. no change in any of the system parameters is required.

Question 112

Year: 2016 | Marks: 1

A “L” shaped robotic arm AB is connected to a motor at end A and a magnetic gripper at B as shown in the figure. If the arm is rotating with an angular velocity of 2 rad/s and an angular acceleration of 3 rad/s2, the magnitude of the acceleration (in m/s2) of the end B is

• A. 2.0
• B. 2.5
• C. 5.0
• D. 6.0

Answer: B. 2.5

Question 113

Year: 2016 | Type: NAT | Marks: 1

A block of weight 100 N is in static equilibrium on an inclined plane which makes an angle 15º with the horizontal. The coefficient of friction between the inclined plane and the block is 0.3. The magnitude of friction force (in N) acting on the block is __________ 0.3 m 0.4 m A B 900 2 rad/s 3 rad/s2 15º 100 N

Answer: 25.5 to 26.0

Question 114

Year: 2016 | Marks: 1

The lower end A of the rigid bar AB is moving horizontally on the floor towards right with a constant velocity of 5 m/s and the point B is sliding down the wall. The magnitude of the velocity of point B at the instant θ=30º is

• A. zero
• B. 4.34 m/s
• C. 7.25 m/s
• D. 8.66 m/s

Answer: D. 8.66 m/s

Question 115

Year: 2016 | Type: NAT | Marks: 1

The state of plane stress at a point in a body is shown in the figure. The allowable shear stress of the material of the body is 200 MPa. According to the maximum shear stress theory of failure the maximum permissible value of σ (in MPa) is __________

Answer: 200

Question 116

Year: 2016 | Marks: 1

For a slender steel column of circular cross-section the critical buckling load is cr P . If the diameter of the column is doubled (keeping other material and geometrical parameters same), then the critical buckling load of the column is

• A. cr P /16
• B. 8 cr P
• C. 2 cr P
• D. 16 cr P

Answer: D. 16 cr P

Question 117

Year: 2016 | Type: NAT | Marks: 1

A closed thin cylindrical pressure vessel having an internal diameter of 1000 mm and a thickness of 10 mm is subjected to an internal pressure of 4 MPa. The maximum shear stress (in MPa) induced in the cylinder is __________ (neglect the radial stress).

Answer: 100

Question 118

Year: 2016 | Type: NAT | Marks: 1

A solid circular shaft subjected to pure torsion develops a maximum torsional shear stress of 120MPa. Keeping the torsional moment same, if the diameter of the shaft is doubled then the maximum shear stress (in MPa) induced in the shaft is __________ A B θ Floor Wall

Answer: 15

Question 119

Year: 2016 | Type: NAT | Marks: 1

On a single straight track, a vehicle of mass 500 kg moving with a velocity of 25 m/s strikes another vehicle of mass 250 kg moving with a velocity 10 m/s in the same direction. After the impact, if both the vehicles stick together, the common velocity (in m/s) with which both the vehicles will move together is __________

Answer: 20

Question 120

Year: 2016 | Type: NAT | Marks: 2

A system with three forces and a concentrated moment at A is shown in the figure. The system is replaced by an equivalent force system with a single force and a single couple at point ‘O’. The magnitude (in N-m) of the equivalent couple at ‘O’ is __________

Answer: 150

Question 121

Year: 2016 | Type: NAT | Marks: 2

A block A on a smooth inclined plane is connected to block B as shown in the figure using an inextensible cord which pass over a mass-less and friction-less pulley. Initially, the block B is constrained to be at rest. If the constraint on block B is released, the magnitude of velocity (in m/s) of the block ‘B’ after 2 seconds from its release is __________ (assume g =10 m/s2). 200 N 30o O 100 N 50 N-m 2 m 2 m 50 N 4 m A 1 m A B 30º 10 kg 5 kg

Answer: 10

Question 122

Year: 2016 | Marks: 2

The vibrating system shown in the figure carries a mass of 10 kg at the free end, where the static deflection is 1 mm. This system is to be replaced by an equivalent vibrating spring mass system having equivalent mass of 2 kg (assume g =10 m/s2). The natural frequency (in rad/s) and the stiffness (in kN/m) of the equivalent system respectively are

• A. 10 and 20
• B. 20 and 100
• C. 100 and 20
• D. 1000 and 20

Answer: C. 100 and 20

Question 123

Year: 2016 | Marks: 2

A beam having flexural rigidity EI and length L is subjected to a concentrated end moment 0 M as shown in the figure. For 3 2 4 10 N-m EI =× , 1m L = and 0 8kN-m M = , the strain energy stored (in kN-m) in the beam and the rotation (in rad) at the free end respectively are

• A. 8.00 and 0.02
• B. 8.00 and 2.00
• C. 8.00 and 0.04
• D. 0.80 and 2.00 2 m 1 m L M0

Answer: B. 8.00 and 2.00

Question 124

Year: 2016 | Marks: 2

At a point ‘O’ on a metal sheet a square OABC of a unit side length is drawn. The square undergoes a small uniform elastic deformation and deforms to OA*B*C* (dashed lines) as shown in the figure. All dimensions are in mm and the figure is not to scale. The normal strains xε, y ε and shear strain xy γ developed in the square respectively are

• A. 0.0020, 0.0025 and 0.0020 −
• B. 0.0020, 0.0025 and 0.0020 − −
• C. 0.0025, 0.0020 and 0.0020 −
• D. 0.0020, 0.0025 and 0.0020 − −

Answer: D. 0.0020, 0.0025 and 0.0020 − −

Question 125

Year: 2016 | Marks: 2

Mohr’s circle for the state of plane stress at a point is shown in the figure. Unit of stress is MPa and the circle is drawn not to scale. Which one of the following options (stress values in MPa) is true? (normal stress) (shear stress) A B C O O – Origin of – axis C – Center of Mohr’s circle 20 30 40 ( A, A) ( , ) ( , 0) ( , 0)

• A. 1 2 50, 10, 30, 70 A B σ σ σ σ =− = = =−
• B. 1 2 50, 20, 30, 50 A B σ σ σ σ =− = = =−
• C. 1 2 30, 30, 30, 10 A B σ σ σ σ =− = = =−
• D. 1 2 20, 10, 50, 30 A B σ σ σ σ =− = = =−

Answer: A. 1 2 50, 10, 30, 70 A B σ σ σ σ =− = = =−

Question 126

Year: 2016 | Type: NAT | Marks: 2

As shown in the figure, links AB and CD support the rigid member BD. Links AB and CD are made of aluminum alloy( ) 100GPa E = and each has a cross-sectional area of 2 100mm . All the members are pin connected and all the dimensions are in mm. Neglecting the weights of the members, the elongation (in mm) of the link AB is __________ A B C D 6 kN 400 400 100 200

Answer: 0.16

Question 127

Year: 2016 | Type: NAT | Marks: 2

Figure shows an elastic beam of constant flexural rigidity EI and length . L The transverse deflection ( ) v x for the beam is represented by the equation ( ) 3 2 0 ( ) (4 ), v x M x x L EI L = − where 0 M is the applied couple. If 100mm L = and 0 100 N-mm M = , then the magnitude of the shear force (in N) at the middle of the beam (at / 2) x L = is __________

Answer: 1.5

Question 128

Year: 2016 | Marks: 2

Which one of the following represents the correct bending moment diagram of the beam PQR loaded as shown in the figure?

• A.
• B.
• C.
• D.

Answer: B.

Question 129

Year: 2016 | Marks: 2

A point in a body is subjected to plane state of stress in XY plane. If 140MPa, x σ= 60MPa y σ = and the major principal stress is 150 MPa, the magnitude of the in-plane shear stress xy τ (in MPa) is

• A. 75
• B. 30
• C. 40
• D. 70

Answer: B. 30

Question 130

Year: 2016 | Type: NAT | Marks: 2

A 40 mm diameter rotor shaft of a helicopter transmits a torque T = 0.16 π kN-m and a tensile force P = 24 π kN. The maximum tensile stress (in MPa) induced in the shaft is __________. Use the value of π = 3.1416. T T P P

Answer: 80

Question 131

Year: 2016 | Type: NAT | Marks: 2

A wooden block of length 400 mm, width 50 mm and depth 100 mm is subjected to uniaxial load as shown in the figure. An inclined plane ABCD is shown which makes an angle θ with the XZ plane and the line CD is parallel to the Z-axis. The normal stress on the plane ABCD is 1 n σ when o θ 30 = and the normal stress on the plane ABCD is 2 n σ when o 120 θ= . The value of 2 1 n n σ σ is __________ 10 kN X 10 kN θ A C D B Y Z 400 mm

Answer: 3

Question 132

Year: 2016 | Marks: 2

For the truss shown in the figure, which one of the following statements is true?

• A. AG is the only zero force member.
• B. AG and BH are the only two zero force members.
• C. AG, BH and HF are zero force members.
• D. AG, BH, HF and GC are zero force members. 9 k N 12 k N D F E 3 m 4 m 4 m 4 m A B C G H

Answer: C. AG, BH and HF are zero force members.

Question 133

Year: 2014 | Type: NAT | Marks: 1

A steel wire of diameter 5 mm is bent around a cylindrical drum of radius 0.5 m. The steel wire has modulus of elasticity of 200 GPa. Find the bending moment in the wire in N-m.

Answer: 12.20 to 12.22

Question 134

Year: 2014 | Type: NAT | Marks: 1

A compressed air tank having an inner diameter of 480 mm and a wall thickness of 8 mm is formed by welding two steel hemispheres. If the allowable shear stress in the steel is 40 MPa, find the maximum permissible pressure (in MPa) inside the tank.

Answer: 5.3 to 5.4

Question 135

Year: 2014 | Marks: 1

A point in a body is subjected to a bi-axial state of stress, equal in magnitude but opposite in nature. On a plane inclined at an angle 45° with respect to x-axis (passing through the point), the

• A. shear and normal stresses are zero
• B. normal stress is maximum and shear stress is zero
• C. shear stress is maximum and normal stress is zero
• D. shear stress is maximum and normal stress is non-zero

Answer: C. shear stress is maximum and normal stress is zero

Question 136

Year: 2014 | Marks: 1

The Euler’s buckling load of a column fixed at both the ends is P. If one of the ends is made free, the buckling load shall change to

• A. P/16
• B. P/8
• C. P/4
• D. P/2 R = 0.5 m d = 5 mm XE (GATE 2014)

Answer: A. P/16

Question 137

Year: 2014 | Type: NAT | Marks: 1

For the pin jointed truss, find the axial force (in kN) in the member 2-5.

Answer: 0

Question 138

Year: 2014 | Marks: 1

A weightless beam subjected to two point loads is shown in the figure below. The shear force diagram of the beam is

• A.
• B.
• C.
• D. 10 kN 10 kN 2 m 1 m 1 m 10 kN 10 kN 10 kN 10 kN 5 kN 5 kN 5 kN 10 kN XE (GATE 2014)

Answer: C.

Question 139

Year: 2014 | Marks: 1

The supporting structure of a water tank is made of reinforced concrete (RC) with a tubular cross section of inner diameter di, outer diameter do, height l, and Young’s modulus E. The mass of the tank is m. If mass of the supporting structure is neglected, then the natural frequency of the water tank in transverse direction is

• A. ට3𝜋𝜋𝜋𝜋൫𝑑𝑑𝑜𝑜 − 4 𝑑𝑑𝑖𝑖 4൯ 64𝑙𝑙3𝑚𝑚
• B. ට𝜋𝜋𝜋𝜋൫𝑑𝑑𝑜𝑜 − 4 𝑑𝑑𝑖𝑖 4൯ 8𝑙𝑙3𝑚𝑚
• C. ) ට384𝜋𝜋𝜋𝜋൫𝑑𝑑𝑜𝑜 − 4 𝑑𝑑𝑖𝑖 4൯ 360𝑙𝑙3𝑚𝑚
• D. ට𝜋𝜋𝜋𝜋൫𝑑𝑑𝑜𝑜 − 4 𝑑𝑑𝑖𝑖 4൯ 64𝑙𝑙3𝑚𝑚

Answer: A. ට3𝜋𝜋𝜋𝜋൫𝑑𝑑𝑜𝑜 − 4 𝑑𝑑𝑖𝑖 4൯ 64𝑙𝑙3𝑚𝑚

Question 140

Year: 2014 | Marks: 1

A mass is attached to a spring and placed horizontally in a frictionless surface. A simple pendulum has been pivoted to the mass. The degree of freedom of this system is

• A. 1
• B. 2
• C. 3
• D. 4 X y x l XE (GATE 2014)

Answer: B. 2

Question 141

Year: 2014 | Marks: 1

Consider the following two statements Statement 1: A body of weight W falls from a height h and strikes the ground. If the body starts from rest, the velocity with which it strikes the ground is ඥ2𝑔𝑔ℎ , where g is the acceleration due to gravity. Statement 2: If the same body (initially at rest) slides without friction along an inclined plane PQ (angle of inclination α) starting from an elevation h above point Q, then its velocity at point Q is ඥ2𝑔𝑔ℎ The correct option is

• A. Both statements 1 and 2 are true
• B. Statement 1 is true and 2 is false
• C. Statement 1 is false and 2 is true
• D. Both statements 1 and 2 are false W h h P Q W α XE (GATE 2014)

Answer: A. Both statements 1 and 2 are true

Question 142

Year: 2014 | Type: NAT | Marks: 2

A composite bar of length ‘L’ is made of a centrally placed steel plate (50 mm wide x 10 mm thick) with two copper plates (each 30 mm wide x 5 mm thick) connected rigidly on each side. If the temperature of the composite bar is raised by 50°C, find the stress developed in each copper plate in MPa. (For Steel: Es = 2x105 MPa and αs= 12x10-6 /°C; For Copper: Ec = 1x105 MPa and αc= 17x10-6 /°C)

Answer: 19.1 to 19.3

Question 143

Year: 2014 | Marks: 2

The vertical deflection at the free end of the cantilever beam as shown in figure is

• A. 1400/EI
• B. 1400/3EI
• C. 200/EI
• D. 100/EI

Answer: B. 1400/3EI

Question 144

Year: 2014 | Type: NAT | Marks: 2

A hollow shaft and a solid shaft have the same length and the same outer radius R. The inner radius of the hollow shaft is 0.6 R. Assuming that both the shafts are made of same material and are subjected to the same torque, find the ratio of shear stress in hollow shaft to that in solid shaft.

Answer: 1.14 to 1.16

Question 145

Year: 2014 | Marks: 2

A beam with overhangs carries one point load acting downwards and the other upward. The clockwise moment Pb is applied at each support. The bending moment at the midpoint of the beam is

• A. 0
• B. PL/2
• C. PL
• D. PbL L Steel plate 5 mm 5 mm Copper plate Copper plate 10 mm 100 kN EI = flexural rigidity 1 m 2 m L P P b b Pb Pb XE (GATE 2014)

Answer: A. 0

Question 146

Year: 2014 | Marks: 2

A cantilever beam of length L supports a concentrated load P at the free end. The cross section of the beam is rectangular with constant width b and varying depth h. The depth h of this idealized cantilever beam varies in such a way that the maximum normal stress at every cross section remains equal to the allowable bending stress. Considering only the bending stresses, the depth hx of the fully stressed beam at any distance x from the free end shall vary

• A. with square of x
• B. with square root of x
• C. linearly with x
• D. with cube of x

Answer: B. with square root of x

Question 147

Year: 2014 | Marks: 2

A cantilever beam is subjected to following three different loading conditions: (a) a concentrated load P at its free end, (b) a couple Mo at its free end and (c) both loads acting simultaneously (a) (b) (c) The flexural rigidity of the beam may be assumed as EI. The strain energy due to bending when both loads act simultaneously

• A. can be determined by applying the principle of superposition and the strain energy is 𝑃𝑃2𝐿𝐿3 6𝐸𝐸𝐸𝐸+ 𝑀𝑀𝑜𝑜2𝐿𝐿 2𝐸𝐸𝐸𝐸
• B. can be determined by applying the principle of superposition and the strain energy is 𝑃𝑃2𝐿𝐿2 6𝐸𝐸𝐸𝐸+ 𝑀𝑀𝑜𝑜𝐿𝐿3 2𝐸𝐸𝐸𝐸
• C. cannot be determined by applying the principle of superposition and the strain energy is 𝑃𝑃2𝐿𝐿3 6𝐸𝐸𝐸𝐸+ 𝑀𝑀𝑜𝑜2𝐿𝐿 2𝐸𝐸𝐸𝐸+ 𝑃𝑃𝑀𝑀𝑜𝑜 𝐿𝐿2 2𝐸𝐸𝐸𝐸
• D. cannot be determined by applying the principle of superposition and the strain energy is 𝑃𝑃2𝐿𝐿2 6𝐸𝐸𝐸𝐸+ 𝑀𝑀𝑜𝑜𝐿𝐿3 2𝐸𝐸𝐸𝐸+ 𝑃𝑃𝑀𝑀𝑜𝑜 𝐿𝐿2 2𝐸𝐸𝐸𝐸 P L P L Mo L Mo L P XE (GATE 2014)

Answer: C. cannot be determined by applying the principle of superposition and the strain energy is 𝑃𝑃2𝐿𝐿3 6𝐸𝐸𝐸𝐸+ 𝑀𝑀𝑜𝑜2𝐿𝐿 2𝐸𝐸𝐸𝐸+ 𝑃𝑃𝑀𝑀𝑜𝑜 𝐿𝐿2 2𝐸𝐸𝐸𝐸

Question 148

Year: 2014 | Marks: 2

A tapered rod has diameter d1 at one end which reduces uniformly to a diameter d2 over the length (L). If the modulus of elasticity of the material is E, the change in the length of the rod due to the application of axial force (P) is

• A. 4𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋𝑑𝑑1𝑑𝑑2
• B. 4𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋൫𝑑𝑑12−𝑑𝑑22൯
• C. 𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋𝑑𝑑1𝑑𝑑2
• D. 2𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋൫𝑑𝑑12−𝑑𝑑22൯

Answer: A. 4𝑃𝑃𝑃𝑃 𝜋𝜋𝜋𝜋𝑑𝑑1𝑑𝑑2

Question 149

Year: 2014 | Type: NAT | Marks: 2

For a point in a body subjected to a plane stress condition (σx= 100 MPa, σy= 50 MPa and τxy = τyx = 25 MPa), the maximum principal stress in MPa is_____________

Answer: 110.3 to 110.5

Question 150

Year: 2014 | Marks: 2

An isotropic body is subjected to a state of stress given by: σx= 10 MPa and τxy = τyx = −20 MPa. Assuming G = 0.4E, the volumetric strain is

• A. 5/E
• B. 7.5/E
• C. 10/E
• D. 15/E

Answer: A. 5/E

Question 151

Year: 2014 | Marks: 2

A block of weight Q rests on an inclined plane and it is attached to a string which runs over a frictionless pulley to carry a block of weight P at its other end. The coefficient of friction between the block of weight Q and the inclined plane is µ. Consider the following cases: Case I: weight Q starts moving down the inclined plane Case II: weight P starts falling down The limiting values of ratio P/Q for Case I and Case II respectively are

• A. (sin 𝛼𝛼−𝜇𝜇cos𝛼𝛼) and (sin𝛼𝛼+ 𝜇𝜇cos 𝛼𝛼)
• B. (𝜇𝜇sin 𝛼𝛼−cos 𝛼𝛼) and (𝜇𝜇sin 𝛼𝛼+ cos 𝛼𝛼)
• C. (sin 𝛼𝛼+ 𝜇𝜇cos𝛼𝛼) and (sin𝛼𝛼−𝜇𝜇cos 𝛼𝛼)
• D. (𝜇𝜇sin 𝛼𝛼+ cos 𝛼𝛼) and 𝜇𝜇(sin 𝛼𝛼−cos 𝛼𝛼) XE (GATE 2014)

Answer: A. (sin 𝛼𝛼−𝜇𝜇cos𝛼𝛼) and (sin𝛼𝛼+ 𝜇𝜇cos 𝛼𝛼)

Question 152

Year: 2014 | Marks: 2

To unload an item from a truck a crane boom is raised with a constant angular velocity of 1 rad/s relative to the cab and then the cab is rotated about a vertical axis with constant angular velocity of 0.5 rad/s. If the length of the boom (OP) is 10 m, the velocity of the tip (P) of the boom in m/s is

• A. √2 5 ൫−2𝑖𝑖 ො− 𝑗𝑗̂ + 2𝑘𝑘෠൯
• B. √2 5 ൫−𝑖𝑖 ො− 2𝑗𝑗̂ + 𝑘𝑘෠൯
• C. 5 √2 ൫−2𝑖𝑖 ො− 𝑗𝑗̂ + 2𝑘𝑘෠൯
• D. 5 √2 ൫−𝑖𝑖 ො−2 𝑗𝑗̂ + 2𝑘𝑘෠൯

Answer: D. 5 √2 ൫−𝑖𝑖 ො−2 𝑗𝑗̂ + 2𝑘𝑘෠൯

Question 153

Year: 2014 | Marks: 2

A block of mass 5 kg moves up on a smooth inclined plane with a velocity of 10 m/s in the direction shown. A bullet of mass 60 g travelling at 500 m/s strikes the block centrally and gets embedded in it. The velocity of the block and embedded bullet in m/s immediately after the impact is

• A. 12.54 at 30°
• B. 13.84 at 51.78°
• C. 13.84 at 30°
• D. 15.62 at 51.78° Z X Y Cab Boom 45° O P 30° x y 500 m/s 10 m/s XE (GATE 2014)

Answer: B. 13.84 at 51.78°

Question 154

Year: 2014 | Type: NAT | Marks: 2

A balloon with ballast (weight) inside it has a gross weight of 500 N. It is falling vertically with a constant acceleration of 2 m/s2. If air resistance is negligible, find the weight of ballast (in N) that must be thrown out in order to give the balloon an upward acceleration of 2 m/s2. (Acceleration due to gravity, g = 9.81 m/s2) 500 N XE (GATE 2014)

Answer: 169.3 to 169.4

Question 155

Year: 2007 | Marks: 1

A simply supported beam AB is subjected to a horizontal force, $P$, as shown in the figure. Shear force at section $a$-$a$ is given by

• A. $Ph/L$
• B. $3Ph/(2L)$
• C. $3Ph/L$
• D. $PL/h$

Question 156

Year: 2007 | Marks: 1

The two-dimensional state of stress at a point is given by the figure (10 MPa biaxial tension in $x$ and $y$). The corresponding Mohr's circle is

• A. a circle of radius $10\text{ MPa}$ with center at $10\text{ MPa}$ on the $\sigma$-axis
• B. a circle of radius $10\text{ MPa}$ with center at $-10\text{ MPa}$ on the $\sigma$-axis
• C. a point on the positive $\sigma$-axis
• D. a point on the negative $\sigma$-axis

Question 157

Year: 2007 | Marks: 1

The kind of fracture surface observed in a brittle specimen that fails under a uniaxial tensile test is

• A. cup and cone
• B. flat rough surface perpendicular to the axis of the specimen
• C. helical rough surface with a $45^\circ$ inclination to the axis of the specimen
• D. flat rough surface with a $45^\circ$ inclination to the axis of the specimen

Question 158

Year: 2007 | Marks: 1

Two masses are attached to the two ends of a cable which passes over two frictionless pulleys as shown in the figure. (acceleration due to gravity $= g$). The acceleration of the $100\text{ kg}$ mass is

• A. $g$
• B. $g/2$
• C. $g/3$
• D. $g/4$

Question 159

Year: 2007 | Marks: 1

A block of mass $m$ is released at rest from a point P of a rough circular path of radius $r$ as shown in the figure. There is a spring of stiffness $k$ at the other end of the path. (acceleration due to gravity $= g$). The maximum spring deflection is

• A. independent of $r$
• B. more than $\sqrt{2mgr/k}$
• C. equal to $\sqrt{2mgr/k}$
• D. less than $\sqrt{2mgr/k}$

Question 160

Year: 2007 | Marks: 1

The truss shown in the figure is

• A. stable and indeterminate
• B. stable and determinate
• C. unstable and indeterminate
• D. unstable and determinate

Question 161

Year: 2007 | Marks: 2

A mass $m_1$ with velocity $v_1$ impacts with a mass $m_2$ at rest. After the impact, the mass $m_1$ comes to rest. Then the coefficient of restitution $e$ should be

• A. $e = \frac{m_1}{m_1 + m_2}$
• B. $e = \frac{m_2}{m_1 + m_2}$
• C. $e = \frac{m_1}{m_2}$
• D. $e = \frac{m_2}{m_1}$

Question 162

Year: 2007 | Marks: 2

A beam ABCD with simple supports at A, B and D and an internal hinge at C is subjected to loads as shown in the figure. The reaction at middle support is given by

• A. 7.5 kN
• B. 10 kN
• C. 20 kN
• D. 22.5 kN

Question 163

Year: 2007 | Marks: 2

A bar of cross sectional area, $A$, is loaded as shown in the figure. The normal stress on a section $a$-$a$ inclined at $45^\circ$ to the axis of the bar is

• A. tensile with a magnitude of $3P/2A$
• B. compressive with a magnitude of $3P/2A$
• C. tensile with a magnitude of $P/2A$
• D. compressive with a magnitude of $P/2A$

Question 164

Year: 2007 | Marks: 2

In a two-dimensional state of stress at a point, the maximum shear strain is found to be $\gamma_{\max} = 500\text{ }\mu\text{m/m}$. The sum of the normal stresses on two perpendicular planes at that point is $40\text{ MPa}$. (Young's Modulus $E = 200\text{ GPa}$; Modulus of rigidity $G = 80\text{ GPa}$; Poisson's ratio $\nu = 0.25$). The principal stresses at the point are

• A. $\sigma_1 = 20\text{ MPa}$ and $\sigma_2 = 20\text{ MPa}$
• B. $\sigma_1 = 60\text{ MPa}$ and $\sigma_2 = -20\text{ MPa}$
• C. $\sigma_1 = 50\text{ MPa}$ and $\sigma_2 = -10\text{ MPa}$
• D. $\sigma_1 = 80\text{ MPa}$ and $\sigma_2 = -40\text{ MPa}$

Question 165

Year: 2007 | Marks: 2

A mass of $2\text{ kg}$ is hung by a string of length $1.5\text{ m}$ as shown in the figure. The mass revolves in a horizontal circular path at $50$ revolutions per minute. The tension in the string in Newtons is

• A. $25\pi/3$
• B. $25\pi^2/3$
• C. $50\pi/3$
• D. $50\pi^2/3$

Question 166

Year: 2007 | Marks: 2

A simply supported beam of span $L$ is subjected to a concentrated load $P$ at midspan. The cross-section of the beam is rectangular. The ratio of the shear stress at $1/4\text{th}$ the depth of the beam to the shear stress at the center of the cross-section is

• A. 3/8
• B. 1/2
• C. 3/4
• D. 2

Question 167

Year: 2007 | Marks: 2

An L-shaped bar of square cross-section with sides, $b$, is loaded as shown in the figure. If the value of the stress component $\sigma_{xx}$ at point P is zero, the distance $a$ of the force $F$ from the $x$-axis should be

• A. $b$
• B. $b/2$
• C. $b/3$
• D. $b/6$

Question 168

Year: 2007 | Marks: 2

The rigid links PQ and PR of length $L$ each are connected to 3 sliders P, Q and R, by pin-joints as shown in the figure. The sliders, Q, and R, slide in the $y$-direction while the slider, P, slides in the $x$-direction. When the slider, P, is at the position $x = 40\text{ mm}$, its velocity is $30\text{ mm/s}$ along the positive $x$-direction. At that instant, the velocity of slider, Q, will be ($L = 50\text{ mm}$)

• A. $40\text{ mm/s}$ along the positive $y$-direction
• B. $40\text{ mm/s}$ along the negative $y$-direction
• C. $22.5\text{ mm/s}$ along the positive $y$-direction
• D. $22.5\text{ mm/s}$ along the negative $y$-direction

Question 169

Year: 2007 | Marks: 2

An open ended thin-walled straight pipe is made of a material that can carry a maximum shear stress of $\tau_{\max}$. The pipe is of diameter, $d$, and thickness, $t$. The maximum internal pressure allowable is given by (neglecting the normal stress in the radial direction)

• A. $\left(\frac{t}{d}\right)\tau_{\max}$
• B. $2\left(\frac{t}{d}\right)\tau_{\max}$
• C. $\sqrt{2}\left(\frac{t}{d}\right)\tau_{\max}$
• D. $4\left(\frac{t}{d}\right)\tau_{\max}$

Question 170

Year: 2007 | Marks: 2

A stepped shaft PQR is fixed at both the ends as shown in the figure. A torque $T$ is applied at point Q. The polar moments of inertia of the shafts PQ and QR are $J_1$ and $J_2$ respectively. $G$ is the modulus of rigidity. The angle of twist at point Q due to torque, $T$, is given by (in radians)

• A. $\frac{2}{J_1}\left(\frac{TL}{G}\right)$
• B. $\frac{1}{J_2}\left(\frac{TL}{G}\right)$
• C. $\left(\frac{2}{J_1} + \frac{1}{J_2}\right)\left(\frac{TL}{G}\right)$
• D. $\left(\frac{2}{J_1 + 2J_2}\right)\left(\frac{TL}{G}\right)$

Question 171

Year: 2007 | Marks: 2

Group I contains beams with different types of supports and loading conditions (P: Simply supported beam with central point load $F$; Q: Simply supported beam with UDL $w=F/L$; R: Cantilever beam with tip point load $F$; S: Cantilever beam with UDL $w=F/L$). The beams have the same flexural rigidity $EI$ and span $L$. Group II contains the maximum deflections: (1) $\frac{1}{3}\left(\frac{FL^3}{EI}\right)$, (2) $\frac{1}{8}\left(\frac{FL^3}{EI}\right)$, (3) $\frac{1}{48}\left(\frac{FL^3}{EI}\right)$, (4) $\frac{5}{384}\left(\frac{FL^3}{EI}\right)$. Match the beam from Group I with the maximum deflection given in Group II.

• A. P-3, Q-4, R-1, S-2
• B. P-3, Q-2, R-1, S-4
• C. P-4, Q-3, R-1, S-2
• D. P-3, Q-4, R-2, S-1

Question 172

Year: 2007 | Marks: 2

Two bars of different Young's moduli, $E_1$ and $E_2$, but with the same cross sectional area, $A$, and coefficient of thermal expansion, $\alpha$, are attached together at one end and fixed at the other as shown in the figure. The construction of this setup was carried out at an ambient temperature of $25^\circ\text{C}$. The stress in the bars when the temperature is uniformly increased by $10^\circ\text{C}$ is

• A. $\sigma = 10\alpha(E_1 + E_2)$
• B. $\sigma = 20\alpha(E_1 + E_2)$
• C. $\sigma = 20\alpha\left(\frac{E_1 E_2}{E_1 + E_2}\right)$
• D. $\sigma = 10\alpha\left(\frac{E_1 E_2}{E_1 + E_2}\right)$

Question 173

Year: 2007 | Marks: 2

A cylinder of radius $r$ and mass $m$ is placed with no initial velocity on a conveyor belt as shown in the figure. There is a frictional contact between the cylinder and the belt at B. A vertical stopper with a smooth surface prevents the cylinder from rolling away. ($I$ is the moment of inertia of the cylinder; $\alpha$ is the angular acceleration of the cylinder; $\mu$ is the coefficient of friction between the cylinder and the belt and, $g$, is the acceleration due to gravity). The valid free-body diagram of the cylinder just after placing it on the belt (that shows inertial forces also) is

• A. Diagram A
• B. Diagram B
• C. Diagram C
• D. Diagram D

Question 174

Year: 2007 | Marks: 2

The figure below shows 4 long columns with different support conditions but the same flexural rigidity $EI$. Let P, Q, R, and S be the values of their critical buckling load as shown below the respective columns. P, Q, R and S can be arranged in increasing order as

• A. Q, S, R, P
• B. Q, P, R, S
• C. Q, R, P, S
• D. Q, R, S, P

Question 175

Year: 2007 | Marks: 2

A circular shaft of diameter, $d$, is fixed at one end and subjected to an axial force, $P$, and a torque, $T$, at the other end. The torque, $T$, is equal to $Pd/8$. The tensile yield stress of the shaft material is $\sigma_y$. A point on the surface of the shaft will yield according to the Tresca yield criterion if $P$ is equal to

• A. $\sigma_y\left(\frac{\pi d^2}{4\sqrt{2}}\right)$
• B. $\sigma_y\left(\frac{\pi d^2}{4}\right)$
• C. $\sigma_y(d^2)$
• D. $\sigma_y(\pi d^2)$

Question 176

Year: 2007 | Marks: 2

A single degree freedom system consisting of 2 springs and a mass is shown in the figure. The natural frequency of the system in radians/sec is given by

• A. $\sqrt{\frac{k_1}{m}}$
• B. $\sqrt{\frac{k_2}{m}}$
• C. $\sqrt{\frac{k_1 k_2}{(k_1 + k_2)m}}$
• D. $\sqrt{\frac{k_1 + k_2}{m}}$

Question 177

Year: 2007 | Marks: 2

Common Data: For the truss shown in the figure, both the members have a cross sectional area $A$ and Young's modulus $E$. The vertical deflection at point Q is

• A. $\left(\frac{FL}{AE}\right)$
• B. $2\left(\frac{FL}{AE}\right)$
• C. $2\sqrt{2}\left(\frac{FL}{AE}\right)$
• D. $(1 + 2\sqrt{2})\left(\frac{FL}{AE}\right)$

Question 178

Year: 2007 | Marks: 2

Common Data: For the truss shown in the figure, both the members have a cross sectional area $A$ and Young's modulus $E$. The horizontal deflection at point Q is

• A. $\frac{1}{2}\left(\frac{FL}{AE}\right)$
• B. $\frac{3}{4}\left(\frac{FL}{AE}\right)$
• C. $\left(\frac{FL}{AE}\right)$
• D. $\frac{3}{2}\left(\frac{FL}{AE}\right)$

Question 179

Year: 2007 | Marks: 2

Linked Answer Q.25 & Q.26: A propped cantilever beam of span $4\text{ m}$ is subjected to a uniformly distributed load of $20\text{ kN/m}$ as shown in the figure. The reaction at the roller support is

• A. 20 kN
• B. 30 kN
• C. 40 kN
• D. 80 kN

Question 180

Year: 2007 | Marks: 2

Linked Answer Q.25 & Q.26: A propped cantilever beam of span $4\text{ m}$ is subjected to a uniformly distributed load of $20\text{ kN/m}$ as shown in the figure. The maximum bending moment in the beam is

• A. 7.5 kNm
• B. 22.5 kNm
• C. 67.5 kNm
• D. 97.5 kNm

Question 181

Year: 2007 | Marks: 2

Linked Answer Q.27 & Q.28: A cylinder of mass, $m$, and radius, $r$, is rolling without slipping down an inclined plane that makes an angle, $\theta$, with the horizontal. The radius of gyration, $k$, of the cylinder is half the radius. (The coefficient of friction is, $\mu$; Acceleration due to gravity is, $g$). The frictional force acting on the cylinder is given by

• A. $\frac{1}{5}mg\sin\theta$
• B. $\frac{1}{2}mg\sin\theta$
• C. $\mu mg\sin\theta$
• D. $mg\sin\theta$

Question 182

Year: 2007 | Marks: 2

Linked Answer Q.27 & Q.28: A cylinder of mass, $m$, and radius, $r$, is rolling without slipping down an inclined plane that makes an angle, $\theta$, with the horizontal. The radius of gyration, $k$, of the cylinder is half the radius. (The coefficient of friction is, $\mu$; Acceleration due to gravity is, $g$). The angular acceleration of the cylinder is given by

• A. $4\left(\frac{g\sin\theta}{r}\right)$
• B. $2\left(\frac{g\sin\theta}{r}\right)$
• C. $4\mu\left(\frac{g\sin\theta}{r}\right)$
• D. $\frac{4}{5}\left(\frac{g\sin\theta}{r}\right)$