List of top Questions asked in GATE Engineering Sciences

A bar of length \( L = 1 \, {m} \) is fixed at one end. Before heating its free end has a gap of \( \delta = 0.1 \, {mm} \) from a rigid wall as shown in the figure. Now the bar is heated resulting in a uniform temperature rise of \( 10^\circ {C} \). The coefficient of linear thermal expansion of the material is \( 20 \times 10^{-6} / \degree C \) 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). 

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).

A simply-supported beam has a circular cross-section with a diameter of 20 mm, area of 314.2 mm\(^2\), area moment of inertia of 7854 mm\(^4\), 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).

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:
\[ U = \frac{2}{243} \frac{P^2 L^3}{EI}, \] where the section modulus, \( EI \), is \( 16.66 \times 10^5 \, {Nm}^2 \) and the length of the beam \( L \) is 1 m.
The deflection at the loading point is ............ mm (rounded off to two decimal places).

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 \( \sigma_{xx} = 200y + 500 \, {N/mm}^2 \), where \( y \) 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.

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 \( \varepsilon_{xx} = 500 \times 10^{-6}, \, \varepsilon_{yy} = 100 \times 10^{-6} \) and \( \varepsilon_{xy} = 0 \). 
Which of the following statements is/are CORRECT?

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

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 \( \theta = 60^\circ \). 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/s\(^2\)) of block R? 

Consider two blocks, P of mass 100 kg and Q of mass 150 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 P upward is \( W \). Which one of the following options is closest to the CORRECT magnitude of \( W \) (in N)? 

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). 

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 ........... mm\(^4\) (in integer). 

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? 

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 \( \theta \) from the x-axis, is in a state of pure shear. Which one of the following values of \( \theta \) (in degrees) is CORRECT?

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 represents \( x(t) \)? 

Corrosion of pure iron takes place in an acidic electrolyte by forming \( {Fe}^{2+} \) ions at ambient condition. The corrosion current density is measured to be \( 2 \times 10^{-4} \, {A cm}^{-2} \). The corrosion rate (in mm per year) of iron is (rounded off to one decimal place) ............

An aluminum transmission line of 7 km length is designed to carry 100 A current with no more than 2 MW power loss. The required minimum diameter (in mm) of the transmission line is (rounded to the two decimal places) ........... 

On applying 10 V across the two ends of a 100 cm long copper wire, the average drift velocity (in cm s\(^{-1}\)) in the wire is (rounded off to two decimal places).............

For a pure element with a BCC crystal structure, the surface energies per unit area of \( \{100\} \) and \( \{110\} \) free surfaces are \( S_{100} \) and \( S_{110} \), respectively. The ratio, \( \frac{S_{100}}{S_{110}} \), is (rounded off to one decimal place): 

A diffraction pattern is obtained from a powdered sample of a pure element, which has FCC crystal structure. If \( x \) and \( y \) are the Bragg angles of the first and the third peaks, respectively, then the ratio, \( \frac{\sin y}{\sin x} \), is (rounded off to one decimal place)...........