GATE Technical: Mechanical Engineering (Set 3) questions with solutions

Q1. Consider a linear elastic rectangular thin sheet of metal, subjected to uniform uniaxial tensile stress of 100 MPa along the length direction. Assume plane stress conditions in the plane normal to the thickness. The Young’s modulus E = 200 MPa and Poisson’s ratio ν = 0.3 are given. The principal strains in the plane of the sheet are

1. (0.35, -0.15)
2. (0.5, 0.0)
3. (0.5, -0.15)
4. (0.5, -0.5)

Answer: (0.5, -0.15)

The correct option (0.5, -0.15) represents the principal strains calculated using the relationships derived from the applied stress and material properties, specifically accounting for the effects of Poisson's ratio on the lateral strain due to the uniaxial tensile stress.

Q2. A spur gear has pitch circle diameter D and number of teeth T. The circular pitch of the gear is

1. πD / T
2. T / D
3. D / T
4. 2πD / T

Answer: πD / T

The circular pitch is defined as the distance from one point on a tooth to the corresponding point on the next tooth, which is calculated by dividing the circumference of the pitch circle (πD) by the number of teeth (T). This gives the formula πD / T.

Q3. Endurance limit of a beam subjected to pure bending decreases with

1. decrease in the surface roughness and decrease in the size of the beam
2. increase in the surface roughness and decrease in the size of the beam
3. increase in the surface roughness and increase in the size of the beam
4. decrease in the surface roughness and increase in the size of the beam

Answer: increase in the surface roughness and increase in the size of the beam

The endurance limit of a beam decreases with increased surface roughness because rough surfaces can initiate fatigue cracks more easily. Additionally, larger beam sizes can lead to higher stress concentrations, further reducing the endurance limit.

Q4. A two-dimensional incompressible frictionless flow field is given by u = xî − yĵ. If ρ is the density of the fluid, the expression for pressure gradient vector at any point in the flow field is given as

1. ρ (xî + yĵ)
2. −ρ (xî + yĵ)
3. ρ (xî − yĵ)
4. −ρ (x²î + y²ĵ)

Answer: −ρ (xî + yĵ)

The correct option is derived from the application of the Euler's equation for inviscid flow, which states that the pressure gradient is equal to the negative of the density times the velocity vector. Since the velocity vector is given as u = xî − yĵ, the pressure gradient vector is thus calculated as −ρ (xî + yĵ).

Q5. Sphere 1 with a diameter of 0.1 m is completely enclosed by another sphere 2 of diameter 0.4 m. The view factor F12 is

1. 0.0625
2. 0.25
3. 0.5
4. 1.0

Answer: 1.0

All radiation leaving sphere 1 must strike the completely enclosing sphere 2, so F12 = 1.0. The value 0.0625 is A1/A2 = (0.1/0.4)^2 = F21, not F12. Correct answer is 1.0.

Q6. One-dimensional steady state heat conduction takes place through a solid whose cross-sectional area varies linearly in the direction of heat transfer. Assume there is no heat generation in the solid and the thermal conductivity of the material is constant and independent of temperature. The temperature distribution in the solid is

1. Linear
2. Quadratic
3. Logarithmic
4. Exponential

Answer: Logarithmic

In one-dimensional steady state heat conduction with a varying cross-sectional area, the temperature distribution is logarithmic due to the relationship between heat flow and area. As the area changes, the heat flux must adjust accordingly, leading to a logarithmic profile rather than a linear or polynomial one.

Q7. For a simple compressible system, v, s, p and T are specific volume, specific entropy, pressure and temperature, respectively. As per Maxwell’s relations, (∂v/∂s) p is equal to

1. (∂s/∂T) p
2. (∂p/∂v) T
3. −(∂T/∂v) p
4. (∂T/∂p) s

Answer: (∂T/∂p) s

From the enthalpy potential dH=TdS+vdP, the Maxwell relation gives (dT/dp)_s=(dv/ds)_p. Therefore (dv/ds)_p equals (dT/dp)_s. The stored option is not a valid Maxwell pairing.

Q8. Which one of the following modifications of the simple ideal Rankine cycle increases the thermal efficiency and reduces the moisture content of the steam at the turbine outlet?

1. Increasing the boiler pressure.
2. Decreasing the boiler pressure.
3. Increasing the turbine inlet temperature.
4. Decreasing the condenser pressure.

Answer: Increasing the turbine inlet temperature.

Increasing the turbine inlet temperature raises the average temperature at which heat is added to the cycle, thereby improving thermal efficiency and reducing the moisture content of the steam at the turbine outlet.

Q9. In an electrical discharge machining process, the breakdown voltage across inter electrode gap (IEG) is 200 V and the capacitance of the RC circuit is 50 μF. The energy (in J) released per spark across the IEG is

1. 0.25
2. 0.50
3. 1.00
4. 2.00

Answer: 1.00

Energy released per spark in EDM is E = 0.5*C*V^2 = 0.5*(50e-6 F)*(200 V)^2 = 0.5*50e-6*40000 = 1.0 J. The correct option is 1.00 J (index 2), not 0.50 J.

Q10. The crank of a slider-crank mechanism rotates counter-clockwise (CCW) with a constant angular velocity ω, as shown. Assume the length of the crank to be r. Using exact analysis, the acceleration of the slider in the y-direction, at the instant shown, where the crank is parallel to x-axis, is given by

1. −ω²r
2. 2ω²r
3. ω²r
4. −2ω²r

Answer: −ω²r

The acceleration of the slider in the y-direction is determined by the centripetal acceleration due to the rotation of the crank. When the crank is parallel to the x-axis, the vertical component of the acceleration is negative, resulting in an expression of -ω²r, which reflects the downward acceleration of the slider.