GATE Technical: MAIN PAPER - ME questions with solutions

Q1. The transverse shear stress acting in a beam of rectangular cross-section, subjected to a transverse shear load, is

1. variable with maximum at the bottom of the beam
2. variable with maximum at the top of the beam
3. uniform
4. variable with maximum on the neutral axis

Answer: variable with maximum on the neutral axis

Transverse shear stress in a beam varies across the height of the cross-section, with the maximum occurring at the neutral axis due to the distribution of shear forces, which leads to higher stress concentrations there.

Q2. A straight rod of length L(t), hinged at one end and freely extensible at the other end, rotates through an angle θ(t) about the hinge. At time t, L(t) = 1 m, L̇(t) = 1 m/s, θ(t) = π/4 rad and θ̇(t) = 1 rad/s. The magnitude of the velocity of the other end of the rod is

1. 1 m/s
2. √2 m/s
3. √3 m/s
4. 2 m/s

Answer: √2 m/s

The end velocity has radial component Ldot = 1 m/s and transverse component L*thetadot = 1*1 = 1 m/s. Magnitude = sqrt(1^2 + 1^2) = sqrt(2) m/s (index 1).

Q3. A planar mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mechanism, using Gruebler’s criterion, is

1. 0
2. 1
3. 2
4. 3

Answer: 1

Using Gruebler's criterion, the degrees of freedom (DOF) for a planar mechanism can be calculated with the formula DOF = 3(N - 1) - 2J, where N is the number of links and J is the number of joints. In this case, substituting N = 8 and J = 10 gives DOF = 3(8 - 1) - 2(10) = 21 - 20 = 1, indicating that the mechanism has 1 degree of freedom.

Q4. An axial residual compressive stress due to a manufacturing process is present on the outer surface of a rotating shaft subjected to bending. Under a given bending load, the fatigue life of the shaft in the presence of the residual compressive stress is

1. decreased
2. increased or decreased, depending on the external bending load
3. neither decreased nor increased
4. increased

Answer: increased

Residual compressive stress on the outer surface of a rotating shaft counteracts the tensile stresses that occur during bending, which enhances the fatigue resistance of the material and ultimately increases the fatigue life.

Q5. For flow of fluid over a heated plate, the following fluid properties are known: viscosity = 0.001 Pa.s; specific heat at constant pressure = 1 kJ/kg.K; thermal conductivity = 1 W/m.K. The hydrodynamic boundary layer thickness at a specified location on the plate is 1 mm. The thermal boundary layer thickness at the same location is

1. 0.001 mm
2. 0.01 mm
3. 1 mm
4. 1000 mm

Answer: 1 mm

The thermal boundary layer thickness is often comparable to the hydrodynamic boundary layer thickness in flow over a heated plate, especially when the fluid properties are similar. Given that both layers are influenced by the same flow conditions and the thermal properties are not drastically different, it is reasonable to conclude that the thermal boundary layer thickness is also 1 mm.

Q6. A displacement sensor (a dial indicator) measures the lateral displacement of a mandrel mounted on the taper hole inside a drill spindle. The mandrel axis is an extension of the drill spindle taper hole axis and the protruding portion of the mandrel surface is perfectly cylindrical. Measurements are taken with the sensor placed at two positions P and Q as shown in the figure. The readings are recorded as Rₓ = maximum deflection minus minimum deflection, corresponding to sensor position at X, over one rotation. If R_P = R_Q > 0, which one of the following would be consistent with the observation?

1. The drill spindle rotational axis is coincident with the drill spindle taper hole axis
2. The drill spindle rotational axis intersects the drill spindle taper hole axis at point P
3. The drill spindle rotational axis is parallel to the drill spindle taper hole axis
4. The drill spindle rotational axis intersects the drill spindle taper hole axis at point Q

Answer: The drill spindle rotational axis is parallel to the drill spindle taper hole axis

When the readings R_P and R_Q are equal and greater than zero, it indicates that the lateral displacement is consistent across both sensor positions, which occurs when the drill spindle rotational axis is parallel to the taper hole axis. This alignment prevents any variation in displacement measurements as the mandrel rotates.