GATE Technical: M mechanical Engineering (Set 1) questions with solutions

Q1. If σ1 and σ3 are the algebraically largest and smallest principal stresses respectively, the value of maximum shear stress is

1. (σ1 + σ3) / 2
2. (σ1 − σ3) / 2
3. √((σ1 + σ3) / 2)
4. √((σ1 − σ3) / 2)

Answer: (σ1 − σ3) / 2

The maximum shear stress in a material is determined by the difference between the largest and smallest principal stresses, divided by two. This relationship arises from the definition of shear stress in relation to normal stresses, making option B the correct choice.

Q2. The equation of motion for a spring-mass system excited by a harmonic force is M x¨ + K x = F cos(ωt), where M is the mass, K is the spring stiffness, F is the force amplitude and ω is the angular frequency of excitation. Resonance occurs when ω is equal to

1. √(M / K)
2. (1 / 2π) √(K / M)
3. 2π √(K / M)
4. √(K / M)

Answer: √(K / M)

The correct option, √(K / M), represents the natural frequency of the spring-mass system, which is the frequency at which the system oscillates when not subjected to external forces. Resonance occurs when the frequency of the external harmonic force matches this natural frequency, leading to maximum amplitude of oscillation.

Q3. For an Oldham coupling used between two shafts, which among the following statements are correct? I. Torsional load is transferred along shaft axis. II. A velocity ratio of 1:2 between shafts is obtained without using gears. III. Bending load is transferred transverse to shaft axis. IV. Rotation is transferred along shaft axis.

1. I and III
2. I and IV
3. II and III
4. II and IV

Answer: I and IV

The correct options I and IV are accurate because an Oldham coupling transfers torsional loads and rotational motion along the shaft axis, ensuring that the shafts maintain their alignment while accommodating angular misalignment.

Q4. For a two-dimensional incompressible flow field given by u = A(x î − y ĵ), where A > 0, which one of the following statements is FALSE?

1. It satisfies continuity equation.
2. It is unidirectional when x → ∞ and y → ∞.
3. Its streamlines are given by x = y.
4. It is irrotational.

Answer: Its streamlines are given by x = y.

The statement is false because the streamlines of the flow field are determined by the velocity components, which indicate that the streamlines are not simply given by the equation x = y; instead, they follow a more complex relationship based on the flow's characteristics.

Q5. Which one of the following statements is correct for a superheated vapour?

1. Its pressure is less than the saturation pressure at a given temperature.
2. Its temperature is less than the saturation temperature at a given pressure.
3. Its volume is less than the volume of the saturated vapour at a given temperature.
4. Its enthalpy is less than the enthalpy of the saturated vapour at a given pressure.

Answer: Its pressure is less than the saturation pressure at a given temperature.

A superheated vapor exists at a temperature above its saturation temperature for a given pressure, which means its pressure must be lower than the saturation pressure at that temperature. This characteristic allows superheated vapors to be used in various thermodynamic processes without condensing.

Q6. In a linearly hardening plastic material, the true stress beyond initial yielding

1. increases linearly with the true strain
2. decreases linearly with the true strain
3. first increases linearly and then decreases linearly with the true strain
4. remains constant

Answer: increases linearly with the true strain

In linearly hardening plastic materials, the true stress continues to increase proportionally with true strain after the initial yield point, reflecting the material's ability to strengthen as it deforms.

Q7. Using the Taylor's tool life equation with exponent n = 0.5, if the cutting speed is reduced by 50%, the ratio of new tool life to original tool life is

1. 4
2. 2
3. 1
4. 0.5

Answer: 4

According to Taylor's tool life equation, tool life is inversely proportional to the cutting speed raised to the power of n. When the cutting speed is halved (50% reduction), the new tool life increases by a factor of 2 raised to the power of 2 (since n = 0.5), resulting in a fourfold increase in tool life.

Q8. A grinding ratio of 200 implies that the

1. grinding wheel wears 200 times the volume of the material removed
2. grinding wheel wears 0.005 times the volume of the material removed
3. aspect ratio of abrasive particles used in the grinding wheel is 200
4. ratio of volume of abrasive particle to that of grinding wheel is 200

Answer: grinding wheel wears 200 times the volume of the material removed

A grinding ratio of 200 indicates that for every unit volume of material removed, the grinding wheel itself wears away 200 times that volume, highlighting the efficiency and wear characteristics of the grinding process.

Q9. Interpolator in a CNC machine

1. controls spindle speed
2. coordinates axes movements
3. operates tool changer
4. commands canned cycle

Answer: coordinates axes movements

The interpolator in a CNC machine is responsible for calculating the precise movements of the machine's axes to create the desired path for machining, ensuring accurate positioning and smooth transitions.

Q10. The time series forecasting method that gives equal weightage to each of the m most recent observations is

1. Moving average method
2. Exponential smoothing with linear trend
3. Triple Exponential smoothing
4. Kalman Filter

Answer: Moving average method

The moving average method calculates the forecast by averaging a specified number of the most recent data points, treating each observation equally, which helps to smooth out short-term fluctuations and highlight longer-term trends.

Q11. A carpenter glues a pair of cylindrical wooden logs by banding their end faces at an angle of θ = 30° as shown in the figure. The glue used at the interface fails if Criterion 1: the maximum normal stress exceeds 2.5 MPa. Criterion 2: the maximum shear stress exceeds 1.5 MPa. Assume that the interface fails before the logs fail. When a uniform tensile stress of 4 MPa is applied, the interface

1. fails only because of criterion 1
2. fails only because of criterion 2
3. fails because of both criteria 1 and 2
4. does not fail

Answer: fails because of both criteria 1 and 2

Both criteria are violated under the applied tensile stress of 4 MPa. The normal stress exceeds the maximum limit of 2.5 MPa, and the shear stress also exceeds the limit of 1.5 MPa due to the angle of the logs, leading to failure according to both criteria.

Q12. A self-aligning ball bearing has a basic dynamic load rating (C10, for 10⁶ revolutions) of 35 kN. If the equivalent radial load on the bearing is 45 kN, the expected life (in 10⁶ revolutions) is

1. below 0.5
2. 0.5 to 0.8
3. 0.8 to 1.0
4. above 1.0

Answer: below 0.5

The expected life of a bearing is inversely proportional to the load it experiences compared to its load rating. Since the equivalent radial load of 45 kN exceeds the basic dynamic load rating of 35 kN, the bearing will have a significantly reduced lifespan, likely falling below 0.5 million revolutions.