List of top Engineering Sciences Questions on Mechanics of Solids asked in GATE XE

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).

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).
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 \( 9.82 \times 10^5 \, {Nm}^2 \) and the cross-section area is \( 7.85 \times 10^{-3} \, {m}^2 \). 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).

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). 

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).

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?

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?

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 \( x(t) \)? 

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 \( 9.82 \times 10^5 \, {Nm}^2 \) and the cross-section area is \( 7.85 \times 10^{-3} \, {m}^2 \). 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).
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?
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The dry bulb temperature and relative humidity of air inside a storage chamber are 37°C and 50%, respectively. The saturation pressure of water vapour at 37°C and barometric pressure are 6.28 kPa and 101.32 kPa, respectively. The humidity ratio of air inside the chamber is _________ \text{ kg water / kg dry air}. (Round off to three decimal places.)