List of practice Questions

Two identical mass-less beams AB and CD are clamped at their ends. The left end of beam CD rests on the right end of beam AB at point C (no friction). A uniformly distributed load of 800 N/m is applied on beam CD of span 1 m. Find the bending moment at end B of beam AB (in N·m). Round off to the nearest integer.

The energy equation for a reversible non-flow process can be expressed as \(\delta q = du + p\,dv\), where \(q\) is the heat transfer per unit mass, \(u\) is the internal energy per unit mass, \(p\) is the pressure, and \(v\) is the mass specific volume. This energy equation is not in exact differential form. It can be made exact differential by multiplying with the following integrating factor: (The temperature \(T\) is absolute temperature)

An air standard Diesel cycle consists of four processes: 1-2 (isentropic compression), 2-3 (constant pressure heat addition), 3-4 (isentropic expansion) and 4-1 (constant volume heat rejection). $T_4$ is the temperature attained at the end of isentropic expansion (3-4) before constant volume heat rejection. The constant volume heat rejection process (4-1) is replaced by a constant pressure heat rejection process (4a-1) such that $T_{4a$ is the temperature reached at the end of isentropic expansion (3-4a), and the state point 1 remains the same. Then}

The beam in the figure is subjected to a moment \( M_0 \) at mid span as shown. Which of the following is the vertical reaction at B?

A spring–mass system having a mass $m$ and spring constant $k$, placed horizontally on a foundation, is connected to a vertically hanging mass $m$ with the help of an inextensible string as shown. Ignore pulley friction, and neglect pulley, string, and spring inertia. Gravity acts downward. The natural frequency of the system in rad/s is:

One end of a uniform rigid rod $OA$ of length $L$ and mass $m$ is attached to a frictionless hinge at $O$. The other end of the rod is connected to the roof at $B$ with a massless inextensible thread $AB$. Initially the rod is horizontal and at rest. The gravity is acting vertically downward as shown. Immediately after the thread $AB$ is cut, the reaction on the rod at $O$ is:

A circular shaft is rigidly connected to a wall at one end. The shaft has a solid portion and a hollow portion as shown in the figure. The length of each portion is $L$ and the shear modulus of the material is $G$. The polar moment of inertia of the hollow portion is $J$ and that of the solid portion is $50J$. A torque $T$ is applied at the rightmost end as shown. The rotation of the section $PQ$ is

A rectangular plate of uniform thickness having initial length \( a \) and width \( b \) is placed between two rigid immovable walls. The temperature of the plate is increased by \( \Delta T \). The plate is free to expand along the \( y \) and \( z \) directions. The mid-surface of the plate remains in the \( xy \)-plane. The Poisson’s ratio is \( \nu \) and the coefficient of thermal expansion is \( \alpha \). Assuming that the plate is initially free of stresses, the change in length of the plate after the increase in temperature is given by:

A mass \( m = 10 \, \text{kg} \) is attached to a spring as shown in the figure. The coefficient of friction between the mass and the inclined plane is 0.25. Assume that the acceleration due to gravity is \( 10 \, \text{m/s}^2 \) and that static and kinetic friction coefficients are the same. Equilibrium of the mass is impossible if the spring force is:

The frame shown is subjected to a uniformly distributed load of 1000 N/m over 0.6 m. Neglecting the frame weight, the maximum shear force (in N) in the region between supports A and B is _________.

A massless semicircular rod held fixed at end A is in the xy-plane as shown. A force P along the negative z direction is acting at point B on the rod. The unit vectors along x, y and z directions are denoted by i, j and k. Due to the applied force P, the cross-section of the rod at point D will be subjected to

In the truss shown in the figure, all the members are pin jointed to each other. The members AB, BD, DE and DC have the same length. For the given loading, which of the following is the correct statement?

End B of the 2 m long rigid rod $AB$ is constrained to move horizontally in the slot as shown in the figure and has a velocity of $1.0\,\mathbf{i}$ m/s. The angular velocity of the rod at the instant shown is $2$ rad/s. The unit vectors along $x$ and $y$ directions are denoted by $\mathbf{i}$ and $\mathbf{j}$. The velocity of point A in m/s is then given by:

The assembly of four masses connected by rigid mass-less rods is kept on a smooth horizontal floor as shown in the figure. Under the applied force $2F$, the magnitude of angular acceleration of the assembly at the instant shown is:

A particle is constrained to move at a constant speed on an inclined plane (ABCD) along the curved path shown in the figure. Edges AD and BC are parallel to the y-axis. The inclined plane makes an angle $\theta$ with the $xy$-plane. The velocity vector of the particle makes an angle $\phi$ with the dotted line which is parallel to edge AB. If the speed of the particle is 2 m/s, $\phi = 30^\circ$, and $\theta = 40^\circ$, then the z-component of the velocity of the particle in m/s is

A uniform elastic rod of constant cross-section is fixed at its left end as shown. An axial force $P$ acts as shown. Assume plane sections remain plane. The ratio of axial displacement at point $A$ $(x=4L)$ to that at point $B$ $(x=L)$ is _________ (rounded off to one decimal place).

A thin-walled spherical pressure vessel has radius 500 mm and thickness 10 mm. Yield strength is 500 MPa. The internal pressure at which yielding occurs according to Tresca criterion is _________ MPa (rounded off to one decimal place).

A two-phase ($\alpha + \beta$) mixture of an A–B binary system has the following properties:

[(i)] Phase $\alpha$ has equal weight percentages of A and B.
[(ii)] Phase $\beta$ has twice the mole fraction of A compared to B.
[(iii)] The two-phase mixture has equal amounts of $\alpha$ and $\beta$.
[(iv)] Atomic mass of A is twice that of B.
The mole fraction of A in the resultant two-phase mixture is _________. (Round off to one decimal)

Component A diffuses into a solid to a depth of 10 μm in 1 hour at 300 K. Treat diffusion in one dimension. Find the time (in seconds) for A to diffuse to the same depth at 600 K. (Round off to 1 decimal). Diffusivity is given by: \[ D_A = D_A^0 \exp\left( -\frac{E_a}{k_B T} \right) \] Given: \[ E_a = 0.3 \text{ eV}, \quad k_B = 8.62\times10^{-5}\,\text{eV/K} \] Depth is same → use relation: \[ x \sim \sqrt{Dt} \Rightarrow Dt = \text{constant} \] Thus: \[ \frac{t_2}{t_1} = \frac{D_1}{D_2} \]

A spherical $\beta$ particle nucleates from the $\alpha$ matrix on a non-deformable substrate $\delta$, forming a contact angle $\theta$ as shown. The value of $\dfrac{\Delta G^{}_{het}}{\Delta G^{}_{hom}}$ is _________. (Round off to three decimal places)

Given: $\alpha$–$\beta$ interfacial energy $= 0.4$ J/m$^{2}$ $\alpha$–$\delta$ interfacial energy $= 0.3$ J/m$^{2}$ $\beta$–$\delta$ interfacial energy $= 0.02$ J/m$^{2}$

The resistivity of a pure semiconductor at 298 K is \(3000\,\Omega\text{m}\). Assume the number of electrons excited \((n_e)\) across the band gap is \[ n_e = N_A \exp\left(-\frac{E_g}{k_B T}\right) \] Given: \[ N_A = 6.02 \times 10^{23}\ \text{mol}^{-1},\quad k_B = 8.62 \times 10^{-5}\ \text{eV/K},\quad T = 298\ \text{K} \] Mobilities: \[ \mu_e = 0.14\ \text{m}^2/(\text{V·s}),\quad \mu_h = 0.06\ \text{m}^2/(\text{V·s}) \] Absolute electron charge: \[ q = 1.60 \times 10^{-19}\ \text{C} \]

A new glass material is developed to minimize the transmission of light through a window with a glass panel of thickness 5 mm. The refractive index of the glass material is 1.5 and the absorption coefficient can be changed from 0.3 cm$^{-1}$ to 1 cm$^{-1}$. In the given range of absorption coefficients, the ratio of the maximum to the minimum fraction of the light coming out of the other side of the glass panel is _________. (Round off to two decimal places)

The third peak in the X-ray diffraction pattern of a face-centered cubic (FCC) crystal occurs at a \(2\theta = 45^\circ\). The wavelength of the monochromatic X-ray beam is \(1.54\ \text{\AA}\). Considering first-order reflection, find the lattice parameter (in \AA). (Round off to two decimal places)

A force \( F \) is applied at an angle \( \theta = 30^\circ \) on an elastic column as shown in the figure. \(E\) and \(I\) are respectively the Young's modulus and the area moment of inertia. The smallest magnitude of \(F\) needed to cause buckling is:

The shear stress due to a transverse shear force in a linear elastic isotropic beam of rectangular cross-section