List of practice Questions

Consider a rod of uniform thermal conductivity whose one end \( (x = 0) \) is insulated and the other end \( (x = L) \) is exposed to the flow of air at temperature \( T_{\infty} \) with convective heat transfer coefficient \( h \). The cylindrical surface of the rod is insulated so that the heat transfer is strictly along the axis of the rod. The rate of internal heat generation per unit volume inside the rod is given as \[ \dot{q} = \cos \left( \frac{2 \pi x}{L} \right). \] The steady-state temperature at the mid-location of the rod is given as \( T_A \). What will be the temperature at the same location, if the convective heat transfer coefficient increases to \( 2h \)?

The system of linear equations in real \( (x, y) \) given by \[ \begin{pmatrix} x \\ y \end{pmatrix} \begin{pmatrix} 2 & 5 - 2\alpha \\ \alpha & 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] involves a real parameter \( \alpha \) and has infinitely many non-trivial solutions for special value(s) of \( \alpha \). Which one or more among the following options is/are non-trivial solution(s) of \( (x, y) \) for such special value(s) of \( \alpha \)?

Let a random variable \( X \) follow Poisson distribution such that
\[ \text{Prob}(X = 1) = \text{Prob}(X = 2). \] The value of \( \text{Prob}(X = 3) \) is ________________ (round off to 2 decimal places).

Consider two vectors: \[ \mathbf{a} = 5 \hat{i} + 7 \hat{j} + 2 \hat{k}, \quad \mathbf{b} = 3 \hat{i} - \hat{j} + 6 \hat{k} \] Magnitude of the component of \( \mathbf{a} \) orthogonal to \( \mathbf{b} \) in the plane containing the vectors \( \mathbf{a} \) and \( \mathbf{b} \) is ________________ (round off to 2 decimal places).

A structure, along with the loads applied on it, is shown in the figure. Self-weight of all the members is negligible and all the pin joints are friction-less. AE is a single member that contains pin C. Likewise, BE is a single member that contains pin D. Members GI and FH are overlapping rigid members. The magnitude of the force carried by member CI is ________________ kN (in integer).

Two rigid massless rods PR and RQ are joined at frictionless pin-joint R and are resting on the ground at P and Q, respectively, as shown in the figure. A vertical force \( F \) acts on the pin R as shown. When the included angle \( \theta<90^\circ \), the rods remain in static equilibrium due to Coulomb friction between the rods and ground at locations P and Q. At \( \theta = 90^\circ \), impending slip occurs simultaneously at points P and Q. Then the ratio of the coefficient of friction at Q to that at P \( \left( \frac{\mu_Q}{\mu_P} \right) \) is ________________ (round off to two decimal places).

A cylindrical disc of mass \( m = 1 \, \text{kg} \) and radius \( r = 0.15 \, \text{m} \) was spinning at \( \omega = 5 \, \text{rad/s} \) when it was placed on a flat horizontal surface and released (refer to the figure). Gravity \( g \) acts vertically downwards as shown in the figure. The coefficient of friction between the disc and the surface is finite and positive. Disregarding any other dissipation except that due to friction between the disc and the surface, the horizontal velocity of the center of the disc, when it starts rolling without slipping, will be ________________ m/s (round off to 2 decimal places).

A planar four-bar linkage mechanism with 3 revolute kinematic pairs and 1 prismatic kinematic pair is shown in the figure, where AB \( \perp \) CE and FD \( \perp \) CE. The T-shaped link CDEF is constructed such that the slider B can cross the point D, and CE is sufficiently long. For the given lengths as shown, the mechanism is

Consider a forced single degree-of-freedom system governed by \[ \ddot{x}(t) + 2 \zeta \omega_n \dot{x}(t) + \omega_n^2 x(t) = \omega_n^2 \cos(\omega t), \] where \( \zeta \) and \( \omega_n \) are the damping ratio and undamped natural frequency of the system, respectively, while \( \omega \) is the forcing frequency. The amplitude of the forced steady state response of this system is given by \[ \left[ (1 - r^2)^2 + (2 \zeta r)^2 \right]^{-1/2}, \quad \text{where} \quad r = \frac{\omega}{\omega_n}. \] The peak amplitude of this response occurs at a frequency \( \omega = \omega_p \). If \( \omega_d \) denotes the damped natural frequency of this system, which one of the following options is true?

A bracket is attached to a vertical column by means of two identical rivets U and V separated by a distance of \( 2a = 100 \, \text{mm} \), as shown in the figure. The permissible shear stress of the rivet material is 50 MPa. If a load \( P = 10 \, \text{kN} \) is applied at an eccentricity \( e = 3\sqrt{7} \, a \), the minimum cross-sectional area of each of the rivets to avoid failure is ________________.

In Fe-Fe₃C phase diagram, the eutectoid composition is 0.8 weight % of carbon at 725 °C. The maximum solubility of carbon in $\alpha$-ferrite phase is 0.025 weight % of carbon. A steel sample, having no other alloying element except 0.5 weight % of carbon, is slowly cooled from 1000 °C to room temperature. The fraction of pro-eutectoid $\alpha$-ferrite in the above steel sample at room temperature is:

Activities A to K are required to complete a project. The time estimates and the immediate predecessors of these activities are given in the table. If the project is to be completed in the minimum possible time, the latest finish time for the activity G is ________________ hours.

A solid spherical bead of lead (uniform density = 11000 kg/m\(^3\)) of diameter \( d = 0.1 \) mm sinks with a constant velocity \( V \) in a large stagnant pool of a liquid (dynamic viscosity = \( 1.1 \times 10^{-3} \) kg·m\(^{-1}\)·s\(^{-1}\)). The coefficient of drag is given by \[ C_D = \frac{24}{Re}, \] where the Reynolds number \( Re \) is defined on the basis of the diameter of the bead. The drag force acting on the bead is expressed as \[ D = (C_D)(0.5 \rho V^2)\left( \frac{\pi d^2}{4} \right), \] where \( \rho \) is the density of the liquid. Neglect the buoyancy force. Using \( g = 10 \) m/s\(^2\), the velocity \( V \) is ________________ m/s.

The figure shows a purely convergent nozzle with a steady, inviscid compressible flow of an ideal gas with constant thermophysical properties operating under choked condition. The exit plane shown in the figure is located within the nozzle. If the inlet pressure (P0) is increased while keeping the back pressure (Pback) unchanged, which of the following statements is/are true?

The plane of the figure represents a horizontal plane. A thin rigid rod at rest is pivoted without friction about a fixed vertical axis passing through O. Its mass moment of inertia is equal to 0.1 kg·cm² about O. A point mass of 0.001 kg hits it normally at 200 cm/s at the location shown, and sticks to it. Immediately after the impact, the angular velocity of the rod is ________________ rad/s (in integer).

A rigid uniform annular disc is pivoted on a knife edge A in a uniform gravitational field as shown, such that it can execute small amplitude simple harmonic motion in the plane of the figure without slip at the pivot point. The inner radius \( r \) and outer radius \( R \) are such that \( r^2 = R^2/2 \), and the acceleration due to gravity is \( g \). If the time period of small amplitude simple harmonic motion is given by \[ T = \beta \pi \sqrt{\frac{R}{g}}, \] where \( \pi \) is the ratio of circumference to diameter of a circle, then \( \beta = \) ________________ (round off to 2 decimal places).

Electrochemical machining operations are performed with tungsten as the tool, and copper and aluminum as two different workpiece materials. Properties of copper and aluminum are given in the table below.
\begin{tabular}{|c|c|c|c|} \hline Material & Atomic mass (amu) & Valency & Density (g/cm\(^3\))
\hline Copper & 63 & 2 & 9
Aluminum & 27 & 3 & 2.7
\hline \end{tabular} Ignore overpotentials, and assume that current efficiency is 100% for both the workpiece materials. Under identical conditions, if the material removal rate (MRR) of copper is 100 mg/s, the MRR of aluminum will be ________________ mg/s. \text{[round off to two decimal places]}

A polytropic process is carried out from an initial pressure of 110 kPa and volume of 5 m³ to a final volume of 2.5 m³. The polytropic index is given by \( n = 1.2 \). The absolute value of the work done during the process is ________________ kJ (round off to 2 decimal places).

A flat plate made of cast iron is exposed to a solar flux of 600 W/m² at an ambient temperature of 25°C. Assume that the entire solar flux is absorbed by the plate. Cast iron has a low temperature absorptivity of 0.21. Use Stefan-Boltzmann constant = \( 5.669 \times 10^{-8} \, \text{W/m}^2 \text{K}^4 \). Neglect all other modes of heat transfer except radiation. Under the aforementioned conditions, the radiation equilibrium temperature of the plate is ________________ °C (round off to the nearest integer).

The value of the integral \[ \int \left( \frac{6z}{2z^4 - 3z^3 + 7z^2 - 3z + 5} \right) dz \] evaluated over a counter-clockwise circular contour in the complex plane enclosing only the pole \( z = i \), where \( i \) is the imaginary unit, is

An L-shaped elastic member ABC with slender arms AB and BC of uniform cross-section is clamped at end A and connected to a pin at end C. The pin remains in continuous contact with and is constrained to move in a smooth horizontal slot. The section modulus of the member is same in both the arms. The end C is subjected to a horizontal force \( P \) and all the deflections are in the plane of the figure. Given the length AB is \( 4a \) and length BC is \( a \), the magnitude and direction of the normal force on the pin from the slot, respectively, are.

A tiny temperature probe is fully immersed in a flowing fluid and is moving with zero relative velocity with respect to the fluid. The velocity field in the fluid is \[ \mathbf{V} = (2x) \hat{i} + (y + 3t) \hat{j}, \] and the temperature field in the fluid is \[ T = 2x^2 + xy + 4t, \] where \(x\) and \(y\) are the spatial coordinates, and \(t\) is the time. The time rate of change of temperature recorded by the probe at \( (x = 1, y = 1, t = 1) \) is ________________.

In the following two-dimensional momentum equation for natural convection over a surface immersed in a quiescent fluid at temperature \( T_\infty \) (g is the gravitational acceleration, \( \beta \) is the volumetric thermal expansion coefficient, \( \nu \) is the kinematic viscosity, \( u \) and \( v \) are the velocities in \( x \) and \( y \) directions, respectively, and \( T \) is the temperature) \[ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = g \beta (T - T_\infty) + \nu \frac{\partial^2 u}{\partial y^2}, \] the term \( g \beta (T - T_\infty) \) represents

Assuming the material considered in each statement is homogeneous, isotropic, linear elastic, and the deformations are in the elastic range, which one or more of the following statement(s) is/are TRUE?

Which of the following heat treatment processes is/are used for surface hardening of steels?