# Recent questions and answers in Applied Mechanics and Design

The number of atoms per unit cell and the number of slip systems, respectively, for a face-centered cubic (FCC) crystal are $3, 3$ $3, 12$ $4, 12$ $4, 48$
An attempt is made to pull a roller of weight $W$ over a curb (step) by applying a horizontal force $F$ as shown in the figure. The coefficient of static friction between the roller and the ground (including the edge of the step) is $\mu$ Identify the correct free body diagram (FBD) of the roller when the roller is just about to climb over the step.
The equation of motion of a spring-mass-damper system is given by $\dfrac{d^2x}{dt^2}+3 \dfrac{dx}{dt} +9x = 10 \sin(5t)$ The damping factor for the system is $0.25$ $0.5$ $2$ $3$
The number of qualitatively distinct kinematic inversions possible for a Grashof chain with four revolute pairs is $1$ $2$ $3$ $4$
The process, that uses a tapered horn to amplify and focus the mechanical energy for machining of glass, is electrochemical machining electrical discharge machining ultrasonic machining abrasive jet machining
Two plates, each of $6$ mm thickness, are to be butt-welded. Consider the following processes and select the correct sequence in increasing order of size of the heat affected zone. Arc welding MIG welding Laser beam welding Submerged arc welding $1-4-2-3$ $3-4-2-1$ $4-3-2-1$ $3-2-4-1$
The members carrying zero force (i.e. zero-force members) in the truss shown in the figure, for any load $P > 0$ with no appreciable deformation of the truss (i.e.with no appreciable change in angles between the members), are $BF$ and $DH$ only $BF, DH,$ and $GC$ only $BF, DH, GC, CD$ and $DE$ only $BF, DH, GC, FG$ and $GH$ only
A single-degree-of-freedom oscillator is subjected to harmonic excitation $F(t) = F_{0}\cos(\omega t)$ as shown in the figure. The non-zero value of $\omega$, for which the amplitude of the force transmitted to the ground will be $F_{0}$, is $\sqrt{\dfrac{k}{2m}} \\$ $\sqrt{\dfrac{k}{m}} \\$ $\sqrt{\dfrac{2k}{m}} \\$ $2\sqrt{\dfrac{k}{m}}$
The stress state at a point in a material under plane stress condition is equi-biaxial tension with a magnitude of $10\: MPa$. If one unit on the $\sigma -\tau$ plane is $1\: MPa$, the Mohr's circle representation of the state-of-stress is given by a circle with a radius ... radius of $10$ units on the $\sigma -\tau$ plane a point on the $\tau$ axis at a distance of $10$ units from the origin
A four bar mechanism is shown below. For the mechanism to be a crank-rocker mechanism, the length of the link $PQ$ can be $80\: mm$ $200\: mm$ $300\: mm$ $350\: mm$
A helical gear with $20^{\circ}$ pressure angle and $30^{\circ}$ helix angle mounted at the mid-span of a shaft that is supported between two bearings at the ends. The nature of the stresses induced in the shaft is normal stress due to bending only ... bending in two planes and axial loading; shear stress due to torsion normal stress due to bending in two planes; shear stress due to torsion
A flywheel is attached to an engine to keep its rotational speed between $100\: rad/s$ and $110\: rad/s$. If the energy fluctuation in the flywheel between these two speeds is $1.05\: kJ$ then the moment of inertia of the flywheel is____________$kg \cdot m^{2}$ (rounded off to $2$ decimal places).
A balanced rigid disc mounted on a rigid rotor has four identical point masses, each of $10\:\text{grams}$, attached to four points on the $100\: mm$ ... the masses gets detached then the magnitude of the resultant unbalance force on the rotor is ________$N$ (rounded off to $2$ decimal places).
Bars of square and circular cross-section with $0.5\: m$ length are made of a material with shear strength of $20\: MPa$. The square bar cross-section dimension is $4\:cm \times$ $4\:cm$ and the cylindrical bar cross-section diameter ... to the applied load as per maximum shear stress theory? Tensile and compressive load specimens Torsional load specimen Bending load specimen None of the specimens
The $2$ kg block shown in the figure (top view) rests on a smooth horizontal surface and is attached to a massless elastic cord that has a stiffness $5\: N/m$. The cord hinged at $\text{O}$ is initially unstretched and always remains elastic. The block is given a velocity $v$ of $1.5\: m/s$ ... in $m/s$ of the block at the instant the cord is stretched by $0.4\: m$ is $0.83$ $1.07$ $1.36$ $1.50$
The truss shown in the figure has four members of length $\text{l}$ and flexural rigidity $\text{EI}$, and one member of length $l\sqrt{2}$ and flexural rigidity $\text{4EI}$. The truss is loaded by a pair of forces of magnitude $\text{P}$, as shown in the figure. The smallest value of $\text{P}$, at ... $\dfrac{2\pi ^{2}EI}{l^{2}} \\$ $\dfrac{\pi ^{2}EI}{2l^{2}}$
A rigid mass-less rod of length $L$ is connected to a disc (pulley) of mass $m$ and radius $r = L/4$ through a friction-less revolute joint. The other end of that rod is attached to a wall through a friction-less hinge. A spring of stiffness $2k$ is attached to the rod at its mid-span. An ... $\sqrt{3}\sqrt{\dfrac{k}{m}} \\$ $\sqrt{\dfrac{k}{m}}$
In a disc-type axial clutch, the frictional contact takes place within an annular region with outer and inner diameters $250\:mm$ and $50\:mm$, respectively. An axial force $F_{1}$ is needed to transmit a torque by a new clutch. However, to transmit the same torque, one ... and the coefficient of friction does not change, then the ratio $F_{1}/F_{2}$ is _________ (round off to $2$ decimal places).
A cam with a translating flat-face follower is desired to have the follower motion $y\left ( \theta \right )=4\left [ 2\pi \theta -\theta ^{2} \right ],\:\:\:0\leq \theta \leq 2\pi .$ Contact stress considerations dictate that the radius of curvature ... profile should not be less than $40\:mm$ anywhere. The minimum permissible base circle radius is _________$mm$ (round off to one decimal place).
A rectangular steel bar of length $500\: mm$, width $100\: mm$, and thickness $15\: mm$ is cantilevered to a $200\: mm$ steel channel using $4$ bolts, as shown For an external load of $10\: kN$ applied at the tip of the steel bar, the resultant shear load on the bolt at $B$, is _______$kN$ (round off to one decimal place).
The barrier shown between two water tanks of unit width $(1\: m)$ into the plane of the screen in modeled as a cantilever. Taking the density of water as $1000\:kg/m^{3}$ , and the acceleration due to gravity as $10\:m/s^{2}$ , the maximum absolute bending moment developed in the cantilever is _________$kN\cdot m$ (round off to the nearest integer).
The magnitude of reaction force at joint $C$ of the hinge-beam shown in the figure is __________$kN$ (round off to $2$ decimal places).
Three vendors were asked to supply a very high precision component. The respective probabilities of their meeting the strict design specifications are $0.8$, $0.7$ and $0.5$. Each vendor supplies one component. The probability that out of total three components supplied by the vendors, at least one will meet the design specification is ___________
A four bar mechanism is shown in the figure. The link numbers are mentioned near the links. Input link $2$ is rotating anticlockwise with a constant angular speed $\omega_2$. Length of different links are: $O_2 O_4=O_2A=L,$$AB=O_4B=\sqrt{2}L$ The magnitude of the angular ... $90^{\circ}$ with $O_2O_4$ as shown. The ratio $\frac{\omega_4}{\omega_2}$ is _________ (round off to two decimal places).
A rigid triangular body, PQR, with sides of equal length of $1$ unit moves on a flat plane. At the instant shown, edge QR is parallel to the $x$-axis, and the body moves such that velocities of points P and R are $V_P$ and $V_R$, in the $x$ and $y$ directions, respectively. The magnitude of the angular velocity of the body is $2V_R$ $2V_P$ $V_R/\sqrt{3}$ $V_P/\sqrt{3}$
Consider a linear elastic rectangular thin sheet of metal, subjected to uniform uniaxial tensile stress of $100$ MPa along the length direction. Assume plane stress conditions in the plane normal to the thickness. The Young’s modulus $E=200$ MPa and Poisson’s ratio $v=0.3$ are given. The principal strains in the plane of the sheet are $(0.35, – 0.15)$ $(0.5, 0.0)$ $(0.5, – 0.15)$ $(0.5, – 0.5)$
A spur gear has pitch circle diameter $D$ and number of teeth $T$. The circular pitch of the gear is $\dfrac{\pi D}{T} \\$ $\dfrac{T}{D} \\$ $\dfrac{D}{T} \\$ $\dfrac{2 \pi D}{T}$
Endurance limit of a beam subjected to pure bending decreases with decrease in the surface roughness and decrease in the size of the beam increase in the surface roughness and decrease in the size of the beam increase in the surface roughness and increase in the size of the beam decrease in the surface roughness and increase in the size of the beam
The figure shows an idealized plane truss. If a horizontal force of $300$ N is applied at point A, then the magnitude of the force produced in member CD is ____ N
The state of stress at a point in a component is represented by a Mohr’s circle of radius $100$ MPa centered at $200$ MPa on the thermal stress axis. On a plane passing through the same point, the normal stress is $260$ MPa. The magnitude of the shear stress on the same plane at the same point is _______ MPa
A wire of circular cross-section of diameter $1.0$ mm is bent into a circular arc of radius. $1.0$ m by application of pure bending moments at its ends. The Young’s modulus of the material of the wire is $100$ GPa. The maximum tensile stress developed in the wire is ______ MPa
A ball of mass $3$ kg moving with a velocity of $4 \:m/s$ undergoes a perfectly-elastic direct-central impact with a stationary ball of mass $m$. After the impact is over, the kinetic energy of the $3$ kg ball is $6$ J. The possible value(s) of $m$ is/are $1$ kg only $6$ kg only $1$ kg, $6$ kg $1$ kg, $9$ kg
Consider two concentric circular cylinders of different materials $M$ and $N$ in contact with each other at $r=b$, as shown below. The interface at $r=b$ is frictionless. The composite cylinder is subjected to internal pressure $P$. Let $(u_r^M, u_{\theta}^M)$ ... $\sigma_{rr}^M = \sigma_{rr}^N \text{ and } \sigma_{\theta \theta}^M = \sigma_{\theta \theta}^N$
A prismatic, straight, elastic, cantilever beam is subjected to a linearly distributed transverse load as shown below. If the beam length is $L$, Young’s modulus $E$, and area moment of inertia $I$, the magnitude of the maximum deflection is $\dfrac{qL^4}{15EI} \\$ $\dfrac{qL^4}{30EI} \\$ $\dfrac{qL^4}{10EI} \\$ $\dfrac{qL^4}{60EI}$
A slender uniform rigid bar of mass $m$ is hinged at $O$ and supported by two springs, with stiffness, $3k$ and $k$ and a damper with damping coefficient $c$, as shown in the figure. For the system to be critically damped, the ratio $c/\sqrt{km}$ should be $2$ $4$ $2 \sqrt{7}$ $4 \sqrt{7}$
The crank of a slider-crank mechanism rotates counter-clockwise (CCW) with a constant angular velocity $\omega$, as shown. Assume the length of the crank to be $r$. Using exact analysis, the acceleration of the slider in the $y$-direction, at the instant shown, where the crank is parallel to $x$-axis, is given by $- \omega ^2r$ $2 \omega ^2r$ $\omega ^2r$ $-2 \omega ^2r$
A horizontal cantilever beam of circular cross-section, length $1.0$ m and flexural rigidity $EI=200 \: N \cdot m^2$ is subjected to an applied moment $M_A = 1.0 \: N \cdot m$ at the free end as shown in the figure. The magnitude of the vertical deflection of the free end is ______ mm (round off to one decimal place).
Two masses $A$ and $B$ having mass $m_a$ and $m_b$, respectively, lying in the plane of the figure shown, are rigidly attached to a shaft which revolves about an axis through O perpendicular to the plane of the figure. The radii of rotation of the masses $m_a$ and $m_b$ ... $r_b=400$mm, then the balance mass to be placed at a radius of $200$ mm is _________ kg (round off to two decimal places).
A uniform disc with radius $r$ and a mass $m$ kg is mounted centrally on a horizontal axle of negligible mass and length of $1.5r$. The disc spins counter-clockwise about the axle with angular speed $\omega$, when viewed from the right-hand side bearing, Q. The axle ... $g=10 m/s^2$, the ratio of the larger to the smaller bearing reaction force (considering appropriate signs) is ______
A short shoe external drum brake is shown in the figure. The diameter of the brake drum is $500 \: mm$. The dimensions $a=1000 \: mm, \: b=500 \: mm$ and $c=200 \: mm$. The coefficient of friction between the drum and the shoe is $0.35$. The force ... shown in the figure. The drum is rotating anti-clockwise. The braking torque on the drum is ______ $N \cdot m$ (round off to two decimal places).