List of top Applied Mechanics and Design Questions

A rigid homogeneous uniform block of mass 1 kg, height $h = 0.4$ m and width $b = 0.3$ m is pinned at one corner and placed upright in a uniform gravitational field ($g = 9.81$ m/s$^2$), supported by a roller as shown. A short duration impulsive force $F$, producing an impulse $I_F$, is applied at a height $d = 0.3$ m from the bottom. Assume all joints to be frictionless. The minimum value of $I_F$ required to topple the block is ________________.
A linear elastic structure under plane stress condition is subjected to two sets of loading, I and II. The resulting states of stress at a point corresponding to these two loadings are shown in the figure below. If these two sets of loading are applied simultaneously, then the net normal component of stress $\sigma_{xx$ is ________________.}
A rigid body in the X–Y plane consists of two point masses (1 kg each) attached to the ends of two massless rods, each of 1 cm length, as shown in the figure. It rotates at 30 RPM counter-clockwise about the Z-axis passing through point O. A point mass of $\sqrt{2$ kg, attached to one end of a third massless rod, is used for balancing the body by attaching the free end of the rod to point O. The length of the third rod is ________________ cm.}
A spring–mass–damper system (mass $m$, stiffness $k$, damping coefficient $c$) excited by a force $F(t)=B\sin \omega t$ is shown in the figure. Four different responses of the system (marked as (i)–(iv)) are shown. In the figures, $A$ is the amplitude of the oscillatory response, and the dashed lines show its envelope. The responses represent only qualitative trends. Four different forcing and parameter conditions are also given:
\[ \text{(P)}\ c>0,\quad \omega=\sqrt{\frac{k}{m}}, \qquad \text{(Q)}\ c<0\ \text{and}\ \omega\ne 0, \] \[ \text{(R)}\ c=0,\quad \omega=\sqrt{\frac{k}{m}}, \qquad \text{(S)}\ c=0,\quad \omega \approx \sqrt{\frac{k}{m}}. \] Which option correctly matches each condition to its response (i)–(iv)?
A rope with two mass–less platforms passes over a fixed pulley. Each disc weighs 20 N. For $n=5$ and $m=0$, a downward force $F = 200$ N applied on the right platform (figure (i)) is just sufficient to initiate upward motion of the left platform. If the force $F$ is removed (figure (ii)), the minimum value of $m$ required to prevent downward motion of the left platform is ______________ (in integer).
A square plate is supported in four different ways (configurations (P) to (S) as shown in the figure). A couple moment $C$ is applied on the plate. Assume all the members to be rigid and mass-less, and all joints to be frictionless. All support links of the plate are identical. The square plate can remain in equilibrium in its initial state for which one or more of the following support configurations?
A structural member under loading has a uniform state of plane stress which in usual notations is given by $\sigma_x = 3P$, $\sigma_y = -2P$ and $\tau_{xy} = \sqrt{2}\,P$, where $P>0$. The yield strength of the material is 350 MPa. If the member is designed using the maximum distortion energy theory, then the value of $P$ at which yielding starts (according to the maximum distortion energy theory) is
A massive uniform rigid circular disc is mounted on a frictionless bearing at the end E of a massive uniform rigid shaft AE which is suspended horizontally in a uniform gravitational field by two identical light inextensible strings AB and CD as shown, where G is the center of mass of the shaft-disc assembly and g is the acceleration due to gravity. The disc is then given a rapid spin $\omega$ about its axis in the positive x-axis direction as shown, while the shaft remains at rest. The direction of rotation is defined using the right-hand thumb rule. If the string AB is suddenly cut, assuming negligible energy dissipation, the shaft AE will

The torque provided by an engine is given by \( T(\theta) = 12000 + 2500 \sin(2\theta) \, \text{N.m}, \) where \( \theta \) is the angle turned by the crank from inner dead center. The mean speed of the engine is 200 rpm and it drives a machine that provides a constant resisting torque varying as \( 200 + 200 \cos(\theta) \). If variation of the speed from the mean speed is not to exceed \( \pm 0.5% \), the minimum mass moment of inertia of the flywheel should be \(\underline{\hspace{1cm}}\) kg.m² (round off to nearest integer).

Consider the system shown in the figure. A rope goes over a pulley. A mass, \( m \), is hanging from the rope. A spring of stiffness, \( k \), is attached at one end of the rope. Assume rope is inextensible, massless and there is no slip between pulley and rope. 

The pulley radius is \( r \) and its mass moment of inertia is \( J \). Assume that the mass is vibrating harmonically about its static equilibrium position. The natural frequency of the system is 
 

A block of negligible mass rests on a surface that is inclined at 30° to the horizontal plane as shown in the figure. When a vertical force of 900 N and a horizontal force of 750 N are applied, the block is just about to slide. 

The coefficient of static friction between the block and surface is \(\underline{\hspace{2cm}}\) (round off to two decimal places).
 

The wheels and axle system lying on a rough surface is shown in the figure. 

Each wheel has diameter 0.8 m and mass 1 kg. Assume that the mass of the wheel is concentrated at rim and neglect the mass of the spokes. The diameter of axle is 0.2 m and its mass is 1.5 kg. Neglect the moment of inertia of the axle and assume \( g = 9.8 \, \text{m/s}^2 \). An effort of 10 N is applied on the axle in the horizontal direction shown at mid span of the axle. Assume that the wheels move on a horizontal surface without slip. The acceleration of the wheel axle system in horizontal direction is \(\underline{\hspace{1cm}}\) m/s² (round off to one decimal place).

A plane frame PQR (fixed at P and free at R) is shown in the figure. Both members (PQ and QR) have length \( L \), and flexural rigidity \( EI \). Neglecting the effect of axial stress and transverse shear, the horizontal deflection at free end, \( R \), is

A power transmission mechanism consists of a belt drive and a gear train as shown in the figure. 

Diameters of pulleys of belt drive and number of teeth (T) on the gears 2 to 7 are indicated in the figure. The speed and direction of rotation of gear 7, respectively, are

A steel cubic block of side 200 mm is subjected to hydrostatic pressure of 250 N/mm². The elastic modulus is \( 2 \times 10^5 \) N/mm² and Poisson ratio is 0.3 for steel. The side of the block is reduced by \(\underline{\hspace{2cm}}\) \(\text{mm (round off to two decimal places).}\)
A column with one end fixed and one end free has a critical buckling load of 100 N. For the same column, if the free end is replaced with a pinned end then the critical buckling load will be \(\underline{\hspace{1cm}}\) N (round off to the nearest integer).
The von Mises stress at a point in a body subjected to forces is proportional to the square root of the

The controlling force curves P, Q and R for a spring controlled governor are shown in the figure, where \( r_1 \) and \( r_2 \) are any two radii of rotation. 

The characteristics shown by the curves are

Consider the mechanism shown in the figure. There is rolling contact without slip between the disc and ground. 

Select the correct statement about instantaneous centers in the mechanism. 
 

A plane truss PQRS (PQ = RS, and \( \angle PQR = 90^\circ \)) is shown in the figure. The forces in the members PR and RS, respectively, are \(\underline{\hspace{2cm}}\) 

The figure shows the relationship between fatigue strength (S) and fatigue life (N) of a material. The fatigue strength of the material for a life of 1000 cycles is 450 MPa, while its fatigue strength for a life of \( 10^6 \) cycles is 150 MPa. The life of a cylindrical shaft made of this material subjected to an alternating stress of 200 MPa will then be \(\underline{\hspace{1cm}}\) cycles (round off to the nearest integer). \includegraphics[width=0.5\linewidth]{image55.png}
A cantilever beam with a uniform flexural rigidity \( EI = 200 \times 10^6 \, \text{N.m}^2 \) is loaded with a concentrated force at its free end. The area of the bending moment diagram corresponding to the full length of the beam is 10000 N.m². The magnitude of the slope of the beam at its free end is \(\underline{\hspace{1cm}}\) micro-radian (round off to nearest integer).
A machine of mass 100 kg is subjected to an external harmonic force with a frequency of 40 rad/s. The designer decides to mount the machine on an isolator to reduce the force transmitted to the foundation. The isolator can be considered as a combination of stiffness (\(K\)) and damper (damping factor, \(\xi\)) in parallel. The designer has the following four isolators: 1) \(K = 640 \, \text{kN/m}, \, \xi = 0.70\)
2) \(K = 640 \, \text{kN/m}, \, \xi = 0.07\)
3) \(K = 22.5 \, \text{kN/m}, \, \xi = 0.70\)
4) \(K = 22.5 \, \text{kN/m}, \, \xi = 0.07\) Arrange the isolators in the ascending order of the force transmitted to the foundation.
A band brake shown in the figure has a coefficient of friction of 0.3. The band can take a maximum force of 1.5 kN. The maximum braking force (F) that can be safety applied is ______ N. (Answer in integers)

A horizontal beam of length 1200 mm is pinned at the left end and is resting on the roller at the other end as shown in the figure. A linearly varying distributed load is applied on the beam. The magnitude of maximum bending moment acting on the beam is _____ Nm (Round off to one decimal place)

A horizontal beam of length 1200 mm is pinned at the left