List of top Applied Mechanics and Design Questions asked in GATE ME

Two plates of thickness 10 mm each are to be joined by a transverse fillet weld on one side and the resulting structure is loaded as shown in the figure below. If the ultimate tensile strength of the weld material is 150 MPa and the factor of safety to be used is 3, the minimum length of the weld required to ensure that the weld does NOT fail is ____________ mm (rounded off to 2 decimal places).


 

An offset slider-crank mechanism is shown in the figure below. The length of the stroke of the slider is __________ mm (rounded off to nearest integer).


 

The system shown in the figure below consists of a cantilever beam (with flexural rigidity \( EI \) and negligible mass), a spring (with spring constant \( K \) and negligible mass) and a block of mass \( m \). Assuming a lumped parameter model for the system, the fundamental natural frequency (\( \omega_n \)) of the system is 


 

A shaft carries a helical spur gear. Which one of the following bearings can NOT be used to support it?
In the context of balancing of rotating and reciprocating masses, which one of the following options is true?

A truss structure is loaded as shown in the figure below. Among the options given, which member in the truss is a zero-force member?

\[ {Given: } F = 1000\,{N} \]

A rigid circular disc of radius \(r\) (in m) is rolling without slipping on a flat surface as shown in the figure below. The angular velocity of the disc is \(\omega\) (in rad/ssuperscript{-1}). The velocities (in m/ssuperscript{-1}) at points 0 and A, respectively, are:


 

A pair of spur gears is required to maintain a velocity ratio of 1:2. The module of the gears is 10 mm and the addendum is 10 mm. If the operating pressure angle is 15°, the minimum number of teeth required on the pinion to ensure NO interference/undercutting is (answer in integer).
The endurance limit of a specific grade of steel is same as its yield strength. The ultimate strength of this grade of steel is twice of its yield strength. A component made of this steel is loaded in tension and unloaded periodically. It is required that the component does NOT fail for at least \( 10^6 \) loading cycles, as per the Soderberg law. Considering a factor of safety of 2, the maximum applied tensile principal stress is:
A block of mass 1 kg connected to a spring of stiffness 10 N m\(^{-1}\) is operating in a viscous medium such that the damping ratio (or damping factor) is equal to the ratio of the damped frequency to the natural frequency. The magnitude of the damping ratio for this system is _________ (rounded off to 2 decimal places).
A thin-walled cylindrical pressure vessel has mean wall thickness of \( t \) and nominal radius of \( r \). The Poisson's ratio of the wall material is \( \frac{1}{3} \). When it was subjected to some internal pressure, its nominal perimeter in the cylindrical portion increased by 0.1% and the corresponding wall thickness became \( \bar{t} \). The corresponding change in the wall thickness of the cylindrical portion, i.e. \( 100 \times \frac{\bar{t} - t}{t} \), is ________________% (round off to 3 decimal places).
A schematic of an epicyclic gear train is shown in the figure. The sun (gear 1) and planet (gear 2) are external, and the ring gear (gear 3) is internal. Gear 1, gear 3 and arm OP are pivoted to the ground at O. Gear 2 is carried on the arm OP via the pivot joint at P, and is in mesh with the other two gears. Gear 2 has 20 teeth and gear 3 has 80 teeth. If gear 1 is kept fixed at 0 rpm and gear 3 rotates at 900 rpm counter clockwise (ccw), the magnitude of angular velocity of arm OP is ________________ rpm (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 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).
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 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 ________________.
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 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
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.
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).
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 methods can improve the fatigue strength of a circular mild steel (MS) shaft?
A square threaded screw is used to lift a load \( W \) by applying a force \( F \). Efficiency of square threaded screw is expressed as
The figure shows a schematic of a simple Watt governor mechanism with the spindle \( O_1O_2 \) rotating at an angular velocity \( \omega \) about a vertical axis. The balls at P and S have equal mass. Assume that there is no friction anywhere and all other components are massless and rigid. The vertical distance between the horizontal plane of rotation of the balls and the pivot \( O_1 \) is denoted by \( h \). The value of \( h = 400 \, \text{mm} \) at a certain \( \omega \). If \( \omega \) is doubled, the value of \( h \) will be ________________ mm.