# GATE Questions & Answers of Engineering Mechanics Mechanical Engineering

#### Engineering Mechanics 61 Question(s) | Weightage 03 (Marks)

A bar of uniform cross section and weighing 100 N is held horizontally using two massless and inextensible strings S1 and S2 as shown in the figure.

The tensions in the strings are

A point mass is shot vertically up from ground level with a velocity of 4 m/s at time, $t=0$. It loses 20% of its impact velocity after each collision with the ground. Assuming that the acceleration due to gravity is 10 $m/s^2$ and that air resistance is negligible, the mass stops bouncing and comes to complete rest on the ground after a total time (in seconds) of

Block P of mass 2 kg slides down the surface and has a speed 20 m/s at the lowest point, Q, where the local radius of curvature is 2 m as shown in the figure. Assuming g = 10  m/s2 , the normal force (in N) at Q is _______ (correct to two decimal places).

A force of 100 N is applied to the centre of a circular disc, of mass 10 kg and radius 1 m, resting on a floor as shown in the figure. If the disc rolls without slipping on the floor, the linear acceleration (in m/s2) of the centre of the disc is ________ (correct to two decimal places).

A partical of unit mass is moving on a plane. Its trajectory, in polar coordinates, is given by $\style{font-family:'Times New Roman'}{r(t)=t^2,\theta(t)\;=t}$, where t is time. The kinetic energy of the partical at time t=2 is

The following figure shows the velocity-time plot for a particle traveling along a staright line. The distance coverd by the particle from t = 0 to t = 5 s is ____________ m,

A point mass of 100 kg is dropped onto a massless elastic bar (cross-sectional area = 100 mm2, length = 1 m, Young's moduls = 100 GPa) from a height H of 10 mm as shown (Figure is not to scale). If g = 10 m/s2, the maximum compression of the elastic bar is ____________mm.

Two disks A and B with identical mass (m) and radius (R) are initialy at rest. They roll down from the top of identical inclined planes without slipping. Disk A has all of its mass concentrated at the rim, while Disk B has its mass uniformly distributed. At the bottom of the plane, the ratio of velocity  of the center of disk A to the velocity of the center of disk B is

The rod PQ of length $L=\sqrt{2}$ , and uniformly distributed mass of $M=10\;kg$, is released from rest at the position shown in the figure. The ends slide along the frictionless faces OP and OQ.  Assume acceleration due to gravity, $g=10\;m/s^2$. The mass moment of inertia of the rod about its centre of mass and an axis perpendicular to the figure is $(ML^2/12)$. At this instant, the magnitude of angular acceleration (in radian/s2) of the rod is____________

A rigid ball of weight 100 N is suspended with the help of a string. The ball is pulled by a horizontal force $F$ such that the string makes an angle of $30^\circ$ with the vertical. The magnitude of force $F$ (in N) is __________

A point mass M is released from rest and slides down a spherical bowl (of radius R) from a height H as shown in the figure below. The surface of the bowl is smooth (no friction). The velocity of the mass at the bottom of the bowl is

A block of mass $m$ rests on an inclined plane and is attached by a string to the wall as shown in the figure. The coefficient of static friction between the plane and the block is 0.25. The string can withstand a maximum force of 20 N. The maximum value of the mass $(m)$ for which the string will not break and the block will be in static equilibrium is ____________ kg.

Take $\cos\theta=0.8$ and $\sin\theta=0.6$.

Acceleration due to gravity g = 10 m/$s^2$

A two-member truss $PQR$ is supporting a load $W$. The axial forces in members $PQ$ and $QR$ are respectively

A point mass having mass $M$ is moving with a velocity $V$ at an angle $\theta$ to the wall as shown in the figure. The mass undergoes a perfectly elastic collision with the smooth wall and rebounds. The total change (final minus initial) in the momentum of the mass is

A mass of 2000 kg is currently being lowered at a velocity of 2 m/s from the drum as shown in the figure. The mass moment of inertia of the drum is $150\mathrm{kg}-\mathrm m^2$. On applying the brake, the mass is brought to rest in a distance of 0.5 m. The energy absorbed by the brake (in kJ) is __________

A system of particles in motion has mass center $G$ as shown in the figure. The particle $i$ has mass $m_i$ and its position with respect to a fixed point $O$ is given by the position vector ${\boldsymbol r}_\mathbf i$. The position of the particle with respect to $G$ is given by the vector ${\boldsymbol\rho}_\mathbf i$. The time rate of change of the angular momentum of the system of particles about $G$ is

(The quantity ${\overset{\boldsymbol.\boldsymbol.}{\mathbf\rho}}_\mathbf i\boldsymbol\infty$ indicates second derivative of ${\boldsymbol p}_\mathbf i$ with respect to time and likewise for $\boldsymbol r\boldsymbol i$ ).

A force F is acting on a bent bar which is clamped at one end as shown in the figure.

The CORRECT free body diagram is

A rigid link PQ is undergoing plane motion as shown in the figure ($V_P$ and $V_Q$ are non - zero). $V_{QP}$ is the relative velocity of point Q with respect to point P.

Which one of the following is TRUE?

An inextensible massless string goes over a frictionless pulley. Two weights of 100 N and 200 N are attached to the two ends of the string. The weights are released from rest, and start moving due to gravity. The tension in the string (in N) is __________

A circular disc of radius 100 mm and mass 1 kg, initially at rest at position A, rolls without slipping down a curved path as shown in figure. The speed $v$ of the disc when it reaches position B is _________ m/s.

Acceleration due to gravity g = 10 m/s2.

A rigid rod (AB) of length $L=\sqrt{2}$ m is undergoing translational as well as rotational motion in the x-y plane (see the figure). The point A has the velocity $V_{1\;}=\widehat i+2\widehat j$ m/s. The end B is constrained to move only along the x direction.

The magnitude of the velocity V2 (in m/s) at the end B is __________

A wheel of radius r rolls without slipping on a horizontal surface shown below. If The velocity of point P is 10 m/s in the horizontal direction, the magnitude of velocity of point Q (in m/s) is ________

Two identical trusses support a load of 100 N as shown in the figure. The length of each truss is 1.0 m; cross-sectional area is 200 mm2, Young’s modulus E = 200 GPa. The force in the truss AB (in N) is ______.

For the turss shown in figure, the magnitude of the force in member PR and the support reaction at R are respectively

A ball of mass 0.1 kg, initially at rest, is dropped from height of 1 m. Ball hits the ground and bounces off the ground. Upon impact with the ground, the velocity reduces by 20% . The height (in m) to which the ball will rise is _____.

A small ball of mass 1 kg moving with a velocity of 12 m/s undergoes a direct central impact with a stationary ball of mass 2 kg. The impact is perfectly elastic. The speed (in m/s) of 2 kg mass ball after the impact will be____.

The initial velocity of an object is 40 m/s. The acceleration $a$ of the object is given by the following expression:

$a=-0.1\;v,$

Where $v$ is the instantaneous velocity of the object. The velocity of the object after 3 seconds will be____.

For the truss shown in figure , the magnitude of the force (in kN) in the member SR is

A weight of 500 N is supported by two metallic ropes as shown in the figure. The values of tensions T1 and T2 are respectively

The value of moment of inertia of the section shown in the figure about the axis-XX is

Figure shows a wheel rotating about O2. Two points A and B located along the radius of wheel have speeds of 80 m/s and 140 m/s respectively. The distance between the points A and B is 300 mm. The diameter of the wheel (in mm) is__________

A bullet spins as the shot is fired from a gun. For this purpose, two helical slots as shown in the figure are cut in the barrel. Projections A and B on the bullet engage in each of the slots.

Helical slots are such that one turn of helix is completed over a distance of 0.5 m. If velocity of bullet when it exits the barrel is 20 m/s, its spinning speed in rad/s is _______.

A  circular object of radius r rolls without slipping on a horizontal level floor with center having velocity V. The velocity at the point of contact between the object and floor is

A block R of mass 100 kg  is placed on a block S of mass 150 kg as shown in the figure.Block R is tied to the wall by a massless and  inextensible string PQ.If the coefficient of static friction for all surfaces is 0.4, the minimum force F (in n kN) needed to move the block S is

A block weighing 200 N is in contact with a level plane whose coefficients of static and kinetic friction are 0.4 and 0.2, respectively. The block is acted upon by a horizontal force (in newton) P=10t, where t denotes the time in seconds. The velocity (in m/s) of the block attained after 10 seconds is _______

A two member truss ABC is shown in the figure. The force (in kN) transmitted in member AB is _______

A truck accelerates up a 10° incline with a crate of 100 kg. Value of static coefficient of friction between the crate and the truck surface is 0.3. The maximum value of acceleration (in m/s2) of the truck such that the crate does not slide down is _______

A mass m1 of 100 kg travelling with a uniform velocity of 5 m/s along a line collides with a stationary mass m2 of 1000 kg. After the collision, both the masses travel together with the same velocity. The coefficient of restitution is

A body of mass (M) 10 kg is initially stationary on a 45° inclined plane as shown in figure. The coefficient of dynamic friction between the body and the plane is 0.5. The body slides down the plane and attains a velocity of 20 m/s. The distance travelled (in meter) by the body along the plane is _______

An annular disc has a mass m, inner radius R and outer radius 2R. The disc rolls on a flat surface without slipping. If the velocity of the center of mass is v, the kinetic energy of the disc is

In a statically determinate plane truss, the number of joints (j) and the number of members (m) are related by

A point mass is executing simple harmonic motion with an amplitude of 10 mm and frequency of 4 Hz. The maximum acceleration (m/s2) of the mass is _______

For the truss shown in the figure, the forces F1 and F2 are 9 kN and 3 kN, respectively. The force (in kN) in the member QS is

A frame is subjected to a load P as shown in the figure. The frame has a constant flexural rigidity EI. The effect of axial load is neglected. The deflection at point A due to the applied load P is

A wardrobe (mass 100 kg, height 4 m, width 2 m, depth 1 m), symmetric about the Y-Y axis, stands on a rough level floor as shown in the figure. A force P is applied at mid-height on the wardrobe so as to tip it about point Q without slipping. What are the minimum values of the force (in newton) and the static coefficient of friction μ between the floor and the wardrobe, respectively?

A ladder AB of length 5 m and weight (W) 600 N is resting against a wall. Assuming frictionless contact at the floor (B) and the wall (A), the magnitude of the force P (in newton) required to maintain equilibrium of the ladder is _______

A pin jointed uniform rigid rod of weight W and length L is supported horizontally by an external force F as shown in the figure below. The force F is suddenly removed. At the instant of force removal, the magnitude of vertical reaction developed at the support is

A force of 400 N is applied to the brake drum of 0.5 m diameter in a band-brake system as shown in the figure, where the wrapping angle is 180°. If the coefficient of friction between the drum and the band is 0.25, the braking torque applied, in N.m. is

Two steel truss members, AC and BC, each having cross sectional area of 100 mm2, are subjected to a horizontal force F as shown in figure. All the joints are hinged.

If F = 1 kN, the magnitude of the vertical reaction force developed at the point B in kN is

Two steel truss members, AC and BC, each having cross sectional area of 100 mm2, are subjected to a horizontal force F as shown in figure. All the joints are hinged.

The maximum force F in kN that can be applied at C such that the axial stress in any of the truss members DOES NOT exceed 100 MPa is

The coefficient of restitution of a perfectly plastic impact is

A stone with mass of 0.1 kg is catapulted as shown in the figure. The total force Fx (in N) exerted by the rubber band as a function of distance x (in m) is given by Fx = 300x2. If the stone is displaced by 0.1m from the un-stretched position (x = 0) of the rubber band, the energy stored in the rubber band is

A 1 kg block is resting on a surface with coefficient of friction μ = 0.1. A force of 0.8 N is applied to the block as shown in figure. The friction force is

A block weighing 981 N is resting on a horizontal surface. The coefficient of friction between the block and the horizontal surface is µ = 0.2.A vertical cable attached to the block provides partial support as shown. A man can pull horizontally with a force of 100 N. What will be the tension, T (in N) in the cable if the man is just able to move the block to the right?

A uniform rigid rod of mass M and length L is hinged at one end as shown in the adjacent figure. A force P is applied at a distance of 2L/3 from the hinge so that the rod swings to the right. The reaction at the hinge is

A straight rod of length L(t), hinged at one end and freely extensible at the other end, rotates through an angle about the hinge. At time t, L(t) =1 m,$\dot L\left(t\right)$ = 1 m/s, $\theta \left(t\right)=\frac{\mathrm{\pi }}{4}$ rad and $\dot\theta\left(t\right)$= 1 rad/s. The magnitude of the velocity at the other end of the rod is

A cantilever type gate hinged at Q is shown in the figure. P and R are the centers of gravity of the cantilever part and the counterweight respectively. The mass of the cantilever part is 75 kg. The mass of the counterweight, for static balance, is

A circular disk of radius R rolls without slipping at a velocity v. The magnitude of the velocity at point P (see figure) is

Consider a truss PQR loaded at P with a force F as shown in the figure

The tension in the member QR is

A block of mass M is released from point P on a rough inclined plane with inclination angle $\theta$, shown in the figure below. The coefficient of friction is $\mu$. If $\mu <\mathrm{tan}\theta$, then the time taken by the block to reach another point Q on the inclined plane, where PQ = s, is