Engineering Mechanics

In order to determine the effects of a force acting on a body, we must know

A rigid body is in equilibrium if sum of all the

Free-body diagram means

The unit of power in S.I. units

The coefficient of friction depends upon

A propped cantilever will have _____ redundant reaction.

According to Lami's theorem, the three forces

The term 'virtual work' refers to

Theorem of perpendicular axis is used in obtaining the moment of inertia of a

Which of the following statements is false about trusses?

A machine component 1.5 m long and weight 1000 N is supported by two ropes AB and CD as shown in Fig. 1 given below. Calculate the tensions $ T_1 and T_2 $ in the ropes AB and CD.

Show that the algebraic sum of the resolved part of a number of forces in a given direction, is equal to the resolved part of their resultant in the same direction.

State the principle of virtual work, and explain how it can be used for solving problems in statics. Two beams AE and BD are supported on rollers at B and C as shown in Figure. Determine the reactions at the rollers B and C, using the method of virtual work.

State the laws of motion. Discuss the first law in the light of second law.

A race car travels around a circular track that has a radius of 300 m. If the car increases its speed at a constant rate of $ 7 m/s^2 $ starting from rest, determine the time needed for it to reach an acceleration of $ 10 m/s^2 $.

A body consisting of a cone and a hemisphere of radius r fixed on the same base, rests on a table. Find the greatest height h of the cone, so that the combined body may stand upright.

What is a frame? Discuss its classification. Distinguish between a perfect frame and an imperfect frame.

Find the moment of inertia of a hollow sphere with respect to a diameter if the unit weight of the material is $ gamma $ and if the outer and inner radii are $ r_o $ and $ r_i $, respectively.

The coefficient of static friction between the block A and the cart B is $ mu $. If the assembly is released from rest on the inclined plane, determine the smallest value of $ mu $ that will prevent the block from sliding on the cart. Find the answer as a function of $ heta $.

State and explain D'Alembert's principle.

A uniform disc of radius r is allowed to roll down a rough inclined plane whose angle of inclination with the horizontal is $ \theta $. Prove that the linear acceleration of the disc is given by: $ a = \frac{g \sin \theta}{\frac{r^2 + k^2}{r^2}} $ where k is the radius of gyration.

In order to determine the effects of a force acting on a body, we must know

A rigid body is in equilibrium if sum of all the

Free-body diagram means

The unit of power in S.I. units

The coefficient of friction depends upon

A propped cantilever will have _____ redundant reaction.

According to Lami's theorem, the three forces

The term 'virtual work' refers to

Theorem of perpendicular axis is used in obtaining the moment of inertia of a

Which of the following statements is false about trusses?

A machine component 1.5 m long and weight 1000 N is supported by two ropes AB and CD as shown in Fig. 1 given below. Calculate the tensions $ T_1 and T_2 $ in the ropes AB and CD.

Show that the algebraic sum of the resolved part of a number of forces in a given direction, is equal to the resolved part of their resultant in the same direction.

State the principle of virtual work, and explain how it can be used for solving problems in statics. Two beams AE and BD are supported on rollers at B and C as shown in Figure. Determine the reactions at the rollers B and C, using the method of virtual work.

State the laws of motion. Discuss the first law in the light of second law.

A race car travels around a circular track that has a radius of 300 m. If the car increases its speed at a constant rate of $ 7 m/s^2 $ starting from rest, determine the time needed for it to reach an acceleration of $ 10 m/s^2 $.

A body consisting of a cone and a hemisphere of radius r fixed on the same base, rests on a table. Find the greatest height h of the cone, so that the combined body may stand upright.

What is a frame? Discuss its classification. Distinguish between a perfect frame and an imperfect frame.

Find the moment of inertia of a hollow sphere with respect to a diameter if the unit weight of the material is $ gamma $ and if the outer and inner radii are $ r_o $ and $ r_i $, respectively.

The coefficient of static friction between the block A and the cart B is $ mu $. If the assembly is released from rest on the inclined plane, determine the smallest value of $ mu $ that will prevent the block from sliding on the cart. Find the answer as a function of $ heta $.

State and explain D'Alembert's principle.

A uniform disc of radius r is allowed to roll down a rough inclined plane whose angle of inclination with the horizontal is $ \theta $. Prove that the linear acceleration of the disc is given by: $ a = \frac{g \sin \theta}{\frac{r^2 + k^2}{r^2}} $ where k is the radius of gyration.

$ C_1 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} $ is an identity matrix then it is equivalent to perform rotation

Two cylinders have the same mass and radius. One is hollow and the other is solid. Which one will have the greater moment of inertia about the central axis?

Single force and a couple acting in the same plane upon a rigid body

If the masses of both the bodies, as shown in the figure, are doubled, then the acceleration in the string will be

The total energy possessed by a system of moving bodies

Principle of transmissibility for free body diagrams is:

The maximum frictional force which comes into play when a body just begins to slide over another surface is called

The motion of a particle (distance in metres and time in seconds) is given by the equation $ S = 2t^3 + 3t $. The distance of starting from $ t=0 $, to attain a velocity of $ 9\text{ m/s} $, the particle will have to travel a

A body of weight W is placed on an inclined plane. The angle made by the inclined horizontal, when the body is on the point of moving down is called

When the car moves on road its wheel has

A 5 m ladder weighing 250 N is placed against a smooth vertical wall with its lower end 3 m away from the wall as shown in fig-1 . If the coefficient of friction between the ladder and the floor is 0.3, show that the ladder will remain in equilibrium in this position.

Block weighing 1000 N is resting on a horizontal surface. The coefficient of friction between the block and the horizontal surface is $ \mu=0.2 $. A vertical cable attached to the block provides partial support as shown in fig-2 . 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 wheel of 600 mm diameter, weighing 10KN rests against a rigid rectangular block of 150mm height as shown in fig-3 . Find the least pull, through the centre of the wheel, required just to turn the wheel over the corner A of the block. Also find the reaction of the block. Take the entire surface to be smooth.

The mass of each ball is 200 grams, and connected by a cord. The length of the cord is 80 cm, and the width of the cord is 40 cm. What is the moment of inertia of the balls about the axis of rotation (Ignore cord's mass)?

A beam 3m long weighing 400 N is suspended in a Horizontal position by two vertical strings, each of which can withstand a maximum tension of 350 N only as shown in fig-4 . How far a body of 200N weight be placed on the beam, so that one of the strings may just break?

Smooth circular cylinder of radius 0.25 meter is lying in a triangular groove, one side of which makes $ 30^{\circ} $ angle and the other $ 45^{\circ} $ angle with the horizontal. Find the reactions at the surfaces of contact, if there is no friction and the cylinder weights 100 N.

A 8 m long simply supported beam with overhangs, rests on supports 4m apart. The left end overhanging is 3 m. The beam carries load of 20 kN and 10 kN on the left and the right ends respectively. Draw S.F.D & B.M.D. Locate point of contraflexure, if any.

Obtain the metric tensor for two dimensional plane in polar coordinates.

Show that any tensor of rank 2 can be expressed as sum of a symmetric and an antisymmetric tensors of rank 2.

The angular velocity of the rigid body is defined by the vector: $ W=w_1 i+w_2 j+w_3 k $. Obtain an expression for this angular velocity in terms of the Euler angles, $ \theta $, $ \phi $ and $ \psi $ in the i, j, and k directions.

A car moving with a velocity of 10 m/s shows down in such a manner that the relation between velocity and time is given by: $ v = 10-t^2-\frac{t^3}{2} $. Find the distance travelled in two seconds, average velocity and average retardation of the car in these two seconds.

A man weighing 750 N stands on the floor of a lift. Find the pressure exerted on the floor when (a) the lift moves upwards with an acceleration of $ 3\text{ m/sec}^2 $ and (b) the lift moves downwards with an acceleration of $ 3\text{ m/sec}^2 $.

A solid shaft transmits 200 kW of power at 600 rpm. Determine the suitable diameter of the shaft if the shear stress is not to exceed 70 MPa and total angle of twist is limited to $ 3^{\circ} $ for 4m length of the shaft, Modulus of rigidity (G) = 80 GPa.

$ C_1 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} $ is an identity matrix then it is equivalent to perform rotation

Two cylinders have the same mass and radius. One is hollow and the other is solid. Which one will have the greater moment of inertia about the central axis?

Single force and a couple acting in the same plane upon a rigid body

If the masses of both the bodies, as shown in the figure, are doubled, then the acceleration in the string will be

The total energy possessed by a system of moving bodies

Principle of transmissibility for free body diagrams is:

The maximum frictional force which comes into play when a body just begins to slide over another surface is called

The motion of a particle (distance in metres and time in seconds) is given by the equation $ S = 2t^3 + 3t $. The distance of starting from $ t=0 $, to attain a velocity of $ 9\text{ m/s} $, the particle will have to travel a

A body of weight W is placed on an inclined plane. The angle made by the inclined horizontal, when the body is on the point of moving down is called

When the car moves on road its wheel has

A 5 m ladder weighing 250 N is placed against a smooth vertical wall with its lower end 3 m away from the wall as shown in fig-1 . If the coefficient of friction between the ladder and the floor is 0.3, show that the ladder will remain in equilibrium in this position.

Block weighing 1000 N is resting on a horizontal surface. The coefficient of friction between the block and the horizontal surface is $ \mu=0.2 $. A vertical cable attached to the block provides partial support as shown in fig-2 . 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 wheel of 600 mm diameter, weighing 10KN rests against a rigid rectangular block of 150mm height as shown in fig-3 . Find the least pull, through the centre of the wheel, required just to turn the wheel over the corner A of the block. Also find the reaction of the block. Take the entire surface to be smooth.

The mass of each ball is 200 grams, and connected by a cord. The length of the cord is 80 cm, and the width of the cord is 40 cm. What is the moment of inertia of the balls about the axis of rotation (Ignore cord's mass)?

A beam 3m long weighing 400 N is suspended in a Horizontal position by two vertical strings, each of which can withstand a maximum tension of 350 N only as shown in fig-4 . How far a body of 200N weight be placed on the beam, so that one of the strings may just break?

Smooth circular cylinder of radius 0.25 meter is lying in a triangular groove, one side of which makes $ 30^{\circ} $ angle and the other $ 45^{\circ} $ angle with the horizontal. Find the reactions at the surfaces of contact, if there is no friction and the cylinder weights 100 N.

A 8 m long simply supported beam with overhangs, rests on supports 4m apart. The left end overhanging is 3 m. The beam carries load of 20 kN and 10 kN on the left and the right ends respectively. Draw S.F.D & B.M.D. Locate point of contraflexure, if any.

Obtain the metric tensor for two dimensional plane in polar coordinates.

Show that any tensor of rank 2 can be expressed as sum of a symmetric and an antisymmetric tensors of rank 2.

The angular velocity of the rigid body is defined by the vector: $ W=w_1 i+w_2 j+w_3 k $. Obtain an expression for this angular velocity in terms of the Euler angles, $ \theta $, $ \phi $ and $ \psi $ in the i, j, and k directions.

A car moving with a velocity of 10 m/s shows down in such a manner that the relation between velocity and time is given by: $ v = 10-t^2-\frac{t^3}{2} $. Find the distance travelled in two seconds, average velocity and average retardation of the car in these two seconds.

A man weighing 750 N stands on the floor of a lift. Find the pressure exerted on the floor when (a) the lift moves upwards with an acceleration of $ 3\text{ m/sec}^2 $ and (b) the lift moves downwards with an acceleration of $ 3\text{ m/sec}^2 $.

A solid shaft transmits 200 kW of power at 600 rpm. Determine the suitable diameter of the shaft if the shear stress is not to exceed 70 MPa and total angle of twist is limited to $ 3^{\circ} $ for 4m length of the shaft, Modulus of rigidity (G) = 80 GPa.

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Fig. 2 shows the cross-section of a cast iron beam. Determine the moments of inertia of the section about horizontal and vertical axes passing through the centroid of the section.

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A body of mass m moving with a constant velocity v strikes another body of same mass moving with same velocity but in opposite direction. The common velocity of both the bodies after collision is

The centre of percussion of the homogeneous rod of length L suspended at the top will be

The figure given below shows the three coplanar forces P, Q and R acting at a point O. If these forces are in equilibrium, then

The angle of inclination of the plane at which the body begins to move down the plane, is called

Pick up wrong statement about friction force for dry surfaces. Friction force is

The term 'centroid' is

The CG of a plane lamina will not be at its geometrical centre in the case of a

If the masses of both the bodies, as shown in the figure below are reduced to 50 percent then tension in the string will be

Forces are called coplanar when all of them acting on body lie in

A weight of 1000 N can be lifted by an effort of 80 N. If the velocity ratio is 20, then the machine is

Four forces of magnitude 10 N, 20 N, 30 N and 40 N are acting respectively along the four sides of a square ABCD as shown in Fig. 1. Determine the resultant moment about the point A. Each side of the square is given 2 m.

Three like parallel forces 100 N, 200 N and 300 N are acting at points A, B and C respectively on a straight line ABC as shown in Fig. 2. The distances are AB = 30 cm and BC = 40 cm. Find the resultant and also the distance of the resultant from point A on line ABC.

Fig. 3 shows a sphere resting in a smooth V-shaped groove and subjected to a spring force. The spring is compressed to a length of 100 mm from the free length of 150 mm. If the stiffness of the spring is $ 2\text{ N/mm} $, determine the contact reaction at A and B.

Using the analytical method, determine the centre of gravity of the plane uniform lamina as shown in Fig. 4.

Explain the conditions for equilibrium of forces in space.

show that $ I_O = I_G + Ah^2 $, where $ I_G $ is the moment of inertia of a lamina about an axis through its centroid and lying in its plane and h is the distance from the centroid to a parallel axis in the same plane about which its moment of inertia is $ I_O $. A being the area of the lamina.

Find the least force required to pull a body of weight W placed on a rough horizontal plane, when the force is applied at an angle $ \theta $ with the horizontal.

A cord connects two bodies of weights 500 N and 1000 N. The two bodies are placed on an inclined plane and cord is parallel to inclined plane. The coefficients of friction for the weight of 500 N is 0.20 and that of 1000 N is 0.4. Determine the inclination of the plane to the horizontal and tension in the cord, when the motion is about to take place, down the inclined plane. The body weight 500 N is below the body weighing 1000 N.

A truss of span 9 m is loaded as shown in Fig. 5. Find the reactions and forces in the members marked.

Define and explain the terms 'perfect frame', 'imperfect frame', 'deficient frame' and 'redundant frame'.

Each of the two uniform hinged bars in Fig. 6 has a mass m and a length l, and is supported and loaded as shown below. For a given force P, determine the angle $ \theta $ for equilibrium.

The mass of the uniform bar of length l in Fig. 7 is m while that of the uniform bar of length 2l is 2m shown below. For a given force P, determine the angle for equilibrium.

A wheel is rotating about its axis with a constant angular acceleration of $ 1\text{ rad/s}^2 $. If the initial and final angular velocities are $ 5.25\text{ rad/s} $ and $ 10.5\text{ rad/s} $, determine the total angle turned through during the time interval this change of angular velocity took place.

(i) A flywheel starts rotating from rest and is given an acceleration of $ 1\text{ rad/s}^2 $. Find the angular velocity and speed in r.p.m. after 1.5 minutes. (ii) If the flywheel is brought to rest with a uniform angular retardation of $ 0.5\text{ rad/s}^2 $, determine the time taken by the flywheel in seconds to come to rest.

State and prove Varignon's theorem.

Derive the equation of path of a projectile and hence show the equation of path of projectile is a parabolic curve.

A particle is moving in X-Y plane and its position is defined by $ \vec{r} = (\frac{3}{2}t^2)\hat{i} + (\frac{2}{3}t^3)\hat{j} $. Find radius of curvature, when $ t=2 $ sec.

A body of mass m moving with a constant velocity v strikes another body of same mass moving with same velocity but in opposite direction. The common velocity of both the bodies after collision is

The centre of percussion of the homogeneous rod of length L suspended at the top will be

The figure given below shows the three coplanar forces P, Q and R acting at a point O. If these forces are in equilibrium, then

The angle of inclination of the plane at which the body begins to move down the plane, is called

Pick up wrong statement about friction force for dry surfaces. Friction force is

The term 'centroid' is

The CG of a plane lamina will not be at its geometrical centre in the case of a

If the masses of both the bodies, as shown in the figure below are reduced to 50 percent then tension in the string will be

Forces are called coplanar when all of them acting on body lie in

A weight of 1000 N can be lifted by an effort of 80 N. If the velocity ratio is 20, then the machine is

Four forces of magnitude 10 N, 20 N, 30 N and 40 N are acting respectively along the four sides of a square ABCD as shown in Fig. 1. Determine the resultant moment about the point A. Each side of the square is given 2 m.

Three like parallel forces 100 N, 200 N and 300 N are acting at points A, B and C respectively on a straight line ABC as shown in Fig. 2. The distances are AB = 30 cm and BC = 40 cm. Find the resultant and also the distance of the resultant from point A on line ABC.

Fig. 3 shows a sphere resting in a smooth V-shaped groove and subjected to a spring force. The spring is compressed to a length of 100 mm from the free length of 150 mm. If the stiffness of the spring is $ 2\text{ N/mm} $, determine the contact reaction at A and B.

Using the analytical method, determine the centre of gravity of the plane uniform lamina as shown in Fig. 4.

Explain the conditions for equilibrium of forces in space.

show that $ I_O = I_G + Ah^2 $, where $ I_G $ is the moment of inertia of a lamina about an axis through its centroid and lying in its plane and h is the distance from the centroid to a parallel axis in the same plane about which its moment of inertia is $ I_O $. A being the area of the lamina.

Find the least force required to pull a body of weight W placed on a rough horizontal plane, when the force is applied at an angle $ \theta $ with the horizontal.

A cord connects two bodies of weights 500 N and 1000 N. The two bodies are placed on an inclined plane and cord is parallel to inclined plane. The coefficients of friction for the weight of 500 N is 0.20 and that of 1000 N is 0.4. Determine the inclination of the plane to the horizontal and tension in the cord, when the motion is about to take place, down the inclined plane. The body weight 500 N is below the body weighing 1000 N.

A truss of span 9 m is loaded as shown in Fig. 5. Find the reactions and forces in the members marked.

Define and explain the terms 'perfect frame', 'imperfect frame', 'deficient frame' and 'redundant frame'.

Each of the two uniform hinged bars in Fig. 6 has a mass m and a length l, and is supported and loaded as shown below. For a given force P, determine the angle $ \theta $ for equilibrium.

The mass of the uniform bar of length l in Fig. 7 is m while that of the uniform bar of length 2l is 2m shown below. For a given force P, determine the angle for equilibrium.

A wheel is rotating about its axis with a constant angular acceleration of $ 1\text{ rad/s}^2 $. If the initial and final angular velocities are $ 5.25\text{ rad/s} $ and $ 10.5\text{ rad/s} $, determine the total angle turned through during the time interval this change of angular velocity took place.

(i) A flywheel starts rotating from rest and is given an acceleration of $ 1\text{ rad/s}^2 $. Find the angular velocity and speed in r.p.m. after 1.5 minutes. (ii) If the flywheel is brought to rest with a uniform angular retardation of $ 0.5\text{ rad/s}^2 $, determine the time taken by the flywheel in seconds to come to rest.

State and prove Varignon's theorem.

Derive the equation of path of a projectile and hence show the equation of path of projectile is a parabolic curve.

A particle is moving in X-Y plane and its position is defined by $ \vec{r} = (\frac{3}{2}t^2)\hat{i} + (\frac{2}{3}t^3)\hat{j} $. Find radius of curvature, when $ t=2 $ sec.

Choose the correct answer from the following (any seven):

Prove that the projection of a sum of vectors onto any axis equals the sum of the projections of the vectors onto the same axis.

Two smooth (frictionless) cylinders $A$ and $B$ of weight $W$ and radius $r$ each are kept in a horizontal channel of width ($b < 4r$) as shown in Fig. 1 below: Find the reaction forces coming from the two sides and the bottom of the channel as well as the forces exerted by the cylinders to each other, assuming the channel walls also to be smooth. Take $r = 250\text{ mm}$, $b = 900\text{ mm}$ and $W = 100\text{ kN}$.

Show that the sum of the moments of inertia of a body, $I_{xx} + I_{yy} + I_{zz}$, is independent of the orientation of the $x, y, z$ axes and thus depends only on the location of its origin.

Rod $CD$ presses against $AB$, giving it an angular velocity. If the angular velocity of $AB$ is maintained at $\omega = 5\text{ rad/s}$, determine the required magnitude of the velocity $v$ of $CD$ as a function of the angle $\theta$ of rod $AB$ (Fig. 2):

Determine the maximum shear stress developed in the $40\text{ mm}$ diameter shaft shown in Fig. 3 below:

Determine the internal normal force, shear force and moment at points $D$ and $E$ in the compound beam. Point $F$ is located just to the left of the $10\text{ kN}$ concentrated load. Assume the support at $A$ is fixed and the connection at $B$ is a pin (Fig. 4):

The coefficient of friction between the block ($80\text{ kg}$) and the inclined rail shown in Fig. 5 below are $\mu_s = 0.25$ and $\mu_k = 0.20$: Determine the smallest values of $P$ required, for the following conditions: (a) To start the block up the rail (b) To keep it moving (c) To prevent it moving down

Write short notes on the following with suitable mathematical expressions: (a) Kinetic friction (b) Newton-Euler laws of motion (c) Gyroscopic motions (d) Principal moment of inertia and axes of inertia

Choose the correct option (any seven):

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The solid cylindrical rotor $B$ has a mass of $43\text{ kg}$ and is mounted on its central axis $C-C$ as shown in Fig. 2. The frame $A$ rotates about the fixed vertical axis $O-O$ under the applied torque $M = 30\text{ N-m}$. The rotor may be unlocked from the frame by withdrawing the locking pin $P$. Calculate the angular acceleration $\alpha$ of the frame $A$ if the locking pin is (a) in place and (b) withdrawn. Neglect all friction and the mass of the frame.

What is space truss? Determine the force in each member of the truss shown in Fig. 3 and state if the members are in tension or compression.

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Determine the centroid and moment of inertia about both $x$ and $y$ axis for the beam's cross-section shown in Fig. 5.

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The coefficient of friction depends upon

Which of the following is a vector quantity?

Moment of inertia of a hollow rectangular section as shown in the figure below about X-X axis is

The moment of a force

A heavy string attached at two ends at same horizontal level and when central dip is very small approaches

The centre of gravity, a T-section $ 100\text{ mm} \times 150\text{ mm} \times 50\text{ mm} $ from its bottom is

Kinetic friction is the

The mechanical advantage of a lifting machine is the ratio of

In ideal machines, mechanical advantage is velocity ratio.

Frictional force encountered after commencement of motion is called

A force of 100 N is acting at a point A as shown in Fig. 1. Determine the moments of this force about O.

The cable AB prevents bar OA from rotating clockwise about the pivot O shown in Fig. 2. If the cable tension is 750 N, determine the n- and t-components of this force acting on point A of the bar.

A lamp weighing 5 N is suspended from the ceiling by a chain. It is pulled aside by a horizontal cord until the chain makes an angle of $ 60^{\circ} $ with the ceiling as shown in Fig. 3. Find the tensions in the chain and the cord by applying Lami's theorem.

A roller of radius 40 cm, weighing 3000 N is to be pulled over rectangular block of height 20 cm as shown in Fig. 4, by a horizontal force applied at the end of a string wound round the circumference of the roller. Find the magnitude of the horizontal force which will just turn the roller over the corner of the rectangular block. Also, determine the magnitude and direction of reactions at A and B. All surfaces may be taken as smooth.

In Fig. 5, the coefficient of friction is 0.2 between the rope and fixed pulley and between other surfaces of contact, $ \mu = 0.3 $. Determine the minimum weight W to prevent the downward motion of the 100 N body.

A body of weight 60 N is placed on a rough horizontal plane. To just move the body on the horizontal plane, a push of 18 N inclined at $ 20^{\circ} $ to the horizontal plane is required. Find the coefficient of friction.

Determine the support reactions and nature, and magnitude of forces in the members of truss shown in Fig. 6.

What are the different methods of analyzing (or finding out the forces) a perfect frame? Which one is used where and why?

Prove that the moment of inertia of a circular section about a horizontal axis (in the plane of the circular section) and passing through the CG of the section is given by $ \pi D^4/64 $.

From a rectangular lamina ABCD, 10 cm x 14 cm a rectangular hole of 3 cm x 5 cm is cut as shown in Fig. 7. Find the centre of gravity of the remainder lamina.

The spring of constant k in Fig. 8 is unstretched when force P = 0. Derive an expression for the force P required to deflect the system to an angle $ \theta $. The mass of the bars is negligible.

For link OA in the horizontal position shown in Fig. 9, determine the force P on the sliding collar which will prevent OA from rotating under the action of the couple M. Neglect the mass of the moving parts.

A particle moves in x-y plane with acceleration components $ a_x = -3\text{ m/s}^2 $ and $ a_y = -16t\text{ m/s}^2 $. If its initial velocity is $ V_0 = 50\text{ m/s} $ directed at $ 35^{\circ} $ to the x-axis, compute the radius of curvature of the path at $ t=2 $ sec.

A force of magnitude of 20 kN, acts at point A (3,4,5) m and has its line of action passing through B (5, -3, 4) m. Calculate the moment of this force about a line passing through points S (2,-5,3) m and T (-3,4,6) m.

Three forces F1, F2 and F3 act at the origin of Cartesian coordinate axes system. The force $ F1 (=70N) $ acts along OA whereas $ F2 (=80\text{ N}) $ acts along OB and $ F3 (=100\text{ N}) $ acts along OC. The coordinates of the points A, B and C are (2, 1, 3), (-1, 2, 0) and (4, 1, 5) respectively. Find the resultant of this force system.

A 75 kg person stands on a weighing scale in an elevator. 3 seconds after the motion starts from rest, the tension in the hoisting cable was found to be 8300 N. Find the reading of the scale in kg during this interval. Also, find the velocity of the elevator at the end of this interval. The total mass of the elevator, including mass of the person and weighing scale is 750 kg. If the elevator is now moving in the opposite direction, with same magnitude of acceleration, what will be the new reading of the scale?

The coefficient of friction depends upon

Which of the following is a vector quantity?

Moment of inertia of a hollow rectangular section as shown in the figure below about X-X axis is

The moment of a force

A heavy string attached at two ends at same horizontal level and when central dip is very small approaches

The centre of gravity, a T-section $ 100\text{ mm} \times 150\text{ mm} \times 50\text{ mm} $ from its bottom is

Kinetic friction is the

The mechanical advantage of a lifting machine is the ratio of

In ideal machines, mechanical advantage is velocity ratio.

Frictional force encountered after commencement of motion is called

A force of 100 N is acting at a point A as shown in Fig. 1. Determine the moments of this force about O.

The cable AB prevents bar OA from rotating clockwise about the pivot O shown in Fig. 2. If the cable tension is 750 N, determine the n- and t-components of this force acting on point A of the bar.

A lamp weighing 5 N is suspended from the ceiling by a chain. It is pulled aside by a horizontal cord until the chain makes an angle of $ 60^{\circ} $ with the ceiling as shown in Fig. 3. Find the tensions in the chain and the cord by applying Lami's theorem.

A roller of radius 40 cm, weighing 3000 N is to be pulled over rectangular block of height 20 cm as shown in Fig. 4, by a horizontal force applied at the end of a string wound round the circumference of the roller. Find the magnitude of the horizontal force which will just turn the roller over the corner of the rectangular block. Also, determine the magnitude and direction of reactions at A and B. All surfaces may be taken as smooth.

In Fig. 5, the coefficient of friction is 0.2 between the rope and fixed pulley and between other surfaces of contact, $ \mu = 0.3 $. Determine the minimum weight W to prevent the downward motion of the 100 N body.

A body of weight 60 N is placed on a rough horizontal plane. To just move the body on the horizontal plane, a push of 18 N inclined at $ 20^{\circ} $ to the horizontal plane is required. Find the coefficient of friction.

Determine the support reactions and nature, and magnitude of forces in the members of truss shown in Fig. 6.

What are the different methods of analyzing (or finding out the forces) a perfect frame? Which one is used where and why?

Prove that the moment of inertia of a circular section about a horizontal axis (in the plane of the circular section) and passing through the CG of the section is given by $ \pi D^4/64 $.

From a rectangular lamina ABCD, 10 cm x 14 cm a rectangular hole of 3 cm x 5 cm is cut as shown in Fig. 7. Find the centre of gravity of the remainder lamina.

The spring of constant k in Fig. 8 is unstretched when force P = 0. Derive an expression for the force P required to deflect the system to an angle $ \theta $. The mass of the bars is negligible.

For link OA in the horizontal position shown in Fig. 9, determine the force P on the sliding collar which will prevent OA from rotating under the action of the couple M. Neglect the mass of the moving parts.

A particle moves in x-y plane with acceleration components $ a_x = -3\text{ m/s}^2 $ and $ a_y = -16t\text{ m/s}^2 $. If its initial velocity is $ V_0 = 50\text{ m/s} $ directed at $ 35^{\circ} $ to the x-axis, compute the radius of curvature of the path at $ t=2 $ sec.

A force of magnitude of 20 kN, acts at point A (3,4,5) m and has its line of action passing through B (5, -3, 4) m. Calculate the moment of this force about a line passing through points S (2,-5,3) m and T (-3,4,6) m.

Three forces F1, F2 and F3 act at the origin of Cartesian coordinate axes system. The force $ F1 (=70N) $ acts along OA whereas $ F2 (=80\text{ N}) $ acts along OB and $ F3 (=100\text{ N}) $ acts along OC. The coordinates of the points A, B and C are (2, 1, 3), (-1, 2, 0) and (4, 1, 5) respectively. Find the resultant of this force system.

A 75 kg person stands on a weighing scale in an elevator. 3 seconds after the motion starts from rest, the tension in the hoisting cable was found to be 8300 N. Find the reading of the scale in kg during this interval. Also, find the velocity of the elevator at the end of this interval. The total mass of the elevator, including mass of the person and weighing scale is 750 kg. If the elevator is now moving in the opposite direction, with same magnitude of acceleration, what will be the new reading of the scale?

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Find the moment of a force 5 N directed along one side of a cube of side length 2 m with respect to— (a) all vertices of the cube; (b) all axes going through the sides.

Draw the free-body diagram of the 50 kg paper roll which has a center of mass at $G$ and rests on the smooth blade of the paper hauler (in Fig. 1). Explain the significance of each force acting on the diagram.

Explain, with due mathematical expression, the mass moment of inertia for an object. Derive the mass moment of inertia about the centroidal axes of a solid sphere, a solid cylinder, and a solid right circular cone.

At the instant shown (in Fig. 2), the disk is rotating with an angular velocity of $\omega$ and has an angular acceleration of $\alpha$. Determine the velocity and acceleration of cylinder $B$ at this instant. Neglect the size of the pulley at $C$.

The hollow circular shaft is subjected to an internal torque of $T = 10\text{ kN}\cdot\text{m}$ (in Fig. 3). Determine the shear stress developed at points $A$ and $B$. Represent each state of stress on an element.

Draw shear force and bending moment diagrams for the beam shown (in Fig. 4). Determine the internal normal force, shear force, and moment at points $C$ and $D$ in the simply supported beam. Point $D$ is located just to the left of the 10 kN concentrated load.

If the coefficient of static friction at all contacting surfaces is $\mu_s$ (in Fig. 5), determine the inclination $\theta$ at which the identical blocks, each of weight $W$, begin to slide.

Write short notes on the following with suitable mathematical expressions: (a) Angle of repose; (b) Types of supports and their reactions; (c) Polar moment of inertia; (d) General planar motion.

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The narrow ring of mass $m$ is free to rotate in the vertical plane about $O$ as shown in Fig. 2: If the ring is released from rest at $\theta = 0^\circ$, determine the expression for the $n$ and $t$ components of the force at $O$ in terms of $\theta$.

What is a truss? Determine the force in terms of the load $P$ for each member of the truss shown in Fig. 3 and state if the members are in tension or compression:

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Determine the location of the centroid of the channel's cross-section area and also calculate the moment of inertia of the area about the axis shown in Fig. 5:

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The resultant of two forces P and Q acting at an angle $ \theta $ is equal to

The moment of a force about any point is geometrically equal to ___ area of the triangle whose base is the line representing the force and vertex is the point about which the moment is taken.

A circular hole of radius (r) is cut out from a circular disc of radius (2r) in such a way that the diagonal of the hole is the radius of the disc. The centre of gravity of the section lies at

The moment of inertia of a triangular section of base (b) and height (h) about an axis passing through its vertex and parallel to the base is ___ as that passing through its CG and parallel to the base.

Which of the following statements is correct?

The efficiency of a screw jack is maximum when the helix angle is equal to

The time of flight of a projectile on an upward inclined plane depends upon

The relationship between linear velocity and angular velocity of a cycle

The loss of kinetic energy due to direct impact of two bodies depends on

In order to increase the acceleration of a mass rolling down on a rough inclined plane (without slipping), we have to

What is meant by moment of a force? How will you explain it mathematically?

State the Varignon's principle of moments.

A force F of magnitude 50 N is exerted on the automobile parking-brake lever at the position $ x=250\text{ mm} $ (Fig. 1). Replace the force by an equivalent force-couple system at the pivot point O.

It is known that a force with a moment of 950 N-m about D is required to straighten the fence post CD (Fig. 2). If $ d=2.70\text{ m} $, determine the tension that must be developed in the cable of winch puller AB to create the required moment about point D.

Describe the method of finding the line of action of the resultant of a system of parallel forces.