EM unit wise.pdf

ME2151 Engineering Mechanics Mech,Chem,Bio-Tech 2012-2013

St.Josephs College of Engineering ISO 9001:2008 St.Josephs Institute of Technology

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Engineering Mechanics

Part A 1. Differentiate between scalar and vector quantities.

Scalar quantity : It is a quantity which is completely specified by the magnitude only.

eg ; Mass, Length, time and temperature.

Vector quantity : It is a quantity which is completely specified by the magnitude and also

direction. eg; Force, Velocity etc.

2. Distinguish between Particle and Rigid Body.

A Particle is a body of infinitely small volume and is considered to be concentrated at a

point. Rigid body is which does not deform under the action of the loads or the external

forces. In case of Rigid body, the distance between any two points of the body remains

constant, when this body is subjected to loads.

3. Newton's Law Gravitation

Two particles of mass m1 and m2 are attracted towards each other along the line connecting

them with a force whose magnitude 'F' is proportional to the product of their masses and

inversely proportional to the square of the distance (r) between them.

F = G m1 m2 r2 Where 'G' is the constant of gravitation and its value is (66.73 0.03) x 10-12m3 kgs2

4. Define a force.

It is defined as an agent that changes or tends to change the position of a body which is either

at rest or in motion.

A force can produce push, pull or twist. Force is a vector quantity which has both magnitude

and direction.



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5. State the parallelogram law of forces.

Parallelogram law of forces: ''If two farces acting on a particle are represented by two

adjacent sides of a parallelogram, then, the diagonal of the parallelogram gives the resultant

of the two forces both in magnitude and direction.'' (Include diagram)

6. State Polygon law of resultant of several forces.

If a number of coplanar forces are acting at a point such that they can be represented in magnitude and direction by the sides of a polygon taken in an order, resultant is represented in both magnitude and direction by the closing side of polygon taken in the opposite order. 7. State Principle of Resolution.

The algebraic sum of the resolved parts of a number of forces in a given direction is equal to the resolved part of their resultant in the same direction. 8. Differentiate between concurrent and non - concurrent force system.

Concurrent forces are the forces whose lines of action pass through a common point. Non - concurrent forces are the forces action are parallel and they will not intersect in one point. 9. Distinguish between the Collinear and Co-planar types of forces.

If the lines of action are along the same line, it is called collinear. If the lines of action are in the same plane, it is called coplanar.

10. State and explain Lami's theorem of triangle law of equilibrium.

If three forces acting on a particle are in equilibrium, then each force is proportional to the sine of the angle included between the other two forces. 11. Define resultant of forces.

Resultant of a system of forces is the single force that replaces the original forces without changing the external effect of the system on the body. 12. Distinguish between the resultant and equilibrant.

Resultant is the single equivalent force of a system (group) of forces. Equilibrant is a single force that balances other forces. Thus, equilibrant can be said to be a single force which is equal, collinear and opposite to the resultant. 13. Find the resultant of an 800 N force acting towards eastern direction and a 500 N

force acting towards north eastern direction.



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Resultant force = 5002 + 8002 , = tan-1 (500/800), = 32 degrees

14. State the analytical conditions for equilibrium of a rigid body.

1. Algebraic sum of the horizontal component of all the forces ( H) must be zero 2. Algebraic sum of the vertical component of all the forces ( V) must be zero 3. The resultant moment of all the forces ( M) must be zero

15. State the condition of equilibrium of a body acted upon by (a) two forces (b) three

forces.

Two forces : the two forces must be equal in magnitude collinear and opposite in sense. Three forces : the three forces must be concurrent.

16. Two vectors A and B are given. Determine their cross product and the unit vector

along it A = 2i + 3j + k and B = 3i - 3j + 4k.

Cross product, A x B = 15i - 5j - 15k, Unit vector - 0.688i - 0.229j - 0.688k 17. A force, F - 10i + 8j - 5k N acts at the point A (2,5,6)m. What is the moment of the

force about the point B (3,1,4) m ?

i j k i j k

M = ( 2 - 3) (5 - 1) (6 - 4) = -1 4 2 = -36i + 15j - 48k B 10 8 -5 10 8 -5 18. Distinguish between space diagram and free - body diagram.

Diagram giving the physical representation of a body and the forces acting on it including the distances is known as space diagram. A free - body diagram is a pictorial representation of the significant, isolated body with all the forces acting on it and all the external forces applied to it. 19. Explain the transmissibility of forces.

The condition of equilibriums or of motion of a rigid body will remain unchanged if the point of application of force acting. in the rigid body is transmitted to act at any other point along its line of action. 20. What is moment of a force about an axis ?

Moment of force acting on a rigid body about an axis measures the tendency of the force to rotate the rigid body about that axis.



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21. State Varignon's theorem.

The moment about a given point 'O' the resultant of several concurrent forces is equal to the sum of the moments of various forces about the same point O. 22. Define couple.

A pair of two equal and unlike parallel forces (forces equal in magnitude with lines of action parallel to each other and acting in opposite directions) is known as couple. 23. What is meant by force - couple system ?

A system of forces acting on a rigid body is replaced by single force acting at a given point and a couple. This force and couple is known as equivalent force - couple system. 24. Under what condition can a force acting on a rigid body be moved to any other

point?

Any force acting on a rigid body may be moved to any other point provided a couple of moment equal to the moment of the force about that point is added to the system. 25. What is support and support reaction ?

A body that supports another body acted upon by a system of forces is called a support. The force exerted by support on the supported body is called support reaction. 26. List the different types of beams.

(i) Simply supported beam (ii) cantilever beam (iii) fixed beam (iv) Continuous beam and (v) overhanging beam. 27. State the different types of supports.

(i) roller or rocker support (ii) hinged or pin-joint support (iii) fixed or built-in support (iv) smooth surface or frictionless surface support. 28. What are the different types of loads ?

(i) Point or concentrated load (ii) Uniformly Distributed Load (iii) Uniformly Varying Load. 29. Distinguish between statically determinate and indeterminate support reactions.

The support reactions are said to be statically determinate when they can be determined by using the three equations of equilibrium. If the support reactions cannot be obtained by using the three equations of equilibrium, they are said to be statically indeterminate. 30. Sketch two types of supports and mark their reactions.



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31. What is statically determinate structure ?

A statically determinate structure can be analyzed by static conditions of equilibrium ( H= 0 ; V= 0 ; M = 0 ) alone. 32. Differentiate between simply supported beam, cantilever and fixed beam.

In simply supported beam, both the ends rest on simple supports without any fixity. In cantilever beam, one end is fixed and the other is free. Fixed beam has both its ends fixed. 33. State the types of equilibrium.

Stable equilibrium : The body returns back to its original position after it is slightly displaced from the position of rest. Stable equilibrium : The body doesn't return back to its original position after it is slightly displaced from the position of rest. Neutral equilibrium : The body occupies a new position (and rests) after it is slightly displaced from the position of rest. 34. Differentiate between Centroid and Centre of gravity.

The centre of figures which have only area but no mass is known as Centroid. Centre of gravity is a point where the entire mass or weight of the body is assumed to be concentrated. 35. Define Centroid axis.

The axis which passes through the point, where the entire mass or weight of the body is assumed to be concentrated, is known as the Centroid. axis. 36. Under what conditions do the Centre of mass and Centre of gravity coincide?

The material must be homogeneous and (ii) the gravitational force on a body of mass 'm' must also pass through its centre of mass. 37. Define Moment of Inertia of an area.

The first moment of a force about any point is the product of the force (P) and the perpendicular distance between the point and the line of action. If this first moment is again multiplied by the perpendicular distance, the resulting moment is the second moment of the force or moment of moment of the force. Instead of force, if the area is considered, it is called the second moment of the area or Moment of Inertia.

38. State the second moment of area of a triangle with respect to the base.

IBase = bh3/12 39. Define Parallel axis theorem.

Parallel axis theorem : Moment of inertia of an area any axis is equal to the sum of the moment of inertia about an axis passing through the centroid parallel, to the given axis and



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(b) the product of area and square of the distance between the two parallel axes. IAB = I CG + A h2 Where, IAB = Moment of inertia of an area about any given axis (say AB) ICG = Moment of inertia about an axis passing through the centroid A = area of the section given h = distance between the two parallel axes 40. State perpendicular Axis Theorem.

Moment of inertia of plane Iamina about an axis perpendicular to the Iamina and passing through its centroid is equal to the sum of moment of inertia about two mutually perpendicular axes passing through the centroid and in the plane of the lamina. Izz = Ixx + Iyy 41. Define polar moment of inertia of an area and state its application.

Moment of inertia of an area about an axis perpendicular to the area through a pole point in the area is called polar moment of the inertia. Polar moment of inertia has application in problems relating to the torsion of cylindrical shafts and rotation of slabs. 42. Define the term, radius of gyration.

The radius of gyration 'k' of any lamina about a given axis is the distance from the given axis at which all the elemental parts of the lamina would have to be placed, so as not to alter the Moment of inertia about the given axis. Radius of gyration, k = I/A

43. Define Product of inertia.

Product of inertia of an area with respect to x and y axes is denoted by Ixy = xy dA where x and y are the coordinates of an element dA of the area A. NOTE : Ixy = 0, for a figure, which is symmetrical about either x or y axes. 44. State the salient properties of product of inertia.

(i) Product of inertia Ixy is zero when x axis or y axis or both the x and y axes are axes of symmetry for the given area. (ii) Product of inertia may be either positive or negative. (iii) Product of inertia of the given area with respect to its principal axes is zero. 45. Define Principal Axis and Principal Moment of Inertia.

The axes about which moments of inertia is ZERO are known as principal axes. The moment of inertia w.r. to the principal axes are called principal moments of inertia. 46. State the salient properties of product of inertia.

(i) Product of inertia lxy is zero when x axis or y axis or both the x and y axes are axes of symmetry for the given area. (ii) Product of inertia may be either positive or negative. (iii) Product of inertia of the given area with respect to its principal axes is zero.



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47. Define Principal Axis and Principal Moment of Inertia.

The axes about which moments of inertia is ZERO are known as principal axes. The moment of inertia w.r. to the principal axes are called principal moments of inertia. 48. State Pappus - Guldinus theorems.

Theorem 1 : The area of surface of revolution obtained by revolving a line or curve is equal to the length of the generating line or curve multiplied by the distance travelled by the centroid of the generating line / curve when it is being rotated. Theorem 2: The volume of a body obtained by revolving an area is equal to the generating area multiplied by the distance traveled by the centroid of the generating area when it is being rotated. 49. Define Moment of inertia of mass. Consider a body of mass m. The moment of the body with respect to the axes AA' is defined by the integral, l=r2dm where, dm is the mass of an element of the body situated at a distance r from axes AA' and integration is extended over the entire volume of the body. 50. State the relationship between the second moment of area and mass moment of

Inertia for thin uniform plate.

Mass moment of inertia of a thin plane about an axis x - x, (Ixx) mass = p t (Ixx) area

Where, (Ixx) area is the second moment of the area of the plate about the axis xx, p is the mass density and t, thickness of the plate which is uniform. 51. Define Friction.

An opposing force, which acts in the opposite direction of the movement of the block is called force of friction or simply Friction. 52. What is limiting friction ?

It is the maximum value, up to which Static friction can rise and balance the externally applied force and beyond which it cannot raise. 53. What is dry friction (or) Coulomb friction (or) solid friction ?

Dry friction also called Coulomb friction or solid friction relates to rigid bodies which are in contact with each other along surfaces that are not lubricated. Dry friction assumes the name static friction when the surfaces of contact are stationary. Dry friction gets the name kinetic friction (also called dynamic friction) when there is motion of one body over another.



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54. Differentiate between Static friction and Dynamic friction.

Static friction : It is the friction experienced by the body when it is at rest, or when the body tends to move. Dynamic friction : It is the friction experienced by the body when it is in motion. It is also called kinetic friction. 55. State the Laws of friction.

1. The direction of the frictional force is always opposite to the direction in which a body resting over another has a tendency to move, under the action of external force. 2. Friction always acts along the common surface of contact between two bodies. 3. Magnitude of limiting friction is directly proportional to the normal reaction, F R 4. Limiting friction is independent of the area and shape of the contact surfaces. 5. The limiting friction depends upon the roughness of the surface. 56. Differentiate between angle of friction and coefficient of friction.

Angle of friction : When the angle of inclination () of a plane is gradually increased, the angle at which the body on the plane just starts sliding down the plane, is called angle of friction. Coefficient of friction : It is the ration of limiting friction to the normal reaction between the two bodies, and is generally denoted by - F / R - tan = Angle of friction . F = Limiting friction. R = Normal reaction. 57. Define angle of repose.

Angle of repose is the maximum angle of inclination that an inclined plane may have with the horizontal before a body lying on the plane begins to slide down under the action of its own weight.

58. What is a Wedge.

A wedge is of a triangular or trapezoidal in cross section. It is generally used for slight

adjustments in the position of a body. i.e., for tightening fits or keys shafts. It is also used for

lifting heavy weights.

59. State the relationship between tension in the belt on tight and slack sides.

e = T1/T2

Where ,T1 = Tension in the belt on the tight side. = Angle of contact in radians. T2 = Tension in the belt on the slack side. = Co efficient of friction. 60. What are motion curves?

The curves which are plotted with the position coordinate, the velocity and the acceleration against the time 't' are called motion curves.



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61. Differentiate between rectilinear motion and curvilinear motion.

Rectilinear motion: If the path traced by a point is a straight line, then the resulting motion is termed as rectilinear motion. Curvilinear motion: If the path traced by a point is a curve, then the resulting motion is termed as curvilinear motion. 62. Differentiate between uniform rectilinear motion and uniformly accelerated

rectilinear motion.

In case of uniform rectilinear motion the velocity 'v ' will be a constant (or) the motion of a particle in which the acceleration a of a particle is zero for every value of ' t '. When the acceleration is constant in a body, which moves, in a straight line, then the resulting motion is called uniformly accelerated rectilinear motion. 63. Determine the position of a particle, whose motion is given by x = t3 - 3t2 - 9t + 12,

when velocity becomes zero.

x = t3 - 3t2 - 9t + 12, v = dx/dt = 3t2 - 6t - 9 = 0 by solving we get, t = 3 or t = -1 't' can not be negative, hence velocity becomes zero when t = 3 64. What is projectile motion?

Any object that is given some initial velocity and during the subsequent motion the object is subjected to only the acceleration due to gravity is termed as a projectile. A projectile travels in the horizontal as well as in the vertical directions and traces a curvilinear path. For example: The motion of a bullet fired from a gun. 65. State D' Alembert's principle.

The force system consisting of external forces and inertia force can be considered to keep the particle in equilibrium. Since the resultant force externally acting on the particle is not zero, the particle is said to be in dynamic equilibrium. This principle is known as D' Alembert's principle. 66. State the different forms of energy.

(i) heat energy (ii) electrical energy (iii) mechanical energy (iv) chemical energy (v) nuclear energy (vi) sound energy and (vii) magnetic energy. 67. Distinguish between potential energy and kinetic energy of a body.

Potential energy of a body is its capacity to do work by virtue of its position or location. Potential energy = mgh. Potential energy is a scalar quantity. Kinetic energy of a body is its capacity to do work by virtue of its motion. K.E. = 1/2 mv2 K.E. is also a scalar quantity.



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68. State the work - energy principle.

The work done by force acting on a particle during its displacement is equal to the change in the kinetic energy of the particle during that displacement. Work done = Final Kinetic energy - Initial Kinetic energy = 1/2 (mv22 -mv12) 69. State the law of conservation of energy.

''Energy can neither be created nor be destroyed. It can only be changed from one form into another form.'' 70. Define linear impulse or (impulse).

Linear impulse or impulse is the product of force acting on a body and the time elapsed. 71. What is impulsive force and impulsive motion?

When a large force acts on a particle for a short period of time and produces a definite change in its momentum, then, such a force is called an impulsive force. The motion caused by such an impulsive force is known as impulsive motion. 72. State impulse momentum principle t mv2 - mv1 = F dt 0 Final momentum - Initial Momentum = Impulse of the Force. The equation expresses that the total change in momentum of a particle during a time interval is equal to the impulse of the force during the same interval of time. 73. State the condition for the dynamic equilibrium of a body.

The equation of motion can be written in the form F - ma = 0 F + (-ma) = 0 To write the equation of dynamic equilibrium of a particle, add a fictious force equal to inertial force to the external forces action on the particle, and equate the sum to zero. 74. State the principle of impulse and momentum or write impuise momentum equation.

Principle of impulse and momentum is written in the form of an equation ''Impulse = final momentum - initial momentum'' F t = m (V - u) 75. What is impact or collision?

A collision between two bodies that lasts for a very short interval of time during which period, the two bodies large forces on each other is called an impact or collision. 76. Define line of impact.

A line perpendicular to the surfaces of contact during impact is known as the line of impact.



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77. What is the difference between central and eccentric impact ?

If the mass centres of the two colliding bodies lie on the line of impact, then, the impact is said to be central impact otherwise it is eccentric.

78. Distinguish between direct impact and oblique impact.

If the velocities of the two colliding bodies act along the line of impact, the impact is called direct impact. If the velocities of the two colliding bodies act along lines other than the line of impact , the impact is known as oblique impact. 79. Distinguish between perfectly plastic impact and perfectly elastic impact.

In the case of perfectly plastic impact, e = 0. This means that there is no period of restitution. The two colliding bodies join together and travel with the same velocity. In the case of perfectly elastic impact, e= 1. This means that the relative velocity before the impact is equal to the relative velocity after the impact. 80. Define linear momentum.

The linear momentum of a particle is the product of mass and velocity.

81. State the Principle of conservation of linear momentum.

Principle of conservation of linear momentum states that if the resultant force acting on a particle is zero then the linear momentum of the particle remains constant.i.e., 'Final momentum - Initial momentum'

82. Define co -efficient of restitution.

The ratio of magnitudes of impulses corresponding to the period of restitution and to the period of deformation is called coefficient of restitution. It is also defined as the ration of velocity of separation to the velocity of approach. relative velocity of separation vb1 - v1a coefficient of restitution, e = = relative velocity of approach va - vb 83. What is Inertia force ?

The inertia force can be defined as the resistance to the change in the condition of rest or of uniform motion of a body. The magnitude of the inertia force is equal to product of the mass and acceleration of the particle and it acts in a direction opposite to the direction of acceleration of the particle. The equation of motion of the particle P F = ma, can be written in the form F - ma = 0 84. What is translation ?

The motion of a rigid body in which the velocity of each element in the rigid body remains equal and the acceleration of each element remains equal is called translation.



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85. What is meant by general plane motion ?

General plane motion is neither a translation nor a rotation. It can be considered to be a sum of translation and rotation about an axis perpendicular to the plane of motion. 86. Give two examples of general plane motion.

(i) A cylinder rolling on a flat or a curved surface without slipping. (ii) A rod one end of which slides along a horizontal track and the other end along a vertical track. 87. Explain Instantaneous centre of rotation.

A rigid body in plane motion, at any given instant of time appears as it rotating about a certain point in the plane of the body. The point which is instantaneously at rest and has zero velocity is called as the instantaneous centre of rotation. The body may seem to be rotating about one point at one instant of time and about another point at the next instant. This instantaneous centre is changing every instant and is not a fixed point. The velocity of any point in the body can be determined by assuming that point to be rotating with some angular velocity , about the instantaneous centre at the instant. 88. What is instantaneous centre of rotation in plane motion?

A rigid body in plane motion can be considered to rotate about a point that remains at rest at a particular instant. This point having zero instantaneous velocity is called the instantaneous centre of rotation. 89. If a circular cylinder rolls without slipping, where the instantaneous centre of

rotation will be located?

At the point of contact of the cylinder with the surface. 90. State the principle of work and energy (work energy equation) for the general plane

motion of rigid bodies.

Principle of work and energy for the general plane motion of rigid bodies (or work energy equation is written as follows: 'Work done by a rigid body undergoing general plane motion = change in kinetic energy o the rigid body due to translation from one point to another point + change in kinetic energy of the rigid body due to rotary motion from one position to another position." 91. State the principle of conservation of energy of a rigid body.

Principle of conservation of energy states that the sum of potential energy and kinetic energy of a rigid body or the system of rigid bodies moving under the influence of conservative forces is always a constant.



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92. Write an expression for the total kinetic energy of a rigid body.

Total K.E. of a rigid body = K.E. due to rotation + KE. due to translation. 93. The change in K.E. of a rigid body due to liner motion is 200 Joules. Change in K.E.

of the body due to rotary motion is 1500 Joules. Find the work done by the rigid

body.

Work done by the rigid body = 200 J + 1500 J = 1 700 J 94. A flywheel has a mass moment of inertia of 11 kg.m2 about the axis of rotation. It runs at a constant angular velocity of 94.25 rad/s. Find the kinetic energy of the fly wheel. 1 1 Kinetic energy = --- mV2 = --- X 11 X 94.252 = 48.86k N.m. 2 2 Unit I - Basics and statics of particles Part B Class work problems: Statics of particles in two dimensions Resultant force 01. A system of four forces acting on a body is shown in figure below. Determine the resultant force and its direction.

02. If five forces act on a particle as shown in figure below and the algebraic sum of horizontal

components of all these forces is 324.904 kN, calculate the magnitude of P and the resultant of all the forces.

03. Three forces act on a particle O as shown in figure below. Determine the value of F such that the resultant of these three forces is horizontal. Find the magnitude and direction of the fourth force which when acting along with the given three forces will keep O in equilibrium.



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04. The figure below shows a block of weight 120 N on a smooth inclined plane. The plane makes an angle of 320 with horizontal and the force F is applied parallel to the plane. Find the values of F and normal reaction.

Equilibrium of particles in two dimensions: 05. The bar AC, 10 m long supports a load of 6000 N as shown below. The cable BC is horizontal and 5 m long. Determine the forces in the cable and the bar.

06. Determine the length of cord AC in figure below so that the 8 kg lamp is suspended in the position shown. The undeformed length of the spring AB is lAB = 0.4 m and the spring has a stiffness of kAB = 300 N/m

07. A mass of 45 kg is suspended by a rope from a ceiling. The mass is pulled by a horizontal force until the rope makes an angle of 700 with the ceiling. Find the horizontal force and the tension in the rope. 08. Determine the tension in cables AB and AC required to hold the 40 kg crate shown in fig below.

09. Two identical rollers, each of weight 50 N, are supported by an inclined plane and a vertical wall as shown in figure below. Find the reactions at the points of supports A, B and C. Assume all the surfaces to be smooth.

10. A cylindrical roller has a weight of 10 kN and it is being pulled by a force which is inclined at 300 with the horizontal as shown in figure below. While moving it comes across an obstacle 10 cm high. Calculate the force required to cross this obstacle, if the diameter of the roller is 1 m.



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11. A uniform wheel 600 mm in diameter rests against a rigid rectangular block 150 mm thick as shown in figure below. Find the least pull P, through the centre of the wheel in order to just turn the wheel over the corner of the block. All surfaces are smooth. Find also the reaction of the block. The wheel weighs 900 N.

12. A roller of radius 30 cm weighing 2 kN is to be pulled over a rectangular obstruction of height 15 cm as shown in figure below by a force P applied tangentially at its crest C, through a string wound around the circumference of the roller. Find the force P required just to turn the wheel over the corner of the obstruction. Also determine the magnitude and direction of the reactions at A and B. Surfaces may be taken as smooth.

Forces in space (vector approach) Resultant and equilibrium of particles in three dimensions. 13. A force vector of magnitude 100 N, is represented by a line AB of co-ordinates A(1,2,3) and B(5,8,12). Determine : (i) the components of the force along x,y and z axes. (ii) angles with x, y and z axes and (iii) specify the force vector. 14. Three concurrent forces in space, F1, F2 and F3 are acting at A as shown in figure. An unknown force F, attached to the system makes the particle A in equilibrium. Find the magnitude and direction of the unknown force F.

15. Members OA, OB and OC form a three member space truss. A weight of 10 kN is suspended at the joint O as shown in figure. Determine the magnitude and nature of forces induced in each of the three member of the truss.

16. A tower guy wire, shown below, is anchored by means of a bolt at A. The tension in the wire is 2500 kN. Determine (i) the components Fx, Fy, Fz of the force acting on the bolt and (ii) the angles x, y, z defining the direction of the force.



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17. A rod AB as shown in figure below is held by a ball and socket joint at A and supports a mass C weighing 1000 N at end B. The rod is in xy plane and is inclined to y axis at an angle of 180. The rod is 12 metres long and has negligible weight. Find the forces in the cables DB and EB.

18. A force acts at the origin of a co-ordinate system in a direction defined by the angles x = 69.30 and z = 57.90. Knowing that the y component of the force is 174 N, determine, (i) the angle y (ii) the other components and the magnitude of the force (iii) projection of this force on XZ plane and its magnitude and (iv) moment of this force about a point of co-ordinate(2,3,4) and its magnitude. 19. A supplementary supporting guy wire system for a 200 m tall tower is tightened. The cables are fastened to the ground at points 1200 apart and 100 m from the tower base. What is the equivalent force system acting on the tower base when the tension is 50 kN in cable AT, 75 kN in BT and 25 kN in CT?

20. Forces 32 kN, 24 kN, 24 kN and 120 kN are concurrent at origin and are respectively directed through the points whose coordinates are A(2,1,6), B(4,2,5), C(3,2,1) and D(5,1,2). Determine the resultant of the system. Assignment problems: 01. Five forces are acting on a particle. The magnitude of the forces are 300 N, 600 N, 700 N, 900 N and P and their respective angles with the horizontal are 00, 600, 1350, 2100 and 2700. If the vertical component of all the forces is 1000 N, find the value of P. Also calculate the magnitude and the direction of the resultant, assuming that the first force acts towards the point, while all the remaining forces act away from the point. 02. A and B weighing 40 N and 30 N respectively, rest on smooth planes as shown in figure below. They are connected by a weightless cord passing over a frictionless pulley. Determine the angle and the tension in the chord for equilibrium.



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03. Two rollers, each of weight 50 N and of radius 10 cm rest in a horizontal channel of width 36 cm as shown in figure below. Find the reaction on the point of contacts A, B and C. Assume all the surfaces of contact are smooth.

04. Determine the magnitude and direction of force F shown in figure below so that the particle A is in equilibrium.

05. Three cables are used to support the 10 kg cylinder shown in figure below. Determine the force developed in each cable for equilibrium.

Unit II - Equilibrium of rigid bodies Class work problems: Part B Statics of rigid bodies in two dimensions Resultant force : 01. A coplanar parallel force system consisting of three forces acts on a rigid bar AB as shown in fig. below. (i) Determine the simplest equivalent action for the force system. (ii) If an additional force of 10 kN acts along the bar A to B, what would be simplest equivalent action?

02. In fig. below, a plate of 700 x 375 mm dimension is acted upon by four forces as shown. (i) Find the resultant of these forces. (ii) Locate the two points where the line of action of the resultant intersects the edges of the plate.



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03. In fig. below, two forces act on a circular disc as shown. If the resultant moment of these forces about point D on the disc is zero, determine (i) Magnitude of force P (ii) Magnitude of the resultant of two forces (iii) The point on the Y-axis through which the line of action of the resultant passes through.

04. A plate is acted upon by 3 forces and 2 couples as shown in fig. below. Determine the resultant of these force-couple system and find co-ordinate x of the point on the x-axis through which the resultant passes.

05. Determine the resultant of the coplanar non-concurrent force system shown in fig. below. Calculate its magnitude and direction and locate its position with respect to the sides AB and AD.

Statics of rigid bodies Force couple system : 06. Reduce the given system of forces acting on the beam AB in figure below to (i) an equivalent force-couple system at A and (ii) an equivalent force-couple system at B.

07. Four forces and a couple are applied to a rectangular plate as shown in fig. below. Determine the magnitude and direction of the resultant of the Force-couple system. Also determine the distance x from O along x-axis where the resultant intersects.



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Equilibrium of rigid bodies in two dimensions: 08. A load P of 3500 N is acting on the beam, which is held by a cable BC as shown in figure below. The weight of the beam can be neglected. (i) Draw the freebody diagram of the beam (ii) Find out the tension in cable BC and (iii) Determine the reaction at A.

09. A uniform bar AB shown in the figure below has a mass 50 kg and supports a mass of 200 kg at A. A supporting cable is tied to the bar at C and the other end is fixed to the vertical wall at D. Calculate the tension in the supporting cable and the magnitude of the reaction force at the pin B.

Equilibrium of rigid bodies Support reactions : 10. Determine the horizontal and vertical components of reaction for the beam loaded as shown in figure below. Neglect the weight of the beam in the calculations.

11. Find the pin reaction at A and the knife-edge reaction at B.

12. Find the reactions at the supports A and B of the beam shown in figure below.



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13. Find the reactions at A and B in figure below.

14. A frame supported at A and B is subjected to force of 500 N as shown below. Compute the reactions at the support points for the cases of = 00, = 900 and = 600.

15. A simply supported overhanging beam 20 m long carries a system of loads and a couple as shown in figure below. Determine the reactions at supports A and B.

16. Determine the reactions of the beam shown in figure below.

Forces in space (vector approach) Resultant and equilibrium of rigid bodies in three dimensions. 17. A tension T of magnitude 10 kN is applied to the cable attached to the top A of rigid mast and secured to the ground at B as shown in figure below. Determine moment of the tension T about the z-axis passing through the base O.

18. Determine the tension in the cables AB, AC and AD if the crate shown in figure below is weighing 9.07 kg.



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19. A parallel force system of four forces 130 kN, 26 kN, F1, F2 are shown in figure below. If the resultant R of magnitude 260 kN passes through P(1.4,0,1.2), find F1, F2.

20. The 8 m pipe AB shown in figure below has a fixed end at A. A steel cable is stretched from B to a point C on the vertical wall. If the tension in the cable is 1200 N, determine the moment about A of the force exerted by the cable at B.

Assignment problems: 01. A system of four forces A, B, C, D of magnitudes 10 kN, 15 kN, 18 kN and 12 kN acting on a body are shown in rectangular co-ordinates as shown in fig below. (i) Find the moments about the origin (ii) Find the resultant moment.

02. For the system of forces shown in fig. below, determine the magnitudes of P and Q such that the resultant of the system passes through A and B.

03. Three forces + 20 N, 10 N and + 30 N are acting perpendicular to xz plane as shown in figure below. The lines of action of all the forces are parallel to y axis. The co-ordinates of the point of action of these forces along x and z directions are respectively (2,3), (4,2) and (7,4). All the distances



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being referred in metres. Find out (i) the magnitude of the resultant force and (ii) the location of the resultant.

04. Find the reactions at A.

05. Determine the tension in cables BC and BD and the reactions at the ball and socket at A for the rod shown in figure below.

Unit III - Properties of Surfaces and Solids Part B Class work problems: Centroid (First moment of area) of sections : 01. Determine the centroid of the cross-sectional area of an unequal I-section shown below.

02. Locate the centroid of the plane area shown below.

03. Determine the co-ordinates of the centroid of the shaded area shown below.



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04. For the plane area shown below, locate the centroid of the area.

05. Locate the centroid of area shown below.

06. Determine the centroid of area shown in figure below by taking moment of area about the given aa - axis and bb - axis.

07. In the figure below, a solid is formed by joining a hemi sphere, a cylinder and a cone, all made out of same material. Find the location of the centroid of this solid on the Z axis.

Area Moment of Inertia, Radius of Gyration and Polar Moment of Inertia : 08. Determine the moments of inertia of the area shown with respect to the centroidal axes parallel and perpendicular to the side AB.

09. For the plane area shown below, determine the area moment of inertia and radius of gyration about the x axis.



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10. Determine the moment of inertia and radius of gyration of the T section about centroidal Y axis. Also find polar moment of inertia.

11. Determine the moments of inertia Ix and Iy of the area shown below with respect to centroidal axes respectively parallel and perpendicular to the side AB.

12. Compute the second moment of area of the plane surface shown below about its horizontal and vertical centroidal axes.

13. Find the moment of inertia about 11 and 22 axes for the area shown in figure below.

14. Calculate the moment of inertia of the section shown in figure below about the xx and yy axis through the centroid.

Product of Inertia, Principal Moment of Inertia and Mass Moment of Inertia : 15. Determine the product of inertia of the angle section shown in figure below with respect to centroidal axes.



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16. Determine the product of inertia of the angle section shown in figure below w.r.t aa and bb axes.

17. For the section shown in figure below determine the principal moments of inertia and locate the principal axes. Also find the major and minor principal axes.

18. Find the mass moment of inertia of the rectangular block shown below about the vertical y axis. A cuboid of 20 mm x 20 mm x 20 mm has been removed from the rectangular block as shown below. The mass density of the material of the block is 7850 kg/m3.

19. Determine the mass moment of inertia of the composite body about Z axis shown in figure below. The mass density of the cylinder is 6000 kg/m3 and the rectangular prism is 7000 kg/m3.

20. A structural member in the shape shown in figure below is machined from titanium. Calculate the moment of inertia about the z axis. (Take = 3080 kg/m3 for titanium).



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Assignment problems: 01. Determine the centroidal co-ordinates of the area shown in figure below with respect to the shown x-y coordinate system.

02. Determine Ixx and Iyy about the centroidal axes, as shown in figure below. All dimensions in cm.

03. Derive mass moment of inertia of a rectangular section. 04. Derive mass moment of inertia of a prism. 05. Derive mass moment of inertia of a sphere. Unit IV 1. The motion of a particle is defined by the relation x = 3t4 + 4t3 7t2 - 5t + 8 , where x and t are expressed in millimeters and seconds, respectively. Determine the position, the velocity and the acceleration of the particle when t = 3s. Ans: x = 281 mm, v= 385mm/s, a= 382 mm/s2 2. The motion of a particle is defined by the relation x = 3t3 6t2 12t + 5 , where x and t are expressed in meters and seconds, respectively. Determine (a) when the velocity is zero (b) the position, the acceleration and the total distance traveled when t = 4s. Ans: (a) t = 2s (b) x = 53m, a= 60 m/s2, 96m 3. The motion of a particle is defined by the relation x = t3 9t2 + 24t 8 , where x and t are expressed in meters and seconds, respectively. Determine (a) when the velocity is zero (b) the position and the total distance traveled when the acceleration is zero. 4. A ball is thrown vertically up with an initial velocity of 25 m/s. Calculate the maximum altitude reached by the ball and the time t after throwing for it to return to the ground. Neglect air resistance and take the gravitational acceleration to be constant at 9.81m/s2 Ans: S = 31. 85m, t = 5.1s 5. Two trains A and B leave the same station in parallel lines. Train A starts with a uniform acceleration of 0.15m/sec2 and attains a speed of 27 kmph when the steam is reduced to keep speed constant. Train B leaves 40 seconds later with uniform acceleration of 0.3m/sec2 to attain a maximum speed of 54 kmph. When and where will B overtake A? Ans: t = 105 sec, S = 600m



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6. Two cars are traveling towards each other on a single lane road at the velocities 12 m/sec and 9 m/sec. respectively. When 100m apart, both drivers realize the situation and apply their brakes. They succeed in stopping simultaneously and just short of colliding. Assume constant deceleration for each case and find (a) time required for cars to stop (b) deceleration of each car and (c) the distance traveled by each car while slowing down. Ans: t = 9.524sec, a1 = -1.26 m/sec2, a2 = -0.945 m/sec2, x = 57.14m, 42.86m 7. Two trains A and B leave the same station on parallel lines. A starts with uniform acceleration 1/6 m/s2 and attains a speed of 24kmph when steam is reduced to keep the speed constant. 40 seconds after B leaves with uniform acceleration of 1/3 m/s2 to attain a maximum speed of 48 kmph. When will train B overtake train A. 8. Three marks A, B, C spaced at a distance of 100m are made along a straight road. A car starting from rest and accelerating uniformly passes the mark A and takes 10 seconds to reach the mark B and further 8 seconds to reach mark C. Calculate: (a) the magnitude of the acceleration of the car; (b) the velocity of the car at A, (c) the velocity of the car at B and (d) the distance of mark from starting point. Ans: (a) 0.2778 m/sec2; (b) 8.61 m/sec; (c) 10.833 m/sec; and (d) 133.47 m 9. Two automobiles A and B traveling in the same direction in adjacent lanes are stopped at a traffic signal as the signal turns green, automobile A accelerates at a constant rate of 2 m/s2. Two seconds later, automobile B starts and accelerates at a constant rate of 3.6 m/s2. Determine (a) when and where B will over take A (b) the speed of each automobile at that time. t = 7.85s, 61.7m, va= 15.71m/s, vb= 21.1m/s 10. A bus starts from rest at point A and accelerates at a rate of 0.9m/s2 until it reaches a speed of 7.2 m/s. It then proceeds with the same speed until the brakes are applied. It comes to rest at point B, 18m beyond the point, where the brakes are applied. Assuming uniform acceleration, determine the time required for the bus to travel from point A to B. The distance between the points A and B is 90m. Problems using Principle of Work Energy 1. A block weighing 100N is moving along a horizontal surface of friction coefficient 0.2with a velocity of 5m/s. A push of 80N inclined at 30 to horizontal acts on the block. Find the velocity of the block, Using work-energy principle after it had moved through a distance of 20m. Ans: U1-2 = 825.64 N-m, V2 = 13.67m/s 2. A 70kg block resting on a 30 incline ( = 0.3 ) is released from rest. Determine the speed after it slides down 10m down the incline. U1-2 = 1649.4N-m, V2 = 6.86m/s 3. A 20 kg mass slides down 250mm from rest down the 25plane and hits a spring of constant 1800N/m.If the coefficient of kinetic friction is 0.2 determine the maximum compression of the spring. Ans: 144mm 4. A block of weight 12N falls on at a distance of 0.75m on top of a spring. Determine the spring constant if it is compressed by 150mm to bring the weight momentarily to rest. Ans : K = 960N/m 5. A bullet of mass 81gm moving with a velocity of 300m/s is fired into a block of wood and it penetrates to a depth of 10cm. If the bullet moving with the same velocity were fired into a similar piece of wood 5cm thick, with what velocity would it emerge? Find also the force of resistance assuming it uniform. V= 212.13m/s IMPACT 1. Two bodies of mass 8kg and 6kg move with velocities 6m/s and 2m/s respectively to the right. Find the velocities of these bodies directly after impact, if e=0.6. Ans : 3.26m/s, 5.66 m/s



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2. A sphere A of mass 6kg moving with a velocity of 10m/s to the right impinges with a sphere B of mass 4kg moving with a velocity of 8m/s to the left. If after impact the velocity of sphere B is observed to be 9m/s to the right, determine the coefficient of restitution between the spheres. Ans : -1.33m/s, 0.574 3. A ball of mass 1kg moving with a velocity of 2m/s, strikes directly on a ball B of mass 2kg at rest. The ball A after striking comes to rest. Find the velocity of ball B after striking and coefficient of restitution. Ans: 1m/s, 0.5 4. Three spherical balls of masses 2 kg, 6 kg, 12 kg are moving in the same direction with velocities 12 m/s, 4 m/s, 2m/s respectively. If the ball of mass 2kg impinges with ball of mass 6kg which in turn impinges with ball of mass 12kg. Prove that the balls of masses 2kg and 6kg will be brought to rest, by impact. Assume the balls to be perfectly elastic. 5. A sphere A of weight 10N moving with a velocity of 3m/s to the right impinges with a sphere B of weight 50N moving with a velocity of 0.6 m/s to the right. If the coefficient of restitution between the spheres is 0.75, find the loss of kinetic energy and show that the direction of the motion of sphere A is reversed. Ans: - 0.5m/s, 1.29m/s, 1.136Nm 6. A vehicle of mass 600kg moving with a velocity of 12m/s strikes another vehicle of mass 400kg moving at 9m/s in the same direction. Both the vehicles get coupled together due to impact. Find the common velocity with which the vehicles move. Also find the loss in kinetic energy. Ans: 10.8 m/s,1080 N-m 7. A ball of mass 500gm is dropped on a horizontal floor from a height of 8m. Find the co efficient of restitution between the ball and the floor. Ans: e = 0.6667 Unit V- Friction and Elements of rigid body dynamics Part B Class work problems: Body in a horizontal plane & inclined plane : 01. Determine whether the block shown in figure below having a mass of 40 kg is in equilibrium and find the magnitude and direction of the friction force. Take s = 0.40 and k = 0.30

02. The coefficients of static and kinetic friction between the 100 kg block and inclined plane are 0.3 and 0.2 respectively. Determine (i) the friction force F acting on the block when P is applied with a magnitude of 200 N to the block at rest (ii) the force P required to initiate motion up the incline from rest and (iii) the friction force F acting on the block if P = 600 N.

03. A body having mass of 22 kg rests on a plane inclined at 60 degrees with the horizontal. The coefficient of friction between the body and inclined plane is 1/3. The body is acted upon by a



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horizontal force P. What is the value of P so that the body will not slide down the plane? What is the value of P so that the body will slide up the plane? In between these values, the body will be at rest.

04. A cord is attached to a block of 50 kg mass, the block is positioned on a 200 incline as shown in figure below. The other end of the cord is supporting a cylinder. If the coefficient of friction between the block and the incline is 0.2 and coefficient of friction between the cord and cylindrical support surface is 0.3, determine the range of mass of cylinder for which the system is in equilibrium.

Two bodies in contact : 05. Determine the smallest force P required to move the block B if (a) block A is restrained by cable CD as shown in figure below (b) cable CD is removed. Take the coefficients of frictions as s = 0.30 and k = 0.25 between all surfaces of contact.

06. A block of weight W1 = 1290 N rests on a horizontal surface and supports another block of weight W2 = 570 N on top of it as shown in figure below. Block of weight W2 is attached to a vertical wall by an inclined string AB. Find the force P applied to the lower block, that will be necessary to cause the slipping to impend. Coefficient of friction between blocks 1 and 2 = 0.25. Coefficient of friction between block 1 and horizontal surface = 0.40.

07. What is the least value of P required to cause the motion impend, the system shown in figure below. Assume coefficient of friction on all contact surfaces as 0.2.

Ladder friction :



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08. A ladder is 8m long and weighs 300 N. The centre of gravity of the ladder is 3 m along the length of ladder from the bottom end. The ladder rests against a vertical wall at B and on the horizontal floor at A as shown below. Determine the safe height up to which a man weighing 900 N can climb without making the ladder slip. The coefficient of friction between ladder and floor is 0.4 and ladder top and wall is 0.3.

09. A ladder AB of weight 900 N is held in impending motion towards the right by a rope tied to the wall as shown in figure below. The coefficient of friction between the floor and ladder is 0.25 and that between the wall and ladder is 0.40. Calculate the tension in rope.

Wedge friction : 10. A block overlying a 100 wedge on a horizontal floor and leaning against a vertical wall and weighing 1500 N is to be raised by applying a horizontal force to the wedge. Assuming coefficient of friction between all the surfaces in contact to be 0.3, determine the minimum horizontal force to be applied to raise the block.

11. A concrete block weighing 10 kN is to be shifted away from the wall with the help of a 150 wedge as shown in figure below. Calculate the magnitude of the vertical force that has to be applied to the top of the wedge to shift the block. The coefficient of friction between all the rubbing surfaces is 0.25.

Screw friction : 12. In a screw jack, the pitch of the square threaded screw is 5.5 mm and the mean diameter is 70 mm. The force exerted in turning the screw is applied at the end of a lever 210 mm long measured from the axis of the screw. If the coefficient of friction of the screw jack is 0.07, calculate the force required at the end of the lever to (i) raise a weight of 30 kN (ii) lower the same weight (iii) Find the efficiency of the jack (iv) Is it self locking? Belt friction :



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13. Determine the minimum tension in the rope required to support a cylinder of mass 500 kg shown in figure below when the rope passes (a) once over the rod shown and (b) two times over the rod shown. Take s = 0.2.

14. A 100 kg mass is lifted by a rope, rolling on a cylinder of 150 mm dia as shown in figure below. (i) Determine the force required on the otherside if the coefficient of friction is 0.20. (ii) Calculate the torque and power transmitted, if the velocity is 30 m/s. If the load is lifted by applying a horizontal force as shown in figure below calculate (iii) the necessary force (iv) Torque at the cylinder surface (v) power transmitted. Rolling resistance (Wheel friction): 15. A wheel weighs 1000 N and its diameter is 600 mm. (i) If the coefficient of rolling resistance is 15 mm, calculate the force required to roll the wheel on a horizontal surface without slipping. (ii) If the wheel rolls down a slope of 1 in 55.56 , find the coefficient of rolling resistance. Rotation of Rigid Bodies. 16. The rotation of a fly wheel is governed by the equation = 3t2 2t + 2 where is in radians per second and t is in seconds. After one second from the start the angular displacement was 4 radians. Determine the angular displacement, angular velocity and angular acceleration of the flywheel when t = 3 seconds. Ans: = 26 radians, = 23 rad/sec, = 16 rad/sec2. 17. Power supply was cut off to a power driven wheel when it was rotating at a speed of 900 rpm. It was observed to come to rest after making 360 revolutions. Determine its angular retardation and time it tool to come to rest after power supply was cut off. Ans: = - 1.9635 rad/sec2, t = 48 sec. 18. A wheel is rotating about its axis with a constant acceleration of 1 rad/sec2. If the initial and final velocities are 50 rpm and 100 rpm, determine the time taken and number of revolution made during this period. Ans: t = 5.236 sec and revolutions = 6.545. 19. A rotor of an electric motor is uniformly accelerated to a speed of 1800 rpm from rest for 5 seconds and then immediately power is switched off and the rotors decelerate uniformly. If the total time elapsed from start to stop is 12.5 sec, determine the number of revolutions made while (i) acceleration, (ii) deceleration. Also determine the value of deceleration. Ans: 75 revolutions, 175 revolutions and 25.1327 rad/sec2. 20. A wheel rotating about a fixed axis at 20 revolutions per minute is uniformly accelerated for 70 seconds during which it makes 50 revolutions. Find the (i) angular velocity at the end of this interval and (ii) time required for the velocity to reach 100 revolutions per minute. Ans: = 0.06839 rad/sec2, = 6.8816 rad/sec, t = 122.5 sec. Instantaneous Centre Method 21. The velocity of the point B shown in fig is 2.5 m/s to the right. Determine the velocity of point A by the method of instantaneous centre of rotation. Ans: AB = 0.625 rad/s, vA = 4.33 m/s L.

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A

B

8 m



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22. In the engine system shown in fig the crank AB has a constant angular velocity of 3000 rpm. For

the crank position indicated, find the (i) the angular velocity of the connecting rod. (ii) velocity of the piston. Ans: vB = 23.56 m/s, BC = 77.425 rad/s, vC = 19.62 m/s.

A

B

C200mm75mm

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