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1 ENGINEERING MECHANICS UNIT-I BASICS & STATICS OF PARTICLES 1. What is meant by mechanics? 2. What is meant by Engineering Mechanics? 3. State the different type of mechanics 4. Define Statics 5. Define Dynamics 6. Define Kinematics 7. Define Kinetics 8. What do you understand from the concept of Law of dimensional homogeneity? 9. State Parallelogram law. 10. State triangle law. 11. Define Lami s theorem 12. Define principle of transmissibility of forces 13. Define vectors 14. Define Unit vector. 15. The point of application of a force F = 5i + 10j 15k is displaced from the point I + 3k to the point 3i j 6K. Find the work done by the force 16. Given A = 2i 3j k and B = I + 4j 2k. Find A.B and A X B. 17. The following forces act at a point. Determine the resultant force. 18. Two forces F1 = 5i and F2 = 8.66j pass through a point whose coordinates are (2,1)m. Calculate the moment of the forces about the origin. 19. A couple of moment 60Nm acts in the plane of a paper. Indicate this couple with 30N forces. 20. A horizontal bar ABC is hinged at A and freely support over B. Calculate the reaction a B1 if a force of 6KN acts downwards at C. 21. A horizontal beam ABC is hinged at A and freely supported by B. Calculated the reaction at B due to clockwise moment of 12 Nm applied at C. 22. A force F has the components Fx = 50N, Fy = 75N and Fz = 100N. Find out the angles θx, θx, θx that forms with the x, y and z axes. 23. Find the moment of the force about A. If the forces are acting as shown in figure. Figure (Q & A 6) 24. State Varigon s theorem 25. Define moment of a force about a point. 26. Define a couple. 27. State the conditions for the equilibrium of a two dimensional rigid body. Page 1 of 46

2 28. State the analytical conditions for the equilibrium of coplanar forces in a plane. 29. What is meant by coplanar concurrent force systems? 30. What is the single force that replaces a system of coplanar concurrent forces? 31. Two forces act on a body as shown in fig. Calculate the magnitude of their resultant? Figure 32. How will you resolve a given force into a force and a couple? 33. Differentiate Resultant and equilibrant. 34. A force system has a resultant of 58 KN. The resultant acts at 360 with horizontal. What will be the direction and magnitude of equilibrant. 35. Determine the resultant of the body given below Figure 36. Replace the given system with a couple and force. 37. What is the value of R and S if the system is in equilibrium 38. Find the Value of x Figure 39. Following forces act a point P. a) F1 = 50i b) F2 = -30i -15j c) F3 = -25i -10j + 5k 40. Two forces F1 = 5i and F2 = 8.66j pass through a pint whose co ordinates are (2,1)m. Calculate the moment of the forces about the origin. 41. What are the conditions for the equilibrium of a particle in space? 42. A force 27N makes an angle 30, 45, 80 with x, y, z axes. Find the force vector. 43. A force R = 5i + 2j 8k KN acts through origin. What is the magnitude of the force and angle it makes with x, y and z axis 44. Take two points (1,2,3), (4,5,6). If a force of 25N is applied between this pints. Find the force vector. 45. A force of 125N makes an angle of 30, 60 and 120 with X,Y and Z axis. Find the force vector. 46. What are fundamental and derived units? Give examples 47. Distinguish between Equal vectors and like vectors. 48. State the equations of equilibrium of a coplanar system of forces 49. Find the resultant of a n 800N force acting towards eastern direction and 500N force acting towards north eastern direction. 50. Distinguish the following types of forces with suitable sketch: a) Collinear and b) Co-planar forces 51. State and explain Lami s theorem of triangle law of equilibrium. 52. A man has a mass of 72 Kg is standing on a board inclined 200 with the horizontal. Find the component of man s weight. A) Perpendicular to the plane of the board b) Parallel to the plane of the board. 53. A force is represented by a vector form P = (10i 8j +14k)N. Determine the projection of this force on a line which originates from (2, -5, 3) and passes through point (5,2,-4) 54. A force of magnitude 750N is directed along AB where A is (0.8, 0, 1.2)m and B is (1.4, 1.2,0) m. Write the vector form of the force. Page 2 of 46

3 55. A force (10i + 20j 5k) N applied at A (3,0,2)m is moved to point B(6,3,1)m. Find the work done by the force. 56. How many equations of equilibrium are defined for a concurrent force system and coplanar force system? 57. A force F = 700i j is applied to a bolt A. Determine the magnitude of the force and the angle it forms with the horizontal. 58. Two forces of magnitude 20N and 40N are acting on a particle such that the angle between the two is If both these forces are acting away from the particle, calculate their resultant and find its direction. 59. A force of magnitude 700N is directed along PQ where P is (0.8,0,1.2)m and Q is (1.4,1.2,0)m. Write the vector form of the force. 60. Define equivalent system of forces. 61. A vector A is equal to 2i 3j + 2k. Find the projection of this vector on the line joining the point P(-3,2,1) and Q(2,-2,-1) 62. The sum of two concurrent forces F1 and F2 is 300N and their resultant is 200N. The angle between the force F1 and resultant is 900. Find the magnitude of each force. 63. A 100N force acts at the origin in a direction defined by the angles θx = 750 and θy = 450. Determine θz and the component of the force in the z direction. 64. Find the magnitude and direction cosines of the resultant of two concurrent forces F1 = 4i + 8j 8k and F2 = 5i 5j + 4k 65. Using Lami s theorem calculate the forces in the member CA and CB for the system shown in figure 66. A force F has the components Fx = 20N, Fy = -30N, Fz = 60 N. Find the angle θy it forms with the coordinates axes y. 67. What are the characteristics of forces. 68. Explain principles of transmissibility. 69. Define equilibrant. PART B 1. Two forces F 1 = (-2.0i + 3.3j K) N and F 2 = (-i + 5.2j K) N are concurrent at the point (2,2,-5). Find the magnitude of the resultant of these forces and the angle it makes with positive x-axis. 2. A force is represented by a vector F = 8i - 6j N. Find magnitude of the force and the CCW angle that it makes with +ve x-axis. 3. A force of 200N is directed along the line drawn from the point A(8,2,3)m to the point B(2,- 4.4,7.8). Find the magnitude of moment of this force about z-axis. 4. A 50N force is directed along the line drawn from point whose x,y,z coordinates are (8,2,3)m to the point whose coordinates are (2,-6,5)m. Find the moment of this force about z axis. Page 3 of 46

4 5. 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. 6. Compute the moment of the 200N force about points A and B for the figure shown below. 7. Determine the resultant of three forces F = 3i + 2 j + 5, F 2 = 2i j + 4k and F3 = 7i 3 j + 6 which are concurrent at the point (1, -6,5). The forces are in N and the distances are in metres. 8. A force F = 5i + 2 j + 4 is acting at a point A whose position vector is given by )4i + 2j 3k). find the moment of the force about the origin. 9. The coordinates of the initial and terminal points of a vectors are (4,5,2) and (6, -3,9) respectively. Determine the components of the vector and its angles with the axes. 10. An inclined plane makes 45 0 with the horizontal. A body of weight 1000N is acted by three forces such F = 150N, T =1300N and R = 500N as shown in figure. Find the resultant force acting on that body. in terms of I, j, k and it magnitude. Take A =i-2j-3k; B 11. Find the vector + + =4i- 3j+2k; =0.5i +j-k 12. Find the vector (- -2 B -3 C ) in terms of i,j,k and its magnitude. Take A =4i-3j+7k; B =3i- 2j-5k; = 2i-j-4k 13. Find the unit vector along the line which originates at the point (3,4, -3) and passes through the point (2,1,6). 14. Resolve the force 250N acting on the joint as shown in figure. Into components in the (a) x and y directions and (b) x 1 and y 1 directions. Page 4 of 46

5 15. A moment vector of magnitude 90Kgf. Starts at the point (2,3,4) and terminates at the point (5,-3,7). Find the vector expression for the moment. 16. A cantilever beam is loaded as shown in figure. Determine the moment of the forces about O. 17. A force F = 6i j + 2 is applied at point A(6,4.35,2). Find the moment of the force about the point B (6.5,3.75,-0.5) 18. Find the dot product and cross product of the following vectors. =(3.3i -4.6j+5.5k)m =( j-3.7k)m 19. There are 3 concurrent vectors. =(4i+3j+3k)m B = (i-3j+2k)m C = (-2i +4j + 6k)m What will be the volume enclose by these three concurrent vectors? 20. If vector A = I + 3j +2k, B = 2i -2j +3k (a) Find A X B and unit vector along it. (b) Find the included angle between vector A and vector resulting from the cross product. 21. What is the displacement vector from position (3,5,4)m to position (4,-1,2)m? What are its magnitude and direction cosines? 22. On a beam ABC, 10m long, forces are indicated as shown in figure. What is the moment of these forces about points A and B? Page 5 of 46

6 23. A rectangular bar weighing 500N hangs from a point C, by two strings AC and BC. Ac is inclined at 30 0 to the horizontal and BC is 45 0 to the vertical as shown in figure. Determine the forces in the strings AC and BC. 24. Find the magnitude and direction of the resultant R of four concurrent forces acting as shown in figure. 25. A cord supported at A and B carries a load of 20 KN at D and a load of W at C as shown in figure. Find the value of W so that CD remains horizontal. Page 6 of 46

7 26. Two smooth spheres A and B rest in a box as shown in figure. A has a diameter of 250mm and weights 1000N. B has a diameter of 100mm and weights 2KN. The box is 500mm wide at the bottom. Find the reaction at the supporting surfaces. 27. A body is subjected to the five forces as shown in figure. Determine the direction of the force F so that the resultant is in X direction, When (a) F =5000N, (b) F = 3000N. 28. Determine the magnitude and direction of the resultant of forces acting at O as shown in figure. Page 7 of 46

8 29. Five force are acting on a particle. The magnitude of the forces are 300N, 600N, 700N, 900N and P and their respective angles with the horizontal are 0 0,60 0,135 0, and as shown in figure. If the vertical component of all these forces is -1000N. 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. 30. Two spherical balls each of radius 200mm and weight 200N are kept between two vertical walls 600mm apart such that the first ball is resting on the ground and touching one of the vertical walls, the second ball is touching the first ball and the other vertical wall. Determine the reactions at the contact surfaces. 31. Find dot product and cross product of the following vectors: P = I + 2j -3k and Q = 4i -5j +6k 32. If vector A = 5i +3j+2k, B = I j-2k i) Find A X B and unit vector along it. ii) Find the included angle between vectors A and the vector resulting from the cross product. 33. The resultant of two concurrent forces is 1500N and the angle between the forces is The resultant makes an angle of 36 0 with one of the force. Find the magnitude of each force. 34. A force vector of magnitude 40N, is directed from A(1,4) to B(6,7). Determine i) The components of the force along x,y axes. ii) Angles with x and y axes iii) Specify the force vector. Page 8 of 46

9 35. Two cylinders of diameter 60mm and 30mm weighing 160N and 40N respectively are placed as shown in figure. Assuming all the contact surfaces to be smooth, find the reactions at A, B and C. 36. A system of four forces acting on a body as shown in figure. Determine the resultant force and its direction. 37. A force of 300N forms angles of 30 0,45 0 and 155 0, respectively with the x,y and z axes. Find the components Fx, Fy and Fz of the force. 38. Determine a) The x,y and z components of both forces b) The angles θx,θy and θz that the force forms with the coordinate axes. Page 9 of 46

10 39. In figure, a pole 10m high is supporting a wire in the xy plane, exerting a force of 500N on the top at an angle of 20 0 below the horizontal. Two guy wires are affixed as shown. Determine the tension in each guy wire and the compression in the pole assuming that the column is supported by a ball and socket joint at C. UNIT 2 EQUILIBRIUM OF RIGID BODIES 1. A horizontal bar ABCD is hinged at A and freely supported over D. Determine the reaction at C. (fig) 2. Sketch the hinged support and mark the direction of reaction 3. What is the conditions in equilibrium for two dimensional rigid body? 4. What are the types of supports? 5. Sketch a roller support and mark the direction of reaction. 6. Distinguish between particle and rigid body. 7. Explain what is meant by couple in space. 8. How will you resolve a given force into a force and couple in rigid body? 9. Define equivalent system of forces. 10. Find moment about x, y, z axes for the vector F = (20i + 30j + 50k)KN 11. Explain moment of force about a point in space. 12. Explain moment of force about an axis in space. 13. Define couple in space. 14. Give necessary and sufficient conditions for the equilibrium of a rigid body. 15. A simply supported beam of 6m span carries a concentrated load P at 2m from the left end. If the support reaction at the left and support is 8KN, find P. 16. What is the general condition of equilibrium of a rigid body? 17. The dimensions of power is The unit of polar moment of inertia is What are the necessary and sufficient conditions of equilibrium of rigid bodies in two dimensions and in three dimensions? Page 10 of 46

11 19. Three couples +12 Nm, -35Nm and +100Nm are acting in the xy, yz, xz. Write the vector form. 20. What is space diagram and free body diagram? 21. Three couples +16N.m, -45N.m and +120 N.m are acting in the xy, yz, and xz planes respectively. Determine the magnitude of the resultant vector of these three couples. 22. Locate the point where the system of forces given in Fig. is converted into a single force and zero couple. Also find the magnitude of the single force. Figure 23. How do you draw the free-body diagram of a body? 24. With the help of a simple illustration define free body diagram. 25. What is a free body diagram? 26. State Varigon s principle 27. Define equilibrium, and write the various types of equilibrium 28. Define Moment and also write the various types of moments 29. Define equivalent couple 30. What is support, write the various types of supports and their properties 31. What are the various types of beams. 32. State the necessary and sufficient conditions for static conditions of a particle in two dimensions. 33. Why the couple moment is said to be a free vector. 34. State the necessary and sufficient conditions for equilibrium of rigid bodies in two dimensions. 35. Give the equations indicating the principle of equilibrium of rigid bodies. 36. State Varigon s theorem 37. What is couple and moment of a couple? 38. State Newton s law concerning equilibrium of particle. 39. What are the conditions of equilibrium of a stationary body subjected to coplanar forces? m beam is subjected to the forces shown in figure. Reduce the given system of forces to a single force or resultant and the distance of the resultant from point A. Page 11 of 46

12 2. A rigid bar is subjected to a system of parallel forces as shown in figure. Reduce this system to i) Single force ii) A single force moment system at A iii) A single force moment system at B. 3. A system of forces are acting on a rigid bar as shown in figure. Reduce this system to i) A single force ii) A single force and a couple at A iii) A single force and a couple at B. 4. The figure shown, two vertical forces and a couple of moment 200Nm acting on a horizontal rod which fixed at end A. i) Determine the resultant of the system ii) Determine an equivalent system through A. Page 12 of 46

13 5. Four forces are acting along the side of a square ABCD of side 2m as shown in figure. In addition another force is acting along the diagonal CA. Calculate the resultant moment about the corner B. 6. Determine the magnitude and direction of a single force which keeps the system in equilibrium. The system of forces acting is shown in figure. 7. Replace the given system of forces acting on the body by a single force and couple acting at the point A. 8. Forces acting on the Hexagon ABCDEF of side 40cm is shown in figure. Determine the Net moment about A. Page 13 of 46

14 9. A simply supported beam of length 10m, carries the uniformly distributed load and two point loads as shown in figure. Calculate the reactions R A and R B. 10. A beam AB 6m long is loaded as shown in figure. Determine the reactions at A and B by analytical method. Page 14 of 46

15 12. A beam AB of span 4m, over hanging on one side upto a length of 2m, carries a uniformly distributed load of 2Kn/m over the entire length of 6m and a point load of 2 KN/m as shown in figure. Calculate the reactions at A and B. 13. For frame shown in figure. Determine the reactions at A and B. 14. Determine the reactions at the fixed support A for the loaded frame shown in figure. Take the diameter of the pulley as 200mm. Page 15 of 46

16 15. A horizontal beam AB is hinged to a vertical wall at A and supported at the point C by a tie rod CD as shown in figure. Find tension in the tie rod and the reaction at A due to a vertical load of 50N at B. 15. F 1 = (10i + 6j + 7k)N acts at the origin while a force F 2 = -F 1 acts at the length of a rod of length 10m protruding from the origin with direction cosines l = 0.6 and m=0.8. What is the moment of the forces F 1 and F 2 about a point P whose position vector r p = (5i + 10j +15k)m. If the position vector of S is (7.4i + 6.8j + 12K)m, find the moment of the forces about the line PS. 16. A rod AB shown in figure is held by a ball and socket joint at A and supports a mass weighing 1000N at end B. The rod is in xy plane and is inclined to y axis at an angle of The rod is 12m long and has negligible weight. Find the forces in the cable DF and EB. Page 16 of 46

17 17. 5m post shown in figure is acted upon by a 30KN force (F) at C and is held by a ball and socket at A and by the two cable BD and BE. Determine the tension in each cable and the reaction at A. 18. A tension T of magnitude 10KN is applied to the cable attached to the top A of rigid mast and secured to the ground at B as shown in figure. Determine moment of the tension T about the z-axis passing through the base O. Page 17 of 46

18 19. A pole shown in figure is supported by a ball and socket joint at its base and by cables AB and AC. Also it is subjected to forces 300N and 600N and the forces act in a plane parallel to x-y plane. Compute the forces in the cables and reaction at the ball and socket joint. Page 18 of 46

19 UNIT III PROPERTIES OF SURFACES AND SOLIDS 1. Define centre of gravity 2. Define centre of mass 3. Define centroid axis 4. How many centres of gravity, a body has? 5. What is the difference between centre of gravity and centroid? 6. What are the various methods to find the centre of gravity? 7. Where does the centre gravity of the following section lines? a) Semi-circle b) Trapezium 8. State the theorems of Pappus and Guldinus 9. What do you mean by first moment of an are 10. Find out the surface area of the line shown in figure. It is rotated about y axis by using Pappus and guidinus theorem. 11. Define Radius of gyration of a body 12. State parallel axes theorem 13. State perpendicular axes theorem 14. Define polar moment of inertia 15. What is section modulus? 16. Moment of inertia of a triangular about an axis passing through centre of gravity and parallel to the base is 17. The mass moment of inertia of a circular cylinder of radius e and height h about its axis is. 18. What is meant by product of inertia of a given are 19. What is the mass moment of inertia of a thin circular plate of mass M and radius r about its diameter 20. What is the use of routh s rule? 21. The theorem of parallel axis is not used in obtaining the moment of inertia of a Give the expressions for maximum and minimum values of principal moment of inertia. Page 19 of 46

20 23. Define polar axis of a cross section. What is the value of moment of inertia of circular section of 50mm radius about its axis. 24. What do you mean by mass moment of inertia? What is its unit? 25. Give equations for calculating moment of inertia of any plane are. 26. Define centroid and centre of mass. 27. Find the polar moment of inertia at a hollow circular section of external diameter D and internal diameter d. 28. Determine the second moment of area of a triangle with respect to the base. 29. Under what conditions do the following coincide a) Centre of mass and centre of gravity b) Centre of gravity and Centroid 30. A semi circular arc having radius 100mm is located in the xy plane such that its diameteral edge coincides with the y-axis. Determine the x-coordinate of its centroid. 31. Define principal axes and principal moment of inertia 32. Find the product of inertia (Ixy) about centroidal axis for the are formed by subtracting the circle of diameter 40mm for the square of side 80mm as shown in fig. Centroid of circle coincides with centroid of square. 33. Define product of inertia 34. Give the centroid of quarter circular arc. 35. What are principal axes? 36. Express radius of gyration of a body in terms of its mass moment of inertia 37. How would you find out the moment of inertia of a plane area? 38. Define polar axis of a cross section. What is the value of moment of inertial of a circular section of 50mm radius about this axis? 39. Calculate y for the shaded area as shown in figure 40. Calculate moment of inertia Ixx for plane are shown in figure. All dimensions are in mm. Page 20 of 46

21 PART B 1. Find the moments of inertia of the section given in the figure about the horizontal and vertical centroidal axes. Also find the polar moment of inertia and minimum radius gyration (All dimensions are in mm). 2. State and prove perpendicular axis theorem. 3. A rectangle of width 480mm and height 600mm has semi circle of radius 240mm cut out with its diameter coinciding with shorter edge of the rectangle. Locate the centroid of the net area and the moment of inertia about the axis parallel to the shorter edge and passing through the centroid. 4. Find the centre of gravity of the lamina as shown in figure. Page 21 of 46

22 5. Find the moment of inertia of a rectangular section shown in figure about a) Its centroidal axes XX, XY; b) About its short edge (IAA), and c) An axis BB which is 20mm distance from its short edge and parallel to it. 6. Find the centroid of an unequal angle section 100mm X 80mm X 20mm 7. A semi circular area is removed from a trapezium as shown in figure. Determine the centroid of the remaining area. (All dimensions in mm) Page 22 of 46

23 8. Find the moment of inertia about 1-1 and 2-2 axes for the area shown in figure. 9. Find the moment of inertia of the shaded area for the figure about the centroidal axes, the axes being parallel to x and y axes. Page 23 of 46

24 11. Determine the centroid of the cross-sectional area of an unequal I-section shown in figure. Page 24 of 46

25 12. Calculate the moment of inertia and radius of gyration about the x-axis for the sectioned area shown in figure. 13. Determine the product of inertia of the sectioned area about the x-y axes shown in figure. Page 25 of 46

26 14. Determine the centroidal coordinates of thee area shown in figure with respect to the shown x-y coordinate system. 15. Determine the second moment of area of the section shown in figure about its base axis a-a. Page 26 of 46

27 16. Locate the centroid for the plane surface shown in figure. 17. Compute the second moment of area of the plane surface shown in figure about its horizontal centroidal axis. Page 27 of 46

28 18. For the plane area shown in figure, located the centroid of the area. Page 28 of 46

29 19. The cross-section of a culvert is shown in figure. Compute the moment of inertia about the horizontal A-A axis. 20. Determine the moment of inertia of the section shown in figure about centroidal XX and YY axes: Page 29 of 46

30 x = 0.912m y = 2.25m 21. Derive an equation for the mass moment of inertia of cone. 22. Find the second moment of area of the plane lamina shown in figure with respect to the Page 30 of 46

31 23. The cross-section of a hollow circular lamina is as shown in figure. Locate the position of the centroid of the net area and then work out the moments of inertia about both the Page 31 of 46

32 horizontal and vertical centroidal axes. 24. Find the plane section shown in figure determine the moment of inertia about its horizontal and vertical centroidal axes. Page 32 of 46

33 UNIT IV DYNAMICS OF PARTICLES 1. Define rectilinear motion 2. Define curvilinear motion 3. Define average acceleration 4. Define average velocity 5. The particle moving with s = 9t 3 + 2t + 2. Find velocity and acceleration when time t=6 sec. 6. What is a projectile? 7. At angle when a projectile is thrown it gets maximum range. 8. State D Alembert s Principle 9. Distinguish between kinematics and kinetics 10. Define angular momentum 11. Define linear momentum 12. State Newton s second law of motion 13. Mention Newton s law for rectangular coordinates 14. Define inertia force and explain with examples 15. Define work of a force and its unit Page 33 of 46

34 16. Write down expression for work done by a spring force 17. A body if mass 2 Kg is moving with a velocity of 50m/s what will be the kinetic energy of the body? 18. Find potential energy of a body having mass 100 Kg at a height of 20m. 19. Define the term impulse 20. What is meant by impulsive force? 21. How do you define the momentum of the particle 22. Define: Law of conservation of momentum 23. Define angular momentum or Moment of momentum 24. Define law of conservation of angular momentum 25. State the difference between impulse and momentum 26. Write the units for following a) Linear impulse b) Linear momentum c) Angular momentum d) Angular impulse 27. A block of mass 3 Kg is subjected to a time varying force F = t along x direction. Determine the velocity after 3 seconds if its initial velocity is 10 m/s. 28. A 55 Kg man moving horizontally with a velocity of 3 m/s, jumps of the end of a pier into a 300 Kg boat. Determine the horizontal velocity of the boat, if it was approaching the pier with an initial velocity of 0.8 m/sec. 29. A hammer of mass 15 Kg strikes a wedge with a velocity of 12 m/s, and rebounds with a velocity of 2 m/s, The duration of the impact is 0.01 seconds. Calculate the average force exerted on the edge 30. Define an Impact 31. What is meant by elastic impact? 32. What are the two types of impact? 33. Define co-efficient of restitution 34. State Newton s law of collision of elastic bodies 35. State law of conservation of momentum 36. What is the equation used for finding out the loss of kinetic energy due to impact? 37. A body of mass 4 Kg moving with a velocity of 2 m/s, impinges directly on a body of mass 8 Kg at rest. After impinging the first body comes to rest, the second body moving with 1 m/s. Find the co-efficient of restitution. 38. A ball of mass 12 Kg moving with velocity of 5 m/s collides with another ball of mass 4 Kg moving in the opposite direction with a velocity of 1.5 m/s. If they struck together after impact, what will be the common velocity? 39. A ball of 9 N is dropped from the height of 6m. then it strikes the 15 N ball. What will be velocity of first ball? 40. What should be the value of coefficient of restitution, if body is said to be perfectly elastic and perfectly inelastic? 41. A 60 Kg bullet moving with a speed of 500 m/s, strikes a 5 Kg block moving in the same direction with a speed of 30 m/s. What is the resultant speed of the bullet? 42. Two equal billiard balls meet centrally with special 2 m/s and -3 m/s. What will be their final speeds after impact of the co-efficient of restitution is 0.8? 43. Define the term Time of Flight 44. Define line of Impact Page 34 of 46

35 45. A car traverse half of a distance with a velocity of 40 Kmph and the remaining half of distance with a velocity of 60 Kmph. Find the average velocity. 46. Give mathematical definitions of velocity and acceleration. 47. State Newton s law of collision of elastic bodies 48. A ball is dropped on a smooth floor from a height of 1m. If the first rebound height is 81 cm, find the co-efficient of restitution. 49. A body is moving with a velocity of 4 m/s. After 5 seconds the velocity of the body becomes 10 m/s. find the acceleration of the body 50. Define horizontal range and trajectory. 51. A cricket ball of mass 0.20 Kg moving horizontally at 150 m/s hit straight back with a bat with a velocity of 60 m/s. If the contact lasted 1/20 sec, find the impulse force exerted by the bat. 52. What is meant by the term translation? 53. Define angle of repose 54. A stone is dropped from the top of a tower. A stone reaches the ground in 10 seconds. Determine the height of tower. 55. State the principle of work and energy. 56. What general plane motion? 57. Define instantaneous centre of rotation 58. A particle, starting from rest, moves in a straight line and its acceleration is given by a = 50 36t 2 m/sec 2 where t is in sec. Determine the velocity of the particle when it has travelled 52m. 59. A stone is dropped into a well and if the splash is heard 2.50 seconds later, determine the depth of water surface assuming that the velocity of sound is 330 m/sec. 60. What are motion curves? 61. Explain dynamic equilibrium 62. A lift has an upward acceleration of 1 m/s 2. What pressure will a man weighing 600N exert on the floor of the lift? 63. The angular rotation in radians of an accelerated flywheel is given by θ = 9 32 t3. Find its angular acceleration when t = 1.6 s. 64. A ball dropped from a height of 1.6m on a floor rebounds to a height of 0.9m, find the coefficient of restitution. 65. A circular disc has a mass moment of inertia of 12 Kg m 2 about its axis of rotation. If it is initially at rest, find its angular velocity after 3 seconds, if it is acted upon by a torque of magnitude 800Nm. 66. A car starts from rest with a constant acceleration of 4 m/s 2. Determine the distance travelled in the 7 th seconds. 67. The motion of a particle is defined by the relation x = t 3 16t 2 22 where x is expressed in metre and t in seconds. Determine the acceleration of the particle at t -= 3s. 68. A food packet having a mass of 12 Kg is dropped at 5 m/s from a very high altitude. If the atmospheric drag resistance F d is 80N, determine the velocity of the food packet when it has fallen 10m. 69. Write down the equation of principle of work and energy for a rigid body. 70. A particle starting from rest, moves in a straight line and its acceleration is given by a = 40 46t 2 m/sec 2 where t is in sec. Determine the velocity of the particle it has travelled 52m. Page 35 of 46

36 71. A steel ball is thrown vertically upwards from the top of a building 25m above the ground with an initial velocity of 18 m/sec. Find the maximum height reached by the ball from the ground. 72. A body is moving with a velocity of 4 m/s. After five seconds the velocity of body becomes 14 m/s, find the acceleration of the body. 73. A particle moves along x axis and its position is expressed as x = 5t 3-10t 2 where s is in m and t is in sec. Find velocity and acceleration at t = 3sec. 74. State work energy equation of particles. The meaning of each term. 75. Two balls are projected from the same point in the directions inclined at 60 0 and 30 0 to the horizontal. Determine ratio of velocity of projections if they have same maximum height. 76. Explain the terms power and energy. 77. A body is rotating with an angular velocity of 5 radians/sec. After 4 seconds, the angular velocity of the body becomes 13 radians/sec. Determine the angular acceleration of the body. 78. State the impulse momentum principle. Write its equation form, stating the meaning of each term in it. 79. A golf ball is dropped from a height of 10m on to a fixed steel plate. The coefficient of restitution is Find the height to which the ball rebounds at the first bounce. 80. State principle of conservation of energy. 1. A stone falls past a window 2m high in 0.5s. Find the height above the window from where the stone is dropped. 2. A shot is fired with a velocity of 30m/s from a point 15m in front of a vertical wall 6m high. Find the angle of projection with the horizontal to enable the shot to just clear the wall. Explain the double answer. 3. A cage descends a vertical mineshaft with an acceleration of 0.5m/s 2 from rest. After the cage has traveled 25m, a stone is dropped from the top of the shaft. Determine the time taken for the stone to hit the cage and the distance traveled by the before the impact. 4. A particle is projected at such an angle that the horizontal range is thrice the greatest height. 5. A weight of 10N resting on an inclined plane that makes an angle of 30 0 with horizontal is connected by a string passing over a frictionless pulley at the upper end of the plane. On the free end of the string a weight of 20N is connected. If the coefficient of friction between the plane and 10N weight is 0.2, calculate the time taken by the hanging weight to descend by 1m. Adopt work-energy method. 6. A bullet of mass 20g is fired into a body of mass 10Kg which is suspended by a string 0.8m long. Due to this impact, the body swings through an angle of Find the velocity of the bullet. 7. A particle moves along X axis and its position is expressed as S = 4t3 3t2. Page 36 of 46

37 a) Average velocity during t = 2s to t = 4s, and b) Average acceleration during t = 2s to t = 4s. 8. A particle is projected in air with a velocity 100m/sec and at an angle of 30 0 with the horizontal. Find a) The horizontal range. b) The maximum height attained by the particle c) The time of flight. 9. A ball is thrown vertically upward from a point located 20m above the ground. The maximum height reached by the ball is 30m from the ground. Determine the initial velocity of the ball at 20m above the ground and also the velocity with which the ball strikes the ground. 10. A ball of mass 2 Kg, moving with a velocity of 3m/s, impinges on a ball of mass 4 Kg moving with a velocity of 1m/s. The velocities of the two balls are parallel and inclined at 30 0 to the line joining their centres at the instant of impact. If the co-efficient of restitution be 0.5. find a) Direction, in which the 4 Kg ball will move after impact, b) Velocity of the 4 Kg ball after impact, c) Direction, in which the 2Kg ball will move after impact d) Velocity of the 2 Kg ball after impact. 11. A particle under a constant deceleration is moving in a straight line and covers a distance 28m in first 4 seconds and 32m in the next 6 seconds. Calculate the distance it covers in the subsequent next 5 seconds and the total distance covered by it before it comes to rest. 12. What do you mean by projectile, trajectory and range of projectile? 13. A wheel 1.2m in diameter accelerates uniformly from rest to 2000rpm in 20 seconds. Calculate ; 1) Number of revolutions the wheel makes in attaining its speed of 2000 rpm. Ii) Angular acceleration of the wheel and iii) Tangential velocity of a point on the rim of the wheel, 0.6 seconds after it has started from rest. 14. An automobile accelerates uniformly from rest on a straight level load. A second automobile starting from the same point 6 seconds later with initial velocity zero accelerates at 6m/s 2 to overtake the first automobile 400m from the starting point. What is the acceleration of the first automobile? 15. An inextensible cord going around a homogeneous cylinder A of mass 100Kg holds a mass less plate B. The collar C of mass 30 Kg is released from rest in the position shown in figure and drops upon the plate. Determine the velocity of the collar when it has descended an additional 0.5m after striking the plate. Assume that there is no rebound, that is c and b move downwards locked together and the cord remains taut. 16. The equation of motion of a particle moving in a straight line with variable acceleration is given by, s = 15t + 3t 2 t 3, in which, s is the distance measured in m and time t is measured in seconds. Calculate. a) The velocity and acceleration at start. b) The time, at which the particle attain its maximum velocity Page 37 of 46

38 c) The maximum velocity of the particle. 17. A wheel, rotating about a fixed axis at 20 rpm is uniformly accelerated for 70 seconds, during which time it makes 50 revolution. Find a) Angular velocity at the end of this interval b) Time required for the speed to reach 100 revolution per minute. 18. A body is projected at an angle such that its horizontal range is 3 times the maximum height. Find the angle of projection. 19. Two bodies weighting 300N and 450N are hung to the ends of a rope passing over an ideal pulley. With what acceleration the heavier body comes down? What is the tension in the string? 20. An object of mass 5Kg is projected with a velocity of 20m/s at an angle of 60 0 to the horizontal. At the highest point of its path the projectile explodes and brakes up into two fragments of masses 1 Kg and 4 Kg. The fragments separate horizontally after the explosion. The explosion releases internal energy such that the kinetic energy of the system at the highest point is doubled. Calculate the separation between two fragments when they reach the ground. 21. A car starts from rest and accelerates uniformly to a speed of 80 Kmph over a distance of 500m. Find time and acceleration. Further acceleration raises the speed to 96 Kmph in 10 seconds. Find the acceleration and distance. Brakes are applied to bring the car to rest under uniform retardation in 5 seconds. Find the distance covered during braking. 22. A projectile is thrown with a velocity of 5 m/s at an elevation of 60 0 to the horizontal. Find the velocity of another projectile thrown at an elevation of 45 0 which will have (1) equal horizontal range (2) equal maximum height and (3) equal time of flight with the first. 23. An object is thrown vertically upward with a velocity of 30m/s. Four seconds later a second object is project vertically upward with a velocity of 40m/s. Determine (i) the time (after the first object is thrown) when the two objects will meet each other in air (ii) the height from the earth at which the two objects will meet. 24. A 2000N block is hanging from an in extensible string which is wrapped around a drum of radius 400mm rigidly attached to a wheel. Combined mass moment of inertia of drum and when is 20 Kg-m 2. At particular instant as shown in figure, the velocity of the block is 3 m/s downward. If the friction force oppose the motion by giving 80 N-m couple on the bearing, find the velocity of the block after it has moved through 2m downwards, use workenergy relationship. 25. Water drips from a faucet at the rate of five drops per second as shown in figure. Determine the vertical separation between two consecutive drops after the lower drop has attained a velocity of 3 m/s. Page 38 of 46

39 26. In a police investigation of tyre marks, it was concluded that a car while in motion along a straight level road skidded for a total of 60 metres after the brakes were applied. If the coefficient of friction between the tyres and the pavement is estimated as 0.5. What was the probable speed of the car just before the brakes were applied? 27. Two vehicles approach each other in opposite lanes of a straight horizontal roadway as shown in figure. Find the time and positions at which the vehicles meet if both continue to move with constant speed. 28. Two bodies of weight 20N and 10N are connected to the two ends of light inextensible string, passing over a smooth pulley. The weight of 20N is placed on a horizontal surface which the weight of 10N is hanging free in air as shown in figure. The horizontal surface is a rough one, having coefficient between the weight 20N and the plane surface equal to 0.3, using newton s second law of motion determine (i) the acceleration of the system (ii) the tension in the string. Page 39 of 46

40 29. In the oblique central impact shown in figure, the co-efficient of restitution is 0.8. the flat disks shown, slide on a smooth horizontal surface. Determine the final velocity of each disk directly after impact. 30. Two electric trains A and B leave the same station on parallel lines. The train A starts with a uniform acceleration of 0.15 m/s2 and attains a speed of 40 Km/hr when the steam is reduced to keep the speed constant. The train B leaves1 minute after, with a uniform acceleration of 0.3m/sec2 to attain a maximum speed of 70 Km/hr. When the train B will overtake the train A? 31. The angle of rotation of a body is given as the function of time by the equation θ = θ 0 + at + bt 2 where θ 0 is the initial angular displacement, a and b are constants. Obtain the general expression for a) The angular velocity Page 40 of 46

41 b) The angular acceleration of the body. If the initial angular velocity be (3π) radians per second and after two seconds the angular velocity is (8π) radians per second, find the constants a and b. 32. A stone is dropped into a well and the sound of splash is heard in 6 seconds; velocity of sound = 340 m/s, find the depth of surface of water from the top. Explain concepts. 33. A ball is projected vertically upwards with a velocity of 20m/s. Two seconds later, a second ball is projected vertically upwards with a velocity of 16 m/s. Find the height above the surface at which the two balls meet. 34. A ball is dropped from the top of the tower 50m high. At the same instant a second ball is thrown upward from the ground with an initial velocity of 20m/s. when and where do they cross and with what relative velocity. 35. A mass 12 Kg travelling to the right with a speed of 7.5 m/s collides with another mass 24Kg, travelling to the left with a speed of 25 m/s. If the coefficient of restitution is 0.6, find the velocities of the particles after collision and loss in kinetic energy. What is the impulse acting on either particle during the impact? 36. A body weighing 196.2N slides up a 30 0 inclined plane under the action of an applied force of 300N acting parallel to the plane. The co-efficient of friction is 0.2. The body moves from rest. Determine at the end of 4 seconds, the acceleration, distance travelled, velocity, kinetic energy, workdone, momentum and impulse applied on the body. 37. State the law of conservation of energy. Give the proof of this law taking mechanical energy only into account. UNIT V FRICTION AND ELEMENT OF RIGID BODY DYNAMICS 1. What is meant by friction? 2. State the different types of friction 3. Define dry friction 4. What is meant by fluid friction? 5. Define static friction 6. What is meant by Dynamic friction? 7. What are the two types of dynamic friction? 8. What is meant by sliding friction and rolling friction? 9. Define co-efficient of friction 10. Define angle of friction 11. Define angle of repose 12. What is meant by cone of friction? 13. What is friction involved? 14. What is a wedge? 15. What is a ladder Page 41 of 46

42 16. Define rolling resistance 17. What is a belt drive? 18. What is the use of belt friction? 19. A vertical wall of weight 300KN is subjected to lateral force of 3 KN. Will it be safe against string on a horizontal plane on which the co-efficient of friction = 0.01? 20. Determine the minimum tension in the rope required to support a cylinder of mass 500 Kg, when the rope passes once over the rod. 21. A body of weight 130N is placed on a horizontal plane. Horizontal force is given as 53 N. find the angle of friction? 22. A body of weight 100N is placed on rough horizontal plane. If a horizontal force of 55N is acting on the body, determine the co-efficient of friction? 23. State the Columb s law of dry friction 24. Define a rigid body 25. What do you mean by translation? 26. Define rotation about a fixed axis 27. What is meant by general plane motion? 28. Define instantaneous centre of rotation in plane motion. 29. State chashe,s theorem 30. What is the degrees of freedom in rigid body? 31. Define uniform rotation 32. Define uniform accelerated rotation 33. Write the expression for acceleration in the plane motion 34. What is torque? 35. The instantaneous centre is a point identified with the body where the velocity is Define the centrode. 37. Define axode 38. Define Non-centroidal rotation of the rigid body. 39. What is the condition for a when which rolls without slipping and slips without rolling? 40. Define non-centroidal plane motion of the rigid body. 41. Define angular moment of a rotating body 42. Write the expression for kinetic energy in translation and rotation 43. Write the expression for kinetic energy in plane motion 44. State the principle of work and energy in a rigid body. 45. State the principle of conservation of energy 46. Define power 47. The work done by couple is given by 48. The power developed by a force is and that by couple is - 1. A ladder 3m long and weighing 200N is resting on the horizontal floor and leaning against a vertical wall, making 30 0 with the floor. The friction coefficients at the ground and wall contact surfaces are 0.35 and 0.25 respectively. It has to support a weight of 100N at the top. To prevent slipping, a string is tied to the foot of the ladder and attached to the wall in the horizontal position. Determine the minimum tension required in the string for this Page 42 of 46

43 condition. Find also the minimum angle with the floor at which the above ladder with the weight at the top could be placed without slipping in the absence of the string. 2. The tension in the tight side of a belt having an angle of lap of is 500N. Find the tension in the slack side if the coefficient of friction is 0.2. Derive equation used, if any. 3. A uniform ladder of length 10m and weighing 20N is placed against a smooth vertical wall with its lower end 8m from the wall. In this position the ladder is just to slip. Determine a) The coefficient of friction between the ladder and the floor b) Frictional force acting on the ladder at the point of contact between ladder and floor. 4. A uniform ladder of weight 200N of length 4.5m rests on a horizontal ground and leans against a rough vertical wall. The coefficient of friction between the ladder and the floor is 0.4 and between ladder and vertical wall is 0.2. When a weight of 900 N is placed on the ladder at a distance of 1.2m from the top of the ladder, the ladder at the point of sliding. Find (a) The angle made by the ladder with horizontal and (b) Reaction at the foot and top of the ladder. 5. A force of 200N is required to just move a certain body yup an inclined plane of angle 15 0, the force being parallel to the plane. If the angle of inclination of the plane is made 20 0, the force required again parallel to the lane is found to be 230N. Find the weight of the body and the coefficient of friction. The uniform 5 Kg rod shown in figure is acted up on by 30N force which always acts perpendicular to the bar. If the bar has an initial clockwise angular velocity ω1 =10 rad/s when θ = 0 0, determine its angular velocity at the instant θ = Two rough planes inclined at 300 and 600 to horizontal are placed back to back as shown in figure. Two blocks of weight 50N and 100N are placed on the planes and are connected by a cord passing over a friction less pulley. If the coefficient of friction between the planes and blocks is Find the resulting acceleration of the blocks and the tension in the cord. 7. Determine the smallest force P required to move the block B shown in figure. (i) Block A is restrained by cable CD as shown in figure and (ii) cable CD is removed. Take µ s = 0.30 and µ k = Page 43 of 46

44 8. Determine whether the block shown in figure (i) having a mass of 40 Kg is in equilibrium and find the magnitude and direction of the friction force. Take µ s = 0.40 and µ k = A flat belt rolls over a pulley of diameter D, making contact angle β when their respective tensions are T 1 and T 2. Derive a relationship between the tensions, angle of contact and coefficient of friction µ. Show that the diameter of the pulley is immaterial. 10. A block weighing 1350N is placed on an inclined plane whose inclination to the horizontal is A force of 450 N acts on the body in the upward direction parallel to the plane. Determine whether the block is in equilibrium or not, and also find the frictional force between the body and the plane. The co-efficient of static and kinetic frictions are 0.25 and 0.20 respectively. 11. Two bodies of weight 40N and 20N are connected to the two ends of a light inextensible string, passing over a smooth pulley. The weight of 40 N is placed on a smooth horizontal Page 44 of 46

45 surface while the weight of 20N is hanging free in air. Find (i) The acceleration of the system and (ii) The tension in the string. Take g = 9.81 m/sec A block weighing 200N rests on a plane inclined at 450 to the horizontal. The block is tied by a horizontal string as shown in figure. The block is in equilibrium when the tension in the string is 70N. Determine the friction force, the normal reaction of the plane and the coefficient of friction between the block and the plane. 13. A wheel of diameter 560mm rolls without slipping on a flat surface. The centre of wheel is moving a velocity 20m/s. Find the velocity of the point B,D, C as shown in figure. 14. A 60mm radius drum is rigidly attached to a 100mm radius drum as shown in figure. One of the drum rolls without sliding on the surface shown, and a card is wound around the other drum knowing that at the instant shown. The centre point velocity is 160mm/sec. and acceleration 60mm/sec 2. Find the acceleration of the point A, B & C of the drum. Page 45 of 46