Workbook on STATICS Department of Mechanical Engineering LINGAYA S UNIVERSITY Nachauli, Old Faridabad Jasana Road Faridabad , Haryana, India

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1 Workbook on STATICS Department of Mechanical Engineering LINGAYA S UNIVERSITY (Deemed to be University under Section 3 of UGC Act, 1956) Nachauli, Old Faridabad Jasana Road Faridabad 121 2, Haryana, India

2 Contents Chapter Title Page 1 FORCE VECTORS Parallelogram law 1 Addition of a system of coplanar forces 2 Addition of Cartesian vectors 3 Cartesian position and force vectors 4 Dot product 5 2 EQUILIBRIUM OF A PARTICLE Particle equilibrium in two dimensions 7 3 FORCE SYSTEM RESULTANTS Cross product and moment of a force 8 Transmissibility of a force, principles of moments 9 Moment of a force about a specified axis 1 Moment of a couple 11 Resultants of a force & couple system 12 Further reduction of a force/couple system 14 Reduction of a simple distributed loading 16 4 EQUILIBRIUM OF A RIGID BODY Equilibrium of a rigid body in 2D 18 Two and three-force members 19 KEY TO QUESTIONS 2

3 Chapter 1 Force Vectors Parallelogram Law 1) Two forces are to be added to determine the resultant force F r. Compute the angle θ Pythagorean theorem 4) To determine F r shown in the figure, do you use: θ =? θ =? ) Two forces are to be added to determine the resultant force F r. Compute the angle θ ) To determine F r shown in the figure, do you use: Cosine Law Sine Law Pythagorean theorem 5) To determine F r shown in the figure, do you use: Cosine Law Sine Law Cosine Law Sine Law 1

4 Pythagorean theorem Addition of a System of Coplanar Forces -1i N 3) Select the answer that expresses F as a Cartesian Vector. 1) Select the answer that expresses F as a Cartesian Vector. F r =? 25 N -5 N F =? 1j N 1i N -1j N -1i N 2) Select the answer that expresses F as a Cartesian Vector. F =? 1j N 1i N -1j N F =? (-7.7i + 7.7j) N (-7.7i - 7.7j) N (7.7i + 7.7j) N (1i + 1j) N 4) Select the answer that expresses F as a Cartesian Vector. F =? (6i + 8j) N (8i - 6j) N (-8i + 6j) N (-8i - 6j) N 5) Calculate the magnitude of F r. 5 N 6) cont...from qu5, select the equation which is used to determine the direction θ measured from the +x axis. θ =? arctan (-3/4) arctan (4/3) arctan (3/4) Addition of Cartesian Vectors 1) Calculate the magnitude of the force. F = (3i - 4j) N. F =? 25 N 4 N 5 N 2

5 2) Calculate the magnitude of the force. F = (1i + 2j - 2k) N. F =? 25 N 3 N 4 N 3) If F = (4i - 3j) N, select the answer that specifies the cos β. cos β =? ) If F = (-1i + 2j - 2k) N, select the answer that specifies the cos β. cos β =? ) Select the answer that gives the y component for the 1-N force. F y =? 1sin45sin6 1sin45cos6 1cos45cos6 1cos45/sin6 6) Select the answer that gives the y component for the 1-N force. F y =? 1sin6 (4/5) 1sin6 (3/5) 1cos6 (3/5) 1cos6 (4/5) Cartesian Position and Force Vectors 1) Select the answer that represents the position vector r. r =? (7.7i + 7.7j) m (7.7i - 7.7j) m (-7.7i - 7.7j) m (-7.7i + 7.7j) m 2) Select the answer that represents the position vector r. r =? (-2i - 4j + 3k) m (3i + 4j) m (4j + 3k) m (2i + 4j + 4k) m 3) Select the answer that specifies the cosγ. 3

6 cosγ =? ) Select the answer that specifies the cosγ. cosγ =? cosγ =? ) If r has a magnitude of 1 m and acts in a direction defined by the unit vector u = -.77i +.77j, calculate the angle that r makes with the positive x-axis. Answer =? ) If r has a magnitude of 1m calculate γ. 8) Select the answer that expresses r as a Cartesian Vector. r =? (5i - 5j - 7.7k) m (-5i - 7.7j - 5k) m (5i + 5j + 7.7k) m (5i + 7.7j + 5k) m 9) Select the answer that represents the force vector F directed from A towards B. 5) Select the answer that specifies the cosγ. γ =? 4

7 3-5 projection of F along the line OA ) Calculate the angle between r 1 and r 2. Proj F =? N N (3i + 3j - 2k) (2i + 2j - 1k) 34 N N (2i + 1j - 2k) (-2i + 2j - 1k) θ =? 3 6) Calculate the magnitude of the projection of F along the line OA. 45 Dot Product 1) Calculate the dot product. (3i).(-4i) =? ) Calculate the dot product. (1i + 2j + 3k).(1i - 3j - 1k) =? 6 4) If F = (3i + 2j) N and a line is defined by u = (-.8i -.6j), calculate F.u. F.u =? ) Calculate the magnitude of the Proj F =?

8 Chapter 2 Equilibrium of a Particle Particle Equilibrium in Two Dimensions 1) The unstretched length of the elastic cord AB is 3 m. Determine the force in the stretched cord if the cord stiffness is k = 4 N/m. F =? 4) Select the equation that represents the equation of force equilibrium in the x direction. F =? 4 N 75 N 8 N 12 N 2 N 2) Calculate the force F required for the equilibrium of the particle. -6 N -2 N 4 N 3) Select the equation that represents the application of force equilibrium in the x direction. Fcos45 + Pcos45 = Psin45 - Fcos45 = Pcos45 - Fcos = 1 Pcos45 - Fcos45 = Pcos45 - Fcos45 + 1cos3 = -Pcos45 + Fcos75 + 1cos3 = 1cos3 + Fcos75 - Pcos15 = -Pcos45 + Fcos = 5) The force of 1 N acts at point A. Draw the free-body diagram of point A and calculate the force developed in cable AC. 6

9 N F AC =? 7.71 N 1 N N 6) Calculate the stretch in the elastic cable BA if the system is held in the equilibrium position shown. Both AB and AC are elastic cables. Stretch in BA =? 1 m 1.5 m 2 m 7

10 Chapter 3 Force System Resultants Cross Product and Moment of a Force 1) Select the answer that represents the result of the vector cross product. i x k =? j -i -j 2) Select the answer that represents the result of the vector cross product. 2k x 6k =? 12i 12j 12k 3) Select the answer that represents the moment of the force about point. M O =? 2 Nm clockwise 4 Nm clockwise 2 Nm anticlockwise 4) Select the answer that represents the moment of the 1N force about point O. M O =? 2 Nm clockwise 4 Nm clockwise 6 Nm clockwise 8 Nm clockwise 5) Select the answer that represents the moment of the 1N force about point O. M O =? 2 Nm 2 Nm clockwise 6) Use the vector cross product and compute the moment of the 1-N force about point O. M O =? (7.7 k) Nm (141.4 k) Nm ( k) Nm (-7.7 k) Nm 8

11 7) Calculate the moment of the force about point O using the vector cross product and select the correct answer. M O =? (2i - 3k) Nm (6i - 3k) Nm (6i - 3j) Nm (6i + 4j - 3k) Nm Transmissibility of a Force about a Specified System 1) Select the answer that represents the moment of the force about point O. M O =? 2 Nm 1 Nm clockwise 2 Nm clockwise 1 Nm 2) Calculate the resultant moment about point O. M O =? 5 Nm clockwise 1 Nm clockwise 1 Nm 5 Nm 3) Calculate the resultant moment about point O. M O =? 75 Nm clockwise 15 Nm clockwise 2 Nm clockwise 6 Nm 4) Calculate the resultant moment about point O. M O =? 9 2 Nm clockwise 2 Nm 3 Nm clockwise Nm 5) Calculate the resultant moment about point O. M O =? 1 Nm 1 Nm clockwise Nm 2 Nm 6) Calculate the resultant moment about point O. M O =? 5 Nm clockwise 4 Nm clockwise 2 Nm clockwise

Moment of a Force About a Specified Axis 1) For questions 1 to 3, select the answer that represents the moment of the 1 N force about each of the coordinate axes. M x =? -3 Nm -2 Nm 2 Nm 3 Nm 2) cont.

12 Moment of a Force About a Specified Axis 1) For questions 1 to 3, select the answer that represents the moment of the 1 N force about each of the coordinate axes. M x =? -3 Nm -2 Nm 2 Nm 3 Nm 2) cont... M y =? -2 Nm -3 Nm 2 Nm 3) cont... M z =? 3 Nm Nm -2 Nm -3 Nm 4) For questions 4 to 6, select the answer that represents the moment of the 1 N force about each of the coordinate axes. M x =? 5 Nm 1 Nm 2 Nm Nm 5) cont... M y =? 5 Nm 1 Nm 2 Nm 6) cont... M z =? Nm 2 Nm 5 Nm -2 Nm 7) Select the answer that represents the moment of F about the x-axis. Use a Cartesian vector analysis and define the unit vector for the axis as u = i. M x =? 6 Nm -6 Nm 4 Nm -4 Nm 8) Select the possible position vectors r that can be used in the formulation M a = u(r x F). r from A to B, B to D r from A to D, B to C r from A to C, A to B 9) The moment of F is to be determined about the OA axis. Select the terms that are in the second column of the 1

13 determinate used to evaluate M OA = u(r x F).77, 1,.77,, 1 j,, 1 Moment of a Couple 1) Determine the couple moment by computing the moment of the couple's two forces about point A. M c =? 1 Nm (clockwise) 2 Nm (clockwise) 25 Nm (clockwise) 3 Nm (clockwise) 3) Compute the magnitude F for each of the two forces acting on the outer circle, so that they create the same couple moment as that created by the 1-N forces acting on the inner circle. F =? 75 Nm (clockwise) 25 Nm (clockwise) 1 Nm (clockwise) 175 Nm (clockwise) 5) Calculate the moment of the couple using M c = r AC F and M c = r AB F. M c =? 1 Nm (clockwise) 2 Nm (clockwise) 25 Nm (clockwise) 3 Nm (clockwise) 2) Determine the couple moment by computing the moment of the couple's two forces about point A. M c =? 1 N 25 N 4 N 8 N 4) Calculate the magnitude of the resultant couple moment of the two couples acting on the bar. M c =? 11 3j Nm 4j Nm 5j Nm 1j Nm 6) Select the answer that represents the magnitude of the resultant couple moment acting on the block.

14 acting at A. 15 down 5 Nm 3 Nm 4 Nm 7 Nm Resultants of a Force and Couple System 1) Select the answer that represents the equivalent force and couple-moment system acting at A. (-1i)N, (1k)Nm (1i)N, (-1k)Nm (1j)N, (1k)Nm 3) The loading is to be replaced by an equivalent force and couple-moment system acting at A. For questions 3 to 5, calculate the resultants. F Rx =? 5) cont...from question 3, M RA =? 5 Nm Clockwise 25 Nm Clockwise 35 Nm Clockwise 15 Nm Clockwise 6) The loading is to be replaced by an equivalent force and couple-moment system acting at A. For questions 6 to 8, calculate the resultants. F Rx =? 1 N (right), 2 Nm clockwise 1 N (right), 1 N (left), 1 N (right), 2 Nm 2) Select the answer that represents the equivalent force and couple-moment system N 75 N (right) 1 N (right) 1 N (left) 4) cont...from question 3, F Ry =? 1 down 1 up 12 N 8 N (right) 1 N (right) 6 N (right) 7) cont...from question 6, F Ry =? 5 N (up) 6 N (up) 8 N (up) 1 (up)

15 8) cont...from question 6, M RA =? 2 Nm 12 Nm 14 Nm 2 Nm 9) The 1-N force is to be replaced by an equivalent force and couple-moment system at O. For questions 9 to 11, calculate these resultant components. Assume positive components act along the positive x, y, z axes. F Rz =? N 1 N -1 N 1) cont...from question 9, M Rx =? 1 Nm -1 Nm -2 Nm 2 Nm 11) cont...from question 9, M Ry =? Nm -5 Nm -1 Nm -2 Nm 12) cont...from question 9, M Rz =? Nm -5 Nm -1 Nm -2 Nm 13) The 1-N force is to be replaced by an equivalent force and couple-moment system at O. For questions 13 to 16, calculate these resultant components. Assume positive components act along the positive x, y, z axes. F Rz =? N 1 N -1 N 14) cont...from question 13, M Rx =? 1 Nm -1 Nm -2 Nm 2 Nm 15) cont...from question 13, M Ry =? -2 Nm 2 Nm 3 Nm -3 Nm 16) cont...from question 13, M Rz =? Nm -5 Nm 1 Nm -2 Nm 17) The force system is to be replaced by an equivalent force and couple-moment system at O. For questions 17 to 22, calculate these resultant components. Assume positive components act along the positive x,y,z axes. F Rx =? 13

16 4) cont...from question 2, x =? m N -1-1 m 1 m 2 m 3 m 1 N -2 m 2 N 18) cont...from question 17, F Ry =? -1 N 1 N 2 N N 19) cont...from question 17, F Rz =? N 1 N -1 N 2 N Further Reduction of a Force and Couple System 1) Apply the equation Fr(x) = ΣM O in order to determine the position x of the resultant force on the member. Assume moments are positive in the +z direction, ie. Clockwise x =? 2 m 2) For questions 2, 3 and 4, calculate the x and y components of an equivalent single resultant force and specify its location on the member, measured from point O. Assume positive components act along the positive x and y axes. F rx =? N 1 N -1 N 3 N 3) cont...from question 2, F ry =? 1 N -1 N -3 N 14 5) For questions 5, 6 and 7, calculate the x and y components of an equivalent single resultant force and specify its location on the member, measured from point O. Assume positive components act along the positive x and y axes. F rx =? N 5 m +1 N -1 N 6) cont...from question 5, F ry =? -5 N -1 N -15 N 7) cont...from question 5, x =?

17 1 m 2 m 3 m 4 m 8) For questions 8, 9 and 1, calculate the x and y components of an equivalent single resultant force and specify its location on the member, measured from point O. Assume positive components act along the positive x and y axes. F rx =? -6 N -8 N -15 N 28 N 9) cont...from question 8, F ry =? N -18 N 1) cont...from question 8, x =? -1 m -2 m -4 m -8 m 11) For questions 11 to 13, calculate the equivalent single resultant force and specify its x and y coordinates. F rz =? 1 N (down) 2 N (down) 4 N (down) 12) cont...from question 11, x =?.5 m 1 m 1.5 m 2 m 13) cont...from question 11, y =?.5 m 1 m 1.5 m 2 m 15 14) For questions 14 to 16, calculate the equivalent single resultant force and specify its x and y coordinates. F rz =? 2 N (down) 2 N (up) N 15) cont...from question 14, x =? m 1 m 2 m 4 m 16) cont...from question 14, y =? m 1 m 2 m 4 m Reduction of a Simple

Distributed Loading 1) For the next 2 questions, calculate the resultant force F r of the distributed loading and specify its location d measured from point O. F r =? 25 N 1 N 2 N 4 N 2) cont...d =?

18 Distributed Loading 1) For the next 2 questions, calculate the resultant force F r of the distributed loading and specify its location d measured from point O. F r =? 25 N 1 N 2 N 4 N 2) cont...d =? m 1 m 2 m 4 m 3) For the next 2 questions, calculate the resultant force F r of the distributed loading and specify its location d measured from point O. F r =? 33.3 N 15 N 2 N 3 N 4) cont...d =? m.5 m 1 m 1.5 m 5) For the next three questions, calculate the resultant force F r of the distributed load acting on the surface of the plate and specify its x and y coordinates measured from point O. F r =? 1 N 15 N 6 N 12 N 6) cont...x =? 1 m 1.5 m 2 m 3 m 7) cont...y =?.5 m 1 m 1.5 m.75 m 8) For the next 2 questions, use integration to calculate the resultant force F r of the distributed load and specify its location d measured from point O. Select the answer from the list. F r =? 25 N 5 N 33.3 N 66.7 N 9) cont...d =?.5 m.667 m 16

19 .75 m.8 m 17

Chapter 4 Equilibrium of Rigid Body Equilibrium of a Rigid Body in 2D 1) On a sheet of paper draw the free-body diagram of member AB then list the number of unknowns. Number of unknowns =?

20 Chapter 4 Equilibrium of Rigid Body Equilibrium of a Rigid Body in 2D 1) On a sheet of paper draw the free-body diagram of member AB then list the number of unknowns. Number of unknowns =? ) On a sheet of paper draw the free-body diagram of member AB then list the number of unknowns. Number of unknowns =? ) On a sheet of paper draw the free-body diagram of member AB then list the number of unknowns. Number of unknowns =? 2 3 ` 4 4) On a sheet of paper draw the free-body diagram of member AB then list the number of unknowns. Number of unknowns =? ) On a sheet of paper draw the free-body diagram of member AB then list the number of unknowns. Number of unknowns =?

6) The free-body diagram is shown in the figure. Select the equation that represents the moment equilibrium equation about point A. Assume positive moments are clockwise.

21 6) The free-body diagram is shown in the figure. Select the equation that represents the moment equilibrium equation about point A. Assume positive moments are clockwise. Fcos45(2m) + Fsin45(2m) - 6N(6m) = Fcos45(2m) - Fsin45(2m) - 6N(6m) = -6N(6m) - Fcos45(2m) - Fsin45(2m) = 7) The free-body diagram is shown in the figure. Select the equation that represents the moment equilibrium equation about point A. Assume positive moments are clockwise. 2N(4cos3 o m) + F(2m) = 2N(4cos3 o m) - F(2m) = 2N(4m) - Fcos3<sup>o</sup>m(2 m) = 8) The free-body diagram is shown in the figure. Select the equation that represents the moment equilibrium equation about point A. Assume positive moments are. T(3/5)(.5m) + 1Nm = T(3/5)(.5m) - T(4/5) (2m) - 1Nm = T(3/5)(.5m) + T(4/5) (2m) - 1Nm = -T(3/5)(.5m) + T(4/5)(2m) - 1Nm = Two and Three-Force Members equation of equilibrium that should be applied to determine the resultant force acting at the ends of the two-force member AB. ΣM c = ΣM B = ΣF x = 2) On a sheet of paper draw the free-body diagram of member BC, then specify the single equation of equilibrium that should be applied to determine the resultant force acting at the ends of the two-force member AB. ΣM B = ΣF y = ΣM c = 3) Is member BC a two-force member? 2N(4m) - F(2m) = 1) On a sheet of paper draw the free-body diagram of member BC, then specify the single 19

22 Yes No 4) cont...is it necessary to specify where the 6 Nm couple moment acts on member BC? Yes No 5) Is this a two-force member? Yes No 2

23 Key to the questions Chapter 1 Force Vectors Parallelogram law Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Addition of a system of coplanar forces Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Addition of Cartesian vectors Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Cartesian position and force vectors Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Dot product Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Chapter 2 Equilibrium of a particle Particle equilibrium in two dimensions Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Chapter 3 Force System Resultants Cross product and moment of a force Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Transmissibility of a force, principles of moments Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Moment of a force about a specified axis Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Moment of a couple 21

24 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Resultants of a force & couple system Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q2 Further reduction of a force/couple system Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Reduction of a simple distributed loading Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Chapter 4 Equilibrium of a rigid body Equilibrium of a rigid body in 2D Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Two and three-force members Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q1 Source- 22

Engineering Mechanics: Statics in SI Units, 12e Force Vectors

Engineering Mechanics: Statics in SI Units, 1e orce Vectors 1 Chapter Objectives Parallelogram Law Cartesian vector form Dot product and angle between vectors Chapter Outline 1. Scalars and Vectors. Vector