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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition CE 102 Statics Chapter 4 Equilibrium of Rigid Bodies

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Contents Introduction Free-Body Diagram Reactions at Supports and Connections for a Two-Dimensional Structure Equilibrium of a Rigid Body in Two Dimensions Statically Indeterminate Reactions Sample Problem 4.1 Sample Problem 4.2 Sample Problem 4.3 Equilibrium of a Two-Force Body Equilibrium of a Three-Force Body Sample Problem 4.4 Equilibrium of a Rigid Body in Three Dimensions Reactions at Supports and Connections for a Three-Dimensional Structure Sample Problem 4.5

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Introduction The necessary and sufficient condition for the static equilibrium of a body are that the resultant force and couple from all external forces form a system equivalent to zero, Resolving each force and moment into its rectangular components leads to 6 scalar equations which also express the conditions for static equilibrium, For a rigid body in static equilibrium, the external forces and moments are balanced and will impart no translational or rotational motion to the body.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Free-Body Diagram First step in the static equilibrium analysis of a rigid body is identification of all forces acting on the body with a free-body diagram. Select the extent of the free-body and detach it from the ground and all other bodies. Include the dimensions necessary to compute the moments of the forces. Indicate point of application and assumed direction of unknown applied forces. These usually consist of reactions through which the ground and other bodies oppose the possible motion of the rigid body. Indicate point of application, magnitude, and direction of external forces, including the rigid body weight.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Reactions at Supports and Connections for a Two- Dimensional Structure Reactions equivalent to a force with known line of action.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Reactions at Supports and Connections for a Two- Dimensional Structure Reactions equivalent to a force of unknown direction and magnitude. Reactions equivalent to a force of unknown direction and magnitude and a couple.of unknown magnitude

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Equilibrium of a Rigid Body in Two Dimensions For all forces and moments acting on a two- dimensional structure, Equations of equilibrium become where A is any point in the plane of the structure. The 3 equations can be solved for no more than 3 unknowns. The 3 equations can not be augmented with additional equations, but they can be replaced

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Statically Indeterminate Reactions More unknowns than equations Fewer unknowns than equations, partially constrained Equal number unknowns and equations but improperly constrained

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.1 A fixed crane has a mass of 1000 kg and is used to lift a 2400 kg crate. It is held in place by a pin at A and a rocker at B. The center of gravity of the crane is located at G. Determine the components of the reactions at A and B. SOLUTION: Create a free-body diagram for the crane. Determine B by solving the equation for the sum of the moments of all forces about A. Note there will be no contribution from the unknown reactions at A. Determine the reactions at A by solving the equations for the sum of all horizontal force components and all vertical force components. Check the values obtained for the reactions by verifying that the sum of the moments about B of all forces is zero.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.1 Create the free-body diagram. Check the values obtained. Determine B by solving the equation for the sum of the moments of all forces about A. Determine the reactions at A by solving the equations for the sum of all horizontal forces and all vertical forces.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.2 A loading car is at rest on an inclined track. The gross weight of the car and its load is 5500 lb, and it is applied at at G. The cart is held in position by the cable. Determine the tension in the cable and the reaction at each pair of wheels. SOLUTION: Create a free-body diagram for the car with the coordinate system aligned with the track. Determine the reactions at the wheels by solving equations for the sum of moments about points above each axle. Determine the cable tension by solving the equation for the sum of force components parallel to the track. Check the values obtained by verifying that the sum of force components perpendicular to the track are zero.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.2 Create a free-body diagram Determine the reactions at the wheels. Determine the cable tension.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.3 The frame supports part of the roof of a small building. The tension in the cable is 150 kN. Determine the reaction at the fixed end E. SOLUTION: Create a free-body diagram for the frame and cable. Solve 3 equilibrium equations for the reaction force components and couple at E.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.3 Create a free-body diagram for the frame and cable. Solve 3 equilibrium equations for the reaction force components and couple.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Equilibrium of a Two-Force Body Consider a plate subjected to two forces F 1 and F 2 For static equilibrium, the sum of moments about A must be zero. The moment of F 2 must be zero. It follows that the line of action of F 2 must pass through A. Similarly, the line of action of F 1 must pass through B for the sum of moments about B to be zero. Requiring that the sum of forces in any direction be zero leads to the conclusion that F 1 and F 2 must have equal magnitude but opposite sense.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Equilibrium of a Three-Force Body Consider a rigid body subjected to forces acting at only 3 points. Assuming that their lines of action intersect, the moment of F 1 and F 2 about the point of intersection represented by D is zero. Since the rigid body is in equilibrium, the sum of the moments of F 1, F 2, and F 3 about any axis must be zero. It follows that the moment of F 3 about D must be zero as well and that the line of action of F 3 must pass through D. The lines of action of the three forces must be concurrent or parallel.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.4 A man raises a 10 kg joist, of length 4 m, by pulling on a rope. Find the tension in the rope and the reaction at A. SOLUTION: Create a free-body diagram of the joist. Note that the joist is a 3 force body acted upon by the rope, its weight, and the reaction at A. The three forces must be concurrent for static equilibrium. Therefore, the reaction R must pass through the intersection of the lines of action of the weight and rope forces. Determine the direction of the reaction force R. Utilize a force triangle to determine the magnitude of the reaction force R.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.4 Create a free-body diagram of the joist. Determine the direction of the reaction force R tan m 2.313m m tanm 414.1)2545cot( m m cosm445cos 2 1 AE CE BDBFCE CDBD AFAECD ABAF

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.4 Determine the magnitude of the reaction force R.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Equilibrium of a Rigid Body in Three Dimensions Six scalar equations are required to express the conditions for the equilibrium of a rigid body in the general three dimensional case. These equations can be solved for no more than 6 unknowns which generally represent reactions at supports or connections. The scalar equations are conveniently obtained by applying the vector forms of the conditions for equilibrium,

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Reactions at Supports and Connections for a Three- Dimensional Structure

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Reactions at Supports and Connections for a Three- Dimensional Structure

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.5 A sign of uniform density weighs 270 lb and is supported by a ball-and- socket joint at A and by two cables. Determine the tension in each cable and the reaction at A. SOLUTION: Create a free-body diagram for the sign. Apply the conditions for static equilibrium to develop equations for the unknown reactions.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.5 Create a free-body diagram for the sign. Since there are only 5 unknowns, the sign is partially constrain. It is free to rotate about the x axis. It is, however, in equilibrium for the given loading.

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© 2007 The McGraw-Hill Companies, Inc. All rights reserved. Vector Mechanics for Engineers: Statics EighthEdition Sample Problem 4.5 Apply the conditions for static equilibrium to develop equations for the unknown reactions. Solve the 5 equations for the 5 unknowns,

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26 Problem o O D C A B The semicircular rod ABCD is maintained in equilibrium by the small wheel at D and the rollers at B and C. Knowing that = 45 o, determine the reactions at B, C, and D. P

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27 45 o O D C A B 1. Draw a free-body diagram of the body. This diagram shows the body and all the forces acting on it. P Solving Problems on Your Own The semicircular rod ABCD is maintained in equilibrium by the small wheel at D and the rollers at B and C. Knowing that = 45 o, determine the reactions at B, C, and D. Problem 4.6

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28 45 o O D C A B 2. Write equilibrium equations and solve for the unknowns. For two-dimensional structure the three equations might be: F x = 0 F y = 0 M O = 0 where O is an arbitrary point in the plane of the structure or F x = 0 M A = 0 M B = 0 where point B is such that line AB is not parallel to the y axis or M A = 0 M B = 0 M C = 0 where the points A, B, and C do not lie in a straight line. P Solving Problems on Your Own The semicircular rod ABCD is maintained in equilibrium by the small wheel at D and the rollers at B and C. Knowing that = 45 o, determine the reactions at B, C, and D. Problem 4.6

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29 45 o O D C A B Problem 4.6 Solution Draw a free-body diagram of the body. 45 o O D C A B P sin P cos D C R B C/ 2 B/ 2 P P

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30 Problem 4.6 Solution 45 o O D C A B P sin P cos D C R B C/ 2 B/ 2 Write three equilibrium equations and solve for the unknowns. + M O = 0: (P sin ) R _ D (R) = 0 D = P sin + F x = 0: P cos + B/ 2 _ C / 2 = 0 (2) + F y = 0: _ P sin + B/ 2 + C / 2 _ P sin = 0 _ 2P sin + B/ 2 + C / 2 = 0 (3) P

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31 Problem 4.6 Solution 45 o O D C A B P sin P cos D C R B C/ 2 B/ 2 (2) + (3) P(cos _ 2sin ) + 2 B/ 2 = 0 B = (2sin _ cos ) P (4) (2) _ (3) P(cos + 2sin ) _ 2 C/ 2 = 0 C = (2sin + cos ) P (5) P

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32 Problem 4.6 Solution 45 o O D C A B P sin P cos D C R B C/ 2 B/ 2 For = 45 o sin = cos = 1/ 2 EQ. (4) : B = ( _ ) P = P ; B = P 45 o EQ. (5) : C = ( _ ) P = P ; C = P 45 o EQ. (1) : D = P/ 2 D = P / P

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33 4 in 2 in 3 in AB 40 lb 20 lb C D E Problem 4.7 The T-shaped bracket shown is supported by a small wheel at E and pegs at C and D. Neglecting the effect of friction, determine the reactions at C, D, and E when = 30 o.

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34 1. Draw a free-body diagram of the body. This diagram shows the body and all the forces acting on it. 4 in 2 in 3 in AB 40 lb 20 lb C D E Solving Problems on Your Own The T-shaped bracket shown is supported by a small wheel at E and pegs at C and D. Neglecting the effect of friction, determine the reactions at C, D, and E when = 30 o. Problem 4.7

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35 2. Write equilibrium equations and solve for the unknowns. For two-dimensional structure the three equations might be: F x = 0 F y = 0 M O = 0 where O is an arbitrary point in the plane of the structure or F x = 0 M A = 0 M B = 0 where point B is such that line AB is not parallel to the y axis or M A = 0 M B = 0 M C = 0 where the points A, B, and C do not lie in a straight line. 4 in 2 in 3 in AB 40 lb 20 lb C D E Solving Problems on Your Own The T-shaped bracket shown is supported by a small wheel at E and pegs at C and D. Neglecting the effect of friction, determine the reactions at C, D, and E when = 30 o. Problem 4.7

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36 4 in 2 in 3 in AB 40 lb 20 lb C D E 4 in A 40 lb 20 lb C 2 in 3 in B D E E C D 30 o Draw a free-body diagram of the body. Problem 4.7 Solution

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37 4 in A C 40 lb20 lb C 2 in 3 in B D E E D 30 o Write equilibrium equations and solve for the unknowns. F y = 0: E cos 30 o _ 20 _ 40 = 0 E = = lb E = 69.3 lb 60 o 60 lb cos 30 o Problem 4.7 Solution

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38 4 in A C 40 lb 20 lb C 2 in 3 in B D E E D 30 o M D = 0: ( 20 lb)( 4 in) _ ( 40 lb)( 4 in) _ C ( 3 in) + E sin 30 o ( 3 in) = 0 _ 80 _ 3C ( 0.5 )( 3 ) = 0 C = lb C = 7.97 lb Problem 4.7 Solution

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39 Problem 4.7 Solution 4 in A C 40 lb 20 lb C 2 in 3 in B D E E D 30 o + F x = 0: E sin 30 o + C _ D = 0 ( lb )( 0.5 ) lb _ D = 0 D = 42.6 lb

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40 Problem m y 1.5 m z x 5 kN A B C E D 1 m 2 m A 3-m pole is supported by a ball-and-socket joint at A and by the cables CD and CE. Knowing that the line of action of the 5-kN force forms an angle =30 o with the vertical xy plane, determine (a) the tension in cables CD and CE, (b) the reaction at A.

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m y 1.5 m z x 5 kN A B C E D 1 m 2 m 1. Draw a free-body diagram of the body. This diagram shows the body and all the forces acting on it. Solving Problems on Your Own Problem 4.8 A 3-m pole is supported by a ball-and-socket joint at A and by the cables CD and CE. Knowing that the line of action of the 5-kN force forms an angle =30 o with the vertical xy plane, determine (a) the tension in cables CD and CE, (b) the reaction at A.

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42 2. Write equilibrium equations and solve for the unknowns. For three-dimensional body the six scalar equations F x = 0 F y = 0 F z = 0 M x = 0 M y = 0 M z = 0 should be used and solved for six unknowns. These equations can also be written as F = 0 M O = (r x F ) = 0 where F are the forces and r are position vectors. 1.2 m y 1.5 m z x 5 kN A B C E D 1 m 2 m A 3-m pole is supported by a ball- and-socket joint at A and by the cables CD and CE. Knowing that the line of actionof the 5-kN force forms an angle =30 o with the vertical xy plane, determine (a) the tension in cables CD and CE, (b) the reaction at A. Solving Problems on Your Own Problem 4.8

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43 Problem 4.8 Solution Draw a free-body diagram of the body. 1.2 m y 1.5 m z x 5 kN A B C E D 1 m 2 m 1.2 m y 1.5 m z x A B C E D 1 m 2 m 30 o 5 kN T CE T CD A y j A x i A z k

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44 Write equilibrium equations and solve for the unknowns. 1.2 m y 1.5 m z x A B C E D 1 m 2 m 30 o 5 kN T CE T CD A y j A x i A z k 5 unknowns and 6 equations of equilibrium, but equilibrium is maintained, M AC = 0. r B/A = 2 ir C/A = 3 i Load at B, F B = _ ( 5 cos 30 o ) j + ( 5 sin 30 o ) k = _ 4.33 j k CD = _ 3 i+ 1.5 j k CD = m T CD = T CD = ( _ 3 i j k) T CE = T CE = ( _ 3 i j _ 1.2 k) CD T CD T CD CE Problem 4.8 Solution

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m y 1.5 m z x A B C E D 1 m 2 m 30 o 5 kN T CE T CD A y j A x i A z k M A = 0: r C/A x T CD + r C/A x T CE + r B/A x F B = 0 i j k _ i j k _ _ 1.2 i j k _ T CD T CE = 0 Problem 4.8 Solution

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m y 1.5 m z x A B C E D 1 m 2 m 30 o 5 kN T CE T CD A y j A x i A z k Equate coefficients of unit vectors to zero. j: _ _ 5 = 0 _ 3.6 T CD +3.6 T CE _ = 0 (1) T CD T CE k: _ 8.66 = T CD +4.5 T CE = (2) T CD T CE (2) (1): 9T CE _ = 0 ; T CE = 5.90 kN Eq. (1): _ 3.6T CD (5.902) _ = 0 T CD = kN Problem 4.8 Solution

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m y 1.5 m z x A B C E D 1 m 2 m 30 o 5 kN T CE T CD A y j A x i A z k F = 0: A + T CD + T CE + F B = 0 i: A x + ( _ 3) + ( _ 3) = 0 A x = 5.77 kN j: A y + (1.5) + (1.5) _ 4.33 = 0 A y = kN k: A z + (1.2) + ( _ 1.2) = 0 A z = _ kN A = ( 5.77 kN) i + ( kN ) j - ( kN ) k Problem 4.8 Solution

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48 A B C a 20 in 10 in 60 lb Problem 4.9 Rod AC is supported by a pin and bracket at A and rests against a peg at B. Neglecting the effect of friction, determine ( a ) the reactions at A and B when a = 8 in., (b) the distance a for which the reaction at A is horizontal and the corresponding magnitudes of the reactions at A and B.

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49 A B C a 20 in 10 in 60 lb 1. Draw a free-body diagram of the body. This diagram shows the body and all the forces acting on it. Solving Problems on Your Own Problem 4.9 Rod AC is supported by a pin and bracket at A and rests against a peg at B. Neglecting the effect of friction, determine ( a ) the reactions at A and B when a = 8 in., (b) the distance a for which the reaction at A is horizontal and the corresponding magnitudes of the reactions at A and B.

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50 A B C a 20 in 10 in 60 lb 2. For a three-force body, solution can be obtained by constructing a force triangle. The resultants of the three forces must be concurrent or parallel. To solve a problem involving a three-force body with concurrent forces, draw the free-body diagram showing that the three forces pass through the same point. Complete the solution by using a force triangle. Solving Problems on Your Own Problem 4.9 Rod AC is supported by a pin and bracket at A and rests against a peg at B. Neglecting the effect of friction, determine ( a ) the reactions at A and B when a = 8 in., (b) the distance a for which the reaction at A is horizontal and the corresponding magnitudes of the reactions at A and B.

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51 Problem 4.9 Solution Draw a free-body diagram of the body. A B C a 20 in 10 in 60 lb C 10 in 60 lb A 8 in 12 in B B A 2 1 G D F E 10 in (a) a = 8 in tan = = o 1 2

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52 C 10 in 60 lb A 8 in 12 in B B A 2 1 G D F E 10 in 3 - FORCE BODY Reaction at A passes through D where B and 60-lb load intersect AE = EB = (8) = 4 in. EF = BG = 10 _ 4 = 6 in DG = BG = (6) = 3 in. FD = FG _ DG = 8 _ 3 = 5 in. Tan = = ; = o FD AF 5 10 Construct a force triangle. Problem 4.9 Solution

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53 C 10 in 60 lb A 8 in 12 in B B A 2 1 G D F E 10 in FORCE TRIANGLE 60 lb 30 lb = o A B A = B = = lb A = 67.1 lb 26.6 o B = 67.1 lb 26.6 o sin o 30 lb Problem 4.9 Solution

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54 A B C a 20 in 10 in 60 lb Draw a free-body diagram of the body. (b) For A horizontal C 60 lb A B B A 2 1 G F 10 in a = o Problem 4.9 Solution

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55 C 60 lb A B B A 2 1 G F 10 in a = o Problem 4.9 Solution Construct a force triangle. ABF : BF = AF cos BFG : FG = BF sin a = FG = AF cos sin a = (10 in.) cos o sin o a = 4.00 in.

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56 C 60 lb A B B A 2 1 G F 10 in a = o Problem 4.9 Solution FORCE TRIANGLE 60 lb A B = o A = = 120 lb A = lb B = = lb B = lb 26.6 o tan 60 lb sin 60 lb

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57 D C B A aaa 30 o P Problem 4.10 Rod AD supports a vertical load P and is attached to collars B and C, which may slide freely on the rods shown. Knowing that the wire attached at D forms an angle = 30 o with the vertical, determine (a) the tension in the wire, (b) the reactions at B and C.

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58 D C B A aaa 30 o P 1. Draw a free-body diagram of the body. This diagram shows the body and all the forces acting on it. Solving Problems on Your Own Rod AD supports a vertical load P and is attached to collars B and C, which may slide freely on the rods shown. Knowing that the wire attached at D forms an angle = 30 o with the vertical, determine (a) the tension in the wire, (b) the reactions at B and C. Problem 4.10

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59 2. Write equilibrium equations and solve for the unknowns. For two-dimensional structure the three equations might be: F x = 0 F y = 0 M O = 0 where O is an arbitrary point in the plane of the structure or F x = 0 M A = 0 M B = 0 where point B is such that line AB is not parallel to the y axis or M A = 0 M B = 0 M C = 0 where the points A, B, and C do not lie in a straight line. D C B A aaa 30 o P Solving Problems on Your Own Rod AD supports a vertical load P and is attached to collars B and C, which may slide freely on the rods shown. Knowing that the wire attached at D forms an angle = 30 o with the vertical, determine (a) the tension in the wire, (b) the reactions at B and C. Problem 4.10

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60 Problem 4.10 Solution Draw a free-body diagram of the body. D C B A aaa 30 o P D C BA P aaa C B T

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61 Write equilibrium equations and solve for the unknowns. Problem 4.10 Solution D C BA 30 o P aaa C B T F = 0: _ P cos 30 o + T cos 60 o = 0 T = P = P T = 3 P 30 o cos 30 o cos 60 o 3 / 2 1 / 2 + M B = 0: P a _ (C sin 30 o ) a + T cos 30 o (2a) = 0 P a _ ( C ) a + 3 P ( ) 2a = 0 _ C + (1 + 3) P = 0; C = 8 P C = 8 P 30 o

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62 D C BA 30 o P aaa C B T + F = 0: _ B cos 30 o + C cos 30 o _ T sin 30 o = 0 _ B + 8 P _ 3 P ( ) = 0; B = 7 P B = 7 P 30 o Problem 4.10 Solution

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