Explanation The sketch of plate supported by three cables is shown in figure 1. Free body diagram at A is shown in figure 2. Here, T A B is the magnitude of tension in cable A B , T A C is the magnitude of tension in cable A C , T A D is the magnitude of tension in the cable A D , p is the magnitude of force P exerted by the support on point A which is equal to the weight of the rectangular plate. The tension in the cable A C is 60 N . Let T A B , T A C , T A D and P are the tension vector in cable A B , A C , A D and force exerted at A in the upward direction due to the weight of the plate. Let i , j and k are the unit vectors along the of x , y and z direction. Write the equation of vector distance A B . A B → = ( x 2 − x 1 ) i + ( y 2 − y 1 ) j + ( z 2 − z 1 ) k (I) Here, A B → is the vector distance of the cable A B , the variables x 1 , y 1 and z 1 are the coordinates of point A and x 2 , y 2 and z 2 are the coordinates of point B . Write the vector distance of the cable A C . A C → = ( x 3 − x 1 ) i + ( y 3 − y 1 ) j + ( z 3 − z 1 ) k (II) Here, A C → is the vector distance of the cable A C , the variables x 1 , y 1 and z 1 are the coordinates of point A and x 3 , y 3 and z 3 are the coordinates of point C . Write the vector distance of the cable A D . A D → = ( x 4 − x 1 ) i + ( y 4 − y 1 ) j + ( z 4 − z 1 ) k (III) Here, A D → is the vector distance of the cable A D , the variables x 1 , y 1 and z 1 are the coordinates of point A and x 4 , y 4 and z 4 are the coordinates of point D . Write the equation of tension in the cable A B . T A B = λ A B T A B (IV) Here, T A B is the tension in the cable A B , T A B is the magnitude of the tension in the cable A B and λ A B is the unit vector in the direction of A B . Write the equation of λ A B . λ A B = A B → A B (V) Write the equation of tension in the cable A C . T A C = λ A C T A C (VI) Here, T A C is the tension in the cable A C , T A C is the magnitude of the tension in the cable A C and λ A C is the unit vector in the direction of A C . Write the equation of λ A C . λ A C = A C → A C (VII) Write the equation of tension in the cable A D . T A D = λ A D T A D (VIII) Here, T A D is the tension in the cable A D , T A D is the magnitude of the tension in the cable A D and λ A D is the unit vector in the direction of A D . Write the equation of λ A D . λ A D = A D → A D (IX) Write the equation of force exerting at point A along y direction. P = P j (X) Here, P is the force exerted at A in the upward direction due to the weight of the plate. Write the equilibrium condition for the forces at A . ∑ F = 0 Here, F is the force The above equation implies that at equilibrium, total force acting on the cable at A is zero. Refer figure 2 and write the equation of equilibrium of forces at A. T A B + T A C + T A D + P = 0 Conclusion: Substitute 0 mm for x 1 , 480 mm for y 1 , 0 mm for z 1 , 320 mm for x 2 , 0 mm for y 2 and 360 mm for z 2 in equation (I) to get A B → . A B → = − ( 320 mm ) i − ( 480 mm ) j + ( 360 mm ) k Calculate the magnitude of A B → . A B = ( − 320 mm ) 2 + ( − 480 mm ) 2 + ( 360 mm ) 2 = 680 mm Substitute 0 mm for x 1 , 480 mm for y 1 , 0 mm for z 1 , 450 mm for x 3 , 0 mm for y 3 and 360 mm for z 3 in equation (II) to get A C → . A C → = ( 450 mm ) i − ( 480 mm ) j + ( 360 mm ) k Calculate the magnitude of A C → . A C = ( 450 mm ) 2 + ( − 480 mm ) 2 + ( 360 mm ) 2 = 750 mm Substitute 0 mm for x 1 , 480 mm for y 1 , 0 mm for z 1 , 250 mm for x 4 , 0 mm for y 4 and − 360 mm for z 4 in equation (III) to get A D → . A D → = ( 250 mm ) i − ( 480 mm ) j − ( 360 mm ) k Calculate the magnitude of A D → . A D = ( 250 mm ) 2 − ( 480 mm ) 2 − ( 360 mm ) 2 = 650 mm Substitute A B → = − ( 320 mm ) i − ( 480 mm ) j + ( 360 mm ) k for A B → and 680 mm for A B in equation (V) to get λ A B . λ A B = − ( 320 mm ) i − ( 480 mm ) j + ( 360 mm ) k 680 mm = − 8 17 i − 12 17 j + 19 17 k Substitute − 8 17 i − 12 17 j + 19 17 k for λ A B in equation (IV) to get T A B

12th Edition

ISBN: 9781264095032

Author: BEER

Publisher: MCGRAW-HILL HIGHER EDUCATION

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Chapter 2.5, Problem 2.109P

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The weight of the plate as shown in figure P2.109 in the text book , knowing that the tension in cable AC is 60 N.

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The weight of the plate as shown in figure P2 .109 in the text book , knowing that the tension in cable A C is 60 N .

Solution Summary: The author explains the tension in the cable AC is 60n and the weight of the plate supported by three cables.

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The weight of the plate as shown in figure P2.109 in the text book , knowing that the tension in cable AC is 60 N.

Chapter 2 Solutions