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RENSSELAER POLYTECHNIC INSTITUTE TROY, NY MAKEUPEXAM NO. 2 INTRODUCTION TO ENGINEERING ANALYSIS (ENGR-1100) – Spring 13 NAME: Solution Section: ___________ RIN: _______________________________ Wednesday, March 27, 2013 5:00 – 6:50 Please state clearly all assumptions made in order for full credit to be given. Problem Points 1 25 2 25 3 25 4 25 Total 100 Score Problem #1 (25 %) For the following matrices: 3 8 10 0 4 8 2 0 101 A 0 7 0 ; B 1 5 3; C 5; D ; E 8 2 23 1 0 1 3 4 4 5 4 Determine the following expressions. If an expression is not valid, state the reason: a. B b. (D D ) c. I 2 (E E) T C B C T (10) T T T (2) 1 (8) d. Det (A) using the duplicate column method. (5) Note: You need to show all the work to receive full credit. a. 1 0 0 [( + )( ∗ )] = 5 + 5 ∗ [1 5 3] ∗ 5 3 4 4 1 = 10 ∗ [1 ∗ 0 + 5 ∗ 5 + 3 ∗ 4] 7 1 ∗ 32 = 10 ∗ 32 7 ∗ 32 = [32 320 224] b. Invalid, dimensions are not similar (2) (2) (2) (2) (2) (2) c. ∗( ∗ ) 1 0 ∗ 0 1 105 → det 68 ∗( ∗ ) 0.02 = −0.0162 = 1 0 = ∗ 0 1 4 8 8 2 4 5 ∗ 8 4 5 105 68 68 84 68 = 105 ∗ 84 − 68 = 4196 84 1 0 84 −68 = ∗ 0 1 −68 105 −0.0162 0.025 8 2 4 (1) (2) (2) = 84 −68 −68 105 (2) (1) d. 3 8 10 3 8 det( ) = 0 7 0 0 7 1 3 4 1 3 = 3 ∗ 7 ∗ 4 + 8 ∗ 0 ∗ 1 + 10 ∗ 0 ∗ 3 − 1 ∗ 7 ∗ 10 − 3 ∗ 0 ∗ 3 − 4 ∗ 0 ∗ 8 = 14 (2) (2) (1) Problem #2 (25%) A 75- kg container is suspended from a frictionless ring A, to which cables AC and AE are attached and maintained at equilibrium. A force P is applied to end F of a third cable that passes over a frictionless pulley at B and through ring A and then is attached to a support at D. (hint: cable FBAD is a continuous cable passing through a frictionless pulley and a ring and two other cables: CA and EA are tied to the ring) a. Draw the complete and separate free body diagram of the identified particle to determine magnitude of P. (5) b. Write the equilibrium equations for the FBD. (12) c. Determine the tension (force) in each of the three cables and obtain P. (8) Problem #3 (25 %) The triangular plate ABC is supported by ball-and-socket joints at B and D and is held in the position shown by cables AE and CF. If the force exerted by cable CF at C is 33 N, determine the moment of that force about the line joining Points D and B. (All vector answers should be given in the Cartesian coordinates shown) First note: TCF TCF CF CF CF (0.6) 2 (0.9) 2 (0.2) 2 1.1 m 33 N (0.6i 0.9 j 0.2k ) 1.1 3[(6 N)i (9 N) j (2 N)k ] Then TCF Also, DB (1.2) 2 (0.35) 2 (0) 2 Then Now 1.25 m DB DB DB 1 (1.2i 0.35 j) 1.25 1 (24i 7 j) 25 M DB DB (rC/D TCF ) where Then rC/D (0.2 m) j (0.4 m)k M DB 24 7 0 1 (3) 0 0.2 0.4 25 6 9 2 or M DB 9.50 N m 3 (9.6 16.8 86.4) 25 Problem #4 (25%) Beam AD carries a 40-lb load and 300-lb-ft torque as shown. The beam is held by a fixed support at D and by the cable BE that is attached to a counter weight W = 100 lb. (1) Draw complete free-body-diagram for the beam AD. (5) (2) Write equilibrium equations for beam AD with all known and unknown forces and torques. (10) (3) Determine the magnitude and rotation direction of the unknown torque. (5) (4) Determine reactions at D in Cartesian vector form. (5) Solution: (1) (2) ∑ M = 0, ⇒ −M + 300 + ( 40 cos30) 4 = 0 ∑ F = 0, ⇒ D − 40sin30 = 0 (2) ∑ F = 0, ⇒ D − 40 cos30 = 0 (3) D x x y y (3) From Eq.(1), (1) M D = 300 +138.56 = 438.56 lb ⋅ ft clockwise (5) from Eqs (2) and (3): Dx = 40sin30 = 20 lb Dy = 40 cos30 = 34.64 lb D = 20i + 34.64 j lb