statics 3

statics 3 - Problem 3.1 The gure shows the external forces...

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Problem 3.1 The fgure shows the external Forces act- ing on an object in equilibrium. The Forces F 1 =32 N and F 3 =50 N. Determine F 2 and the angle α . y x 30 ° 12 ° α F 1 F 3 F 2 Solution: Write the Forces in component Form. F 1 = 32 sin 30 i + 32 cos 30 j F 1 =16 i +27 . 7 j N F 2 = 50 cos 12 i 50 sin 12 j F 2 = 48 . 9 i 10 . 4 j (N) F 2 = F 2 cos α i F 2 sin α j Sum components in x and y directions ( X F x 48 . 9+ F 2 cos α =0 X F y =27 . 7 10 . 4 F 2 sin α Solving, we get F z =37 . 2 N α . 73 y x F 1 F 2 F 3 30 ° 12 ° Problem 3.2 The Force F 1 = 100 N and the angle α = 60 . The weight oF the ring is negligible. Determine the Forces F 2 and F 3 . x y 30 ° F 3 F 2 F 1 Solution: Write the Forces in component Form. F 1 = F 1 i +0 j F 2 = F 2 cos 30 i + F 2 sin 30 j F 3 = F 3 cos α i F 3 sin α j We know F , thus F x and F y . Writing the equilibrium equations, we have ( X F x = F 1 F 2 cos 30 F 3 cos α X F y = F 2 sin 30 F 3 sin α F 1 = 100 N =60 Solving, we get F 2 =86 . 6 N ,F 3 N x y 30 ° F 3 F 2 F 1
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Problem 3.3 Consider the forces shown in Prob- lem 3.2. Suppose that F 2 = 100 N and you want to choose the angle α so that the magnitude of F 3 is a min- imum. What is the resulting magnitude of F 3 ? Strategy: Draw a vector diagram of the sum of the three forces. Solution: | F 2 | = 100 N, F 1 is horizontal, and F =0 . From the diagram, α =90 and | F 3 | =50 N F 1 F 3 min F 2 α 60 ° 30 ° x Problem 3.4 The beam is in equilibrium. If A x = 77 kN, B = 400 kN, and the beam’s weight is negligible, what are the forces A y and C ? A y B 30 ° C A x 2 m 4 m Solution: + X F x = A x C sin 30 + X F y = A y B + C cos 30 A x =77 kN ,B = 400 kN Solving, we get A y = 267 kN C = 154 kN A y B 30 ° C A x 2 m 4 m Problem 3.5 Suppose that the mass of the beam shown in Problem 3.4 is 20 kg and it is in equilibrium. The force A y points upward. If A y = 258 kN and B = 240 kN, what are the forces A x and C ? Solution: + X F x =0= A x C sin 30 + X F y A y B (20)(9 . 81) + C cos 30 A y = 258 kN = 240 kN Solving, we get A x = 103 kN C = 206 kN A x A y B C 30 ° (20 kg) (9.81)m/s 2
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Problem 3.6 A zoologist estimates that the jaw of a predator, Martes , is subjected to a force P as large as 800 N. What forces T and M must be exerted by the temporalis and masseter muscles to support this value of P ? T 22 ° P M 36 ° Solution: Resolve the forces into scalar components, and solve the equilibrium equations ... Express the forces in terms of horizontal and vertical unit vectors: T = | T | ( i cos 22 + j sin 22 )= | T | (0 . 927 i +0 . 375 j ) P = 800( i cos 270 + j sin 270 )=0 i 800 j M = | M | ( i cos 144 + j sin 144 | M | ( 0 . 809 i . 588 j ) Apply the equilibrium conditions, X F =0= T + M + P =0 Collect like terms: X F x =(0 . 927 | T |− 0 . 809 | M | ) i X F y . 375 | T 0 . 588 | M 800) j Solve the Frst equation, | T | = ( 0 . 809 0 . 927 ) | M | . 873 | M | Substitute this value into the second equation, reduce algebraically, and solve: | M | = 874 N , | T | = 763 . 3 N T 22 ° P M 36 °
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Problem 3.7 The two springs are identical, with un- stretched lengths 250 mm and spring constants k = 1200 N/m.
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statics 3 - Problem 3.1 The gure shows the external forces...

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