Explanation Given information: The system is in equilibrium, when θ 1 = θ 2 = 0 . Calculation: Show the free-body diagram of the arrangement as in Figure 1. Find the horizontal distance ( x B ) using the relation. x B = l sin θ 1 Find the horizontal distance ( x C ) using the relation. x C = l sin θ 2 Find the vertical distance ( y B ) using the relation. y B = − l cos θ 1 Find the vertical distance ( y C ) using the relation. y C = l cos θ 2 When the values are small, sin θ 1 ≈ θ 1 ; sin θ 2 ≈ θ 2 cos θ 1 ≈ 1 − θ 1 2 2 ; cos θ 2 ≈ 1 − θ 2 2 2 Find the potential energy ( V ) using the relation. V = V g + V s = 2 P y B + P y C + 1 2 k ( x C − x B ) 2 Here, the spring constant is k . Substitute − l cos θ 1 for y B , l cos θ 2 for y C , l sin θ 2 for x C , and l sin θ 1 for x B . V = 2 P ( − l cos θ 1 ) + P ( l cos θ 2 ) + 1 2 k ( l sin θ 2 − l sin θ 1 ) 2 = − 2 P l cos θ 1 + P l cos θ 2 + k l 2 2 ( sin θ 2 − sin θ 1 ) 2 Substitute θ 1 for sin θ 1 , θ 2 for sin θ 2 , ( 1 − θ 1 2 2 ) for cos θ 1 , and ( 1 − θ 2 2 2 ) for cos θ 2 . V = − 2 P l ( 1 − θ 1 2 2 ) + P l ( 1 − θ 2 2 2 ) + k l 2 2 ( θ 2 − θ 1 ) 2 (1) Differentiate the Equation (1) with respect to θ 1 . ∂ V ∂ θ 1 = − 2 P l ( − 2 θ 1 2 ) + P l ( 0 ) − 2 k l 2 2 ( θ 2 − θ 1 ) = 2 P l θ 1 − k l 2 ( θ 2 − θ 1 ) (2) Differentiate the Equation (2). ∂ 2 V ∂ θ 1 2 = 2 P l + k l 2 Differentiate the equation (2) with θ 2 to find the derivative of ∂ 2 V ∂ θ 1 ∂ θ 2

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ISBN: 8220106797044

Author: BEER

Publisher: YUZU

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Chapter 10.2, Problem 10.99P

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Find the range of values of P for which the equilibrium of the system is stable.

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Chapter 10.2, Problem 10.98P

Chapter 10.2, Problem 10.100P

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Chapter 10 Solutions

EBK VECTOR MECHANICS FOR ENGINEERS: STA

Ch. 10.1 - Determine the vertical force P that must be...Ch. 10.1 - Determine the horizontal force P that must be...Ch. 10.1 - Prob. 10.3P Ch. 10.1 - 10.3 and 10.4 Determine the couple M that must be...Ch. 10.1 - A spring of constant 15 kN/m connects points C and...Ch. 10.1 - A spring of constant 15 kN/m connects points C and...Ch. 10.1 - The two-bar linkage shown is supported by a pin...Ch. 10.1 - Determine the weight W that balances the 10-lb...Ch. 10.1 - Prob. 10.9P Ch. 10.1 - Prob. 10.10P

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Find the range of values of P for which the equilibrium of the system is stable.

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Chapter 10 Solutions