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

12th Edition

ISBN: 9781259977244

Author: BEER

Publisher: MCG

See similar textbooks

Concept explainers

Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

Videos

Question

Chapter 10.2, Problem 10.99P

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.2, Problem 10.98P

Chapter 10.2, Problem 10.100P

Students have asked these similar questions

CHAPTER 14: Kinetics of a Particle: Conservation of Energy Qu.4 The spring has a stiffness k = 200 N/m and an unstretched length of 0.5 m. If it is attached to the 3- kg smooth collar and the collar is released from rest at A, determine the speed of the collar when it reaches B. Neglect the size of the collar.please show all work step by step

Qu. 2 The 100-kg crate is subjected to the action of two forces. If it is originally at rest, determine the distance it slides in order to attain a speed of 6 m/s. The coefficient of kinetic friction between the crate and the surface is uk = 0.2. i need to show all work step by step problems

(◉ Home - my.uah.edu Homework#5 MasteringEngineering Mastering X + 8 https://session.engineering-mastering.pearson.com/myct/itemView?offset=next&assignmentProblemID=18992146

Chapter 10 Solutions

Vector Mechanics for Engineers: Statics

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

Ch. 10.1 - Prob. 10.11P Ch. 10.1 - Knowing that the line of action of the force Q...Ch. 10.1 - Solve Prob. 10.12 assuming that the force P...Ch. 10.1 - The mechanism shown is acted upon by the force P....Ch. 10.1 - Prob. 10.15P Ch. 10.1 - 10.15 and 10.16 Derive an expression for the...Ch. 10.1 - A uniform rod AB with length l and weight W is...Ch. 10.1 - The pin at C is attached to member BCD and can...Ch. 10.1 - For the linkage shown, determine the couple M...Ch. 10.1 - For the linkage shown, determine the force...Ch. 10.1 - A 4-kN force P is applied as shown to the piston...Ch. 10.1 - A couple M with a magnitude of 100 Nm isapplied as...Ch. 10.1 - Rod AB is attached to a block at A that can...Ch. 10.1 - Solve Prob. 10.23, assuming that the 800-N force...Ch. 10.1 - Prob. 10.25P Ch. 10.1 - Determine the value of corresponding to...Ch. 10.1 - Prob. 10.27P Ch. 10.1 - Determine the value of corresponding to...Ch. 10.1 - Prob. 10.29P Ch. 10.1 - Two rods AC and CE are connected by a pin at Cand...Ch. 10.1 - Solve Prob. 10.30 assuming that force P is movedto...Ch. 10.1 - Two bars AD and DG are connected by a pin at Dand...Ch. 10.1 - Solve Prob. 10.32 assuming that the 900-N...Ch. 10.1 - Two 5-kg bars AB and BC are connected by a pin atB...Ch. 10.1 - A vertical force P with a magnitude of 150 N...Ch. 10.1 - Prob. 10.36P Ch. 10.1 - 10.37 and 10.38 Knowing that the constant of...Ch. 10.1 - Prob. 10.38P Ch. 10.1 - The lever AB is attached to the horizontal shaft...Ch. 10.1 - Solve Prob. 10.39, assuming that P = 350 N, l =250...Ch. 10.1 - Prob. 10.41P Ch. 10.1 - The position of boom ABC is controlled by...Ch. 10.1 - The position of member ABC is controlled by the...Ch. 10.1 - The position of member ABC is controlled by...Ch. 10.1 - The telescoping arm ABC is used to provide...Ch. 10.1 - Solve Prob. 10.45, assuming that the workers...Ch. 10.1 - Denoting the coefficient of static friction...Ch. 10.1 - Knowing that the coefficient of static...Ch. 10.1 - A block with weight W is pulled up a plane forming...Ch. 10.1 - Derive an expression for the mechanical...Ch. 10.1 - Denoting the coefficient of static friction...Ch. 10.1 - Knowing that the coefficient of static...Ch. 10.1 - Using the method of virtual work,...Ch. 10.1 - Using the method of virtual work, determine...Ch. 10.1 - Referring to Prob. 10.43 and using the value...Ch. 10.1 - Prob. 10.56P Ch. 10.1 - Prob. 10.57P Ch. 10.1 - Prob. 10.58P Ch. 10.2 - Using the method of Sec. 10.2C, solve Prob. 10.29....Ch. 10.2 - Using the method of Sec. 10.2C, solve Prob. 10.30....Ch. 10.2 - Using the method of Sec. 10.2C, solve Prob. 10.31....Ch. 10.2 - Using the method of Sec. 10.2C, solve Prob. 10.32....Ch. 10.2 - Using the method of Sec. 10.2C, solve Prob. 10.34....Ch. 10.2 - Prob. 10.64P Ch. 10.2 - Using the method of Sec. 10.2C, solve Prob. 10.37....Ch. 10.2 - Prob. 10.66P Ch. 10.2 - Prob. 10.67P Ch. 10.2 - Show that equilibrium is neutral in Prob. 10.7....Ch. 10.2 - Two uniform rods, each with a mass m, areattached...Ch. 10.2 - Two uniform rods, AB and CD, are attached to gears...Ch. 10.2 - Two uniform rods AB and CD, of the same length...Ch. 10.2 - Two uniform rods, each of mass m and length l, are...Ch. 10.2 - Using the method of Sec. 10.2C, solve Prob....Ch. 10.2 - In Prob. 10.40, determine whether each of...Ch. 10.2 - A load W of magnitude 144 lb is applied to...Ch. 10.2 - Prob. 10.76P Ch. 10.2 - Prob. 10.77P Ch. 10.2 - Prob. 10.78P Ch. 10.2 - A slender rod AB with a weight W is attached to...Ch. 10.2 - A slender rod AB with a weight W is attached totwo...Ch. 10.2 - Prob. 10.81P Ch. 10.2 - A spring AB of constant k is attached to two...Ch. 10.2 - A slender rod AB is attached to two collars A and...Ch. 10.2 - Prob. 10.84P Ch. 10.2 - 10.85 and 10.86 Cart B, which weighs 75 kN, rolls...Ch. 10.2 - 10.85 and 10.86 Cart B, which weighs 75 kN, rolls...Ch. 10.2 - 10.87 and 10.88 Collar A can slide freely on the...Ch. 10.2 - 10.87 and 10.88 Collar A can slide freely on the...Ch. 10.2 - Prob. 10.89P Ch. 10.2 - A vertical bar AD is attached to two springs...Ch. 10.2 - Rod AB is attached to a hinge at A and to two...Ch. 10.2 - Rod AB is attached to a hinge at A and to...Ch. 10.2 - Two bars are attached to a single spring of...Ch. 10.2 - Prob. 10.94P Ch. 10.2 - The horizontal bar BEH is connected to three...Ch. 10.2 - The horizontal bar BEH is connected to three...Ch. 10.2 - Bars AB and BC, each with a length l and of...Ch. 10.2 - Prob. 10.98P Ch. 10.2 - Prob. 10.99P Ch. 10.2 - Prob. 10.100P Ch. 10 - Determine the vertical force P that must be...Ch. 10 - Determine the couple M that must be applied...Ch. 10 - Determine the force P required to maintain...Ch. 10 - Derive an expression for the magnitude of the...Ch. 10 - Derive an expression for the magnitude of the...Ch. 10 - A vertical load W is applied to the linkage at B....Ch. 10 - A force P with a magnitude of 240 N is applied to...Ch. 10 - Two identical rods ABC and DBE are connected bya...Ch. 10 - Solve Prob. 10.108 assuming that the 24-lb load...Ch. 10 - Two uniform rods each with a mass m and length...Ch. 10 - A homogeneous hemisphere with a radius r isplaced...Ch. 10 - A homogeneous hemisphere with a radius r isplaced...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

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

Solution Summary: The author explains how to find the range of values of P for which the equilibrium of the system is stable.

Concept explainers

Videos

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

Want to see the full answer?

Chapter 10 Solutions