VECTOR MECHANICS FOR ENGINEERS: STATICS
VECTOR MECHANICS FOR ENGINEERS: STATICS
12th Edition
ISBN: 9781260912814
Author: BEER
Publisher: MCG
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Chapter 10.1, Problem 10.21P

A 4-kN force P is applied as shown to the piston of the engine system. Knowing that AB = 50 mm and BC = 200 mm, determine the couple M required to maintain the equilibrium of the system when (a) θ = 30°, (b) θ = 150°.

Chapter 10.1, Problem 10.21P, A 4-kN force P is applied as shown to the piston ofthe engine system. Knowing that AB = 50 mm andBC

Fig. P10.21 and P10.22

(a)

Expert Solution
Check Mark
To determine

Find the couple M required to maintain the equilibrium.

Answer to Problem 10.21P

The magnitude of the couple M is 121.8Nm(Counterclockwise)_.

Explanation of Solution

Given information:

The magnitude of the force P is 4 kN.

The distance between the point A and B is 50 mm.

The distance between the point B and C is 200 mm.

The value of the angle θ=30°.

Calculation:

Show the free-body diagram of the engine system as in Figure 1.

VECTOR MECHANICS FOR ENGINEERS: STATICS, Chapter 10.1, Problem 10.21P

Consider the geometry of the Figure 1.

Use the Law of sines;

ABsinϕ=BCsinθsinϕ=ABBCsinθ (1)

Differentiate the equation;

cosϕδϕ=ABBCcosθδθδϕ=ABBCcosθcosϕδθ

Find the horizontal displacement (xC) at point C using the relation.

xC=ABcosθ+BCcosϕ

Differentiate the equation;

δxC=ABsinθδθBCsinϕδϕ

Substitute ABBCsinθ for sinϕ and ABBCcosθcosϕδθ for δϕ.

δxC=ABsinθδθBC(ABBCsinθ)(ABBCcosθcosϕδθ)=ABsinθδθ(ABsinθ)(ABBCcosθcosϕδθ)

Use the principle of virtual work;

δU=0PδxCMδθ=0

Substitute [ABsinθδθ(ABsinθ)(ABBCcosθcosϕδθ)] for δxC.

P[ABsinθδθ(ABsinθ)(ABBCcosθcosϕδθ)]Mδθ=0P[ABsinθ+(ABsinθ)(ABBCcosθcosϕ)]M=0 (2)

Substitute 50 mm for AB, 200 mm for BC, and 30° for θ in Equation (1).

sinϕ=50200×sin30°ϕ=7.181°

Substitute 4 kN for P, 50 mm for AB, 200 mm for BC, 7.181° for ϕ, and 30° for θ in Equation (2).

4[50sin30°+(50sin30°)(50200cos30°cos7.181°)]M=04[25+25(0.21822)]M=0M=121.8kNmm×1,000N1kN×1m1,000mmM=121.8Nm(Counterclockwise)

Therefore, the magnitude of the couple M is 121.8Nm(Counterclockwise)_.

(b)

Expert Solution
Check Mark
To determine

Find the couple M required to maintain the equilibrium.

Answer to Problem 10.21P

The magnitude of the couple M is 78.2Nm(Counterclockwise)_.

Explanation of Solution

Given information:

The magnitude of the force P is 4 kN.

The distance between the point A and B is 50 mm.

The distance between the point B and C is 200 mm.

The value of the angle θ=150°.

Calculation:

Refer part (a) for calculation;

Substitute 50 mm for AB, 200 mm for BC, and 150° for θ in Equation (1).

sinϕ=50200×sin150°ϕ=7.181°

Substitute 4 kN for P, 50 mm for AB, 200 mm for BC, 7.181° for ϕ, and 150° for θ in Equation (2).

4[50sin150°+(50sin150°)(50200cos150°cos7.181°)]M=04[2525(0.21822)]M=0M=78.2kNmm×1,000N1kN×1m1,000mmM=78.2Nm(Counterclockwise)

Therefore, the magnitude of the couple M is 78.2Nm(Counterclockwise)_.

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Chapter 10 Solutions

VECTOR MECHANICS FOR ENGINEERS: STATICS

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