Analyzing a Machine What forces are exerted on the bolt at E in Fig. 6.40 as a result of the 150-N forces on the pliers?

Question

Analyzing a Machine

What forces are exerted on the bolt at E in Fig. 6.40 as a result of the 150-N forces on the pliers?

Strategy
A pair of pliers is a simple example of a machine, a structure designed to move and exert forces. The interconnections of the members are designed to create a mechanical advantage, subjecting an object to forces greater than the forces exerted by the user.

In this case there is no information to be gained from the free-body diagram of the entire structure. We must determine the forces exerted on the bolt by drawing free-body diagrams of the members.

Accepted Answer

We "disassemble" the pliers in Fig. a to obtain the free-body diagrams of the members, labeled (1), (2), and (3). The force R on free-body diagrams (1) and (3) is exerted by the two-force member AB. The angle α = arctan (30/70) = 23.2°. Our objective is to determine the force E exerted by the bolt.

The free-body diagram of member (3) has only three unknown forces and the 150-N load, so we can determine R, [latex] D_x [/latex], and [latex] D_y [/latex], from this free-body diagram alone. The equilibrium equations are

[latex] \sum{F_{x}} = D_{x} + R \cos α = 0 [/latex],

[latex] \sum{F_{y}} = D_{y} - R \sin α + 150 = 0 [/latex],

[latex] \sum{M_{(point B)}} = 30D_{y} - (100)(150) = 0 [/latex],

Solving these equations, we obtain [latex] D_{x} = - 1517 N, D_{y} = 500 N[/latex], and R = 1650 N. Knowing [latex] D_{x}[/latex] we can determine E from the free-body diagram of member (2) by summing moments about C,

[latex] \sum{M_{(point C)}} = -30E - 30D_{x} = 0 [/latex].

The force exerted on the bolt by the pliers is [latex] E = -D_{x} = 1517 N[/latex]. The mechanical advantage of the pliers is (1517 N)/( 150 N) = 10.1.

Discussion
Notice that we did not need to use the free-body diagram of member (1) to determine E. When this happens, you can use the "leftover" free-body diagram to check your work. Using our results for R and E, we can confirm that the sum of the moments about point C of member (1 ) is zero:

[latex] \sum{M_{(point C)}} = (130)(150) - 100R \sin α + 30 E = (130)(150) - (100) (1650) \sin 23.2° + (30)(1517) = 0.[/latex]

Question 6.10: Analyzing a Machine What forces are exerted on the bolt at E...

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