The tension in the cable CB must be of a specific magnitude necessary to provide equilibrium at the concurrent point of C. If the force in the boom AC is 4,000# and Q is 800#, determine the load P (vertical) that can be supported. In addition, find the tension developed in cable CB. Solve this prob

Question

The tension in the cable [latex]CB[/latex] must be of a specific magnitude necessary to provide equilibrium at the concurrent point of [latex]C[/latex] . If the force in the boom [latex]AC [/latex] is [latex]4,000\# [/latex] and [latex]Q[/latex] is [latex] 800\# [/latex], determine the load [latex]P[/latex] (vertical) that can be supported. In addition, find the tension developed in cable [latex]CB[/latex] . Solve this problem analytically as well as graphically using a scale of [latex]1– = 800\# [/latex].

Accepted Answer

Analytical Solution:

Force	[latex]F_{x}[/latex]	[latex]F_{y}[/latex]
Q	[latex]-Q \cos 30^{\circ } =-800\# (0.866)=-693\# [/latex]	[latex]-Q \sin 30^{\circ } =-800\# (0.5)=-400\# [/latex]
AC	[latex]-AC \cos 60^{\circ } =-4,000\# (0.5)=-2,000\# [/latex]	[latex]+AC \sin 60^{\circ } =+4,000\# (0.866)=+3,464\# [/latex]
[latex]P[/latex]	[latex]0[/latex]	[latex]-P[/latex]
CB	[latex]+CB \cos 30^{\circ } =+0.866 \space CB[/latex]	[latex]-CB \sin 30^{\circ } =-0.5 \space CB[/latex]

[latex]\underbrace{R_{x} =\sum{F_{x}} =0 \space \space \space\space R_{y} =\sum{F_{y}} =0}_{For \space equilibrium \space to \space exist} [/latex]

[latex] herefore R_{x} =\left[\sum{F_{x}} =0 ight] -693 \# -2,000\# +0.866 CB=0 [/latex]

[latex] CB=\frac{ +693 \# +2,000\#}{0.866 } =3,110\# [/latex]

[latex]R_{y} =\left[\sum{F_{x}} =0 ight]-400\# + 3,464\# -P-0.5(3,110\# ) =0[/latex]

[latex]P =3,464\# -400\# -0.5(3,110\# ) ; \space \space P=1,509\# [/latex]

Graphical Solution:

Begin the graphical solution by drawing the known force [latex]AC = 4,000 \#[/latex] (a [latex]5^{\prime \prime } [/latex] line at [latex]60^{ \circ } [/latex] from the horizontal axis). Then, to the tip of [latex]AC [/latex], draw force [latex]Q[/latex] ( [latex]1^{\prime \prime } [/latex] long and [latex]30^{ \circ } [/latex] from the horizontal). Force [latex]P [/latex] has a known direction (vertical) but an unknown magnitude. Construct a vertical line from the tip of [latex]Q[/latex] to represent the line of action of force [latex]P [/latex]. For equilibrium to be established, the last force [latex]CB [/latex] must close at the origin point [latex]C [/latex]. Draw [latex]CB [/latex] with a [latex]30^{ \circ } [/latex] inclination passing through [latex]C [/latex]. The intersection of the lines [latex]P [/latex] and [latex]CB [/latex] defines the limits of the forces. Magnitudes of [latex]P [/latex] and [latex]CB [/latex] are obtained by scaling the respective force lines.

Question 2.25: The tension in the cable CB must be of a specific magnitude ...

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