Learning Goal:

To be able to solve three$-$dimensional equitionium problems using the equations of equilibrium.

As with two$-$dimensional problems, a free$-$body diagram is the first step in solving threedimensional equilibrium problems. For the free$-$body diagram, it is important to identify the appropriale reaction forces and couple moments that act in three dimensions. At a support, a force arises when translation of the attached member is restricted and a couple moment arises when rotation is prevented.

For a rigid body to be in equilibrium when subjected to a force system, both the resultant force and the resultant couple moment acting on the body must be zero. These two conditions are expressed as

${\displaystyle \underset{?}{\overset{?}{}}}F=0$

${\displaystyle \underset{?}{\overset{?}{}}}{M}_O=0$

where ${\displaystyle \underset{?}{\overset{?}{}}}F$ is the sum of all external forces acting on the body and ${\displaystyle \underset{?}{\overset{?}{}}}{M}_0$ is the sum of the couple moments and the moments of all the forces about any point $0$

These two equations are equivalent to six scalar equilibrium equations that can be used to find up to six unknowns identified in a free$-$body diagram. These equations require that the sum of the external force components in the $x,y,$ and $z$ directions must be zero and the sum of the moment components about the $x,y,$ and $z$ axes are also zero

Part A

The J $-$shaped member shown in the figure$($Figure $1)$ is supported by a cable $DE$ and a single journal bearing with a square shaft at $A.$ Delermine the reaction forces $A_y$ and $A_z$ at support A required to keep the system in equitibrium. The cylinder has a mass $m_B=2\mathrm{.}80kg,$ and $F=5\mathrm{.}10N$ is a vertical force applied to the member at $C.$ The dimensions of the member are $w=1\mathrm{.}50m,l=6\mathrm{.}00m,$ and $h=2\mathrm{.}00m.$

Express your answers numerically in Newtons to three significant figures separated by a comma.

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Part B $2\mathrm{.}80$ kg ; a vertical force $F=5\mathrm{.}10N$ acts at $C,$ and the member's dimensions are $w=1\mathrm{.}50m,l=6\mathrm{.}00m,$ and $h=2\mathrm{.}00m$

Express your answers numerically in Newton$-$meters to three significant figures separated by commas.

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$$

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