Problem $1$:

$*3-28.$ If the cords suspend the two buckets in the equilibrium position, determine the weight of bucket $B.$ Bucket A has a weight of $60$ lb $.$

To solve this problem, do it this way.

Need two FBDs$.$

First is at F$.$ Show all loads properly located. Put a $xy$ coordinate system and use $i_{?}$ for unit vector in $x$ direction and $j_{?}$ for unit vector in $y$ direction. In the FBD$,$ declare three scalar magnitudes for loads:

$F_E($unknown$)$

$F_C($unknown$)$

$F_A=W_A=60lbf$

And declare three unit vectors that go with them:

One from F to E: $u_{{?}_{?}}(E)$

One from F to C: $u_{{?}_{?}}(C)$

One from F to A: $u_{{?}_{?}}(A)$

Report what the three unit vectors are equal to$.$

Create three vector forces $($each scalar times unit vector$)$ and show equation for resultant so that $F_R=0_{{?}_{?}}.$

Dot the equation with $i_{?}$ to get a scalar equation for $x$ direction. Dot the equation with $j_{?}$ to get a scalar equation for $y$ direction.

Solve the scalar equations for the two unknowns $F_E,F_C$

Alternatively, you can use matrix representation for above.

Repeat steps $(1)$ through $(5)$ for FBD at C$.$ Declare the loads as $F_F,F_D,F_B=W_B$ and unit vectors as $u_{{?}_{?}}(F),u_{{?}_{?}}(D),u_{{?}_{?}}(B).$ With respect to previous FBD$,$ you should state $F_F=F_C$ and $u_{{?}_{?}}(F)=-u_{{?}_{?}}(C)$ from Newton's $3^{rd}$ law $($equal in magnitude, opposite in direction$).$ Unknown scalar loads are $F_D,W_B.$ Solve for them.