Determine the tension developed in each wire used to support the 50-kg chandelier.

#### Solution:

Show me the final answers↓

We will first draw a free body diagram around ring D.

We can now write our equations of equilibrium.

T_{DC}\text{cos}\,(30^0)\,-\,T_{DB}\text{cos}\,(45^0)\,=\,0 (eq.1)

+\uparrow \sum \text{F}_\text{y}\,=\,0

T_{DC}\text{sin}\,(30^0)\,+\,T_{DB}\text{sin}\,(45^0)\,-\,490.5\,=\,0 (eq.2)

Now we can solve for T_{DC} and T_{DB} by first isolating for T_{DC} in eq.1.

(Simplify)

T_{DC}\,=\,0.816T_{DB} (eq.3)

Substitute this value into eq.2 to figure out T_{DC}.

(Solve for T_{DB})

T_{DB}\,=\,439.9 N

Substitute this value back into eq.3 to figure out T_{DC}.

T_{DC}\,=\,358.9 N

Let us now switch our attention to ring B by drawing a free body diagram.

**(Remember we found T_{DB}\,=\,439.9 N)**

Again, we will write our equations of equilibrium, however, this time, we will write our equation of equilibrium for y-axis forces first. Why? It will give us a direct answer to T_{BA}.

(Solve for T_{BA})

T_{BA}\,=\,622.1 N

Now, we can write our equation of equilibrium for the x-axis forces.

(Remember we just found T_{BA}\,=\,622.1 N)

Solve for T_{BC}

T_{BC}\,=\,227.7 N

#### Final Answers:

T_{DC}\,=\,358.9 N

T_{BA}\,=\,622.1 N

T_{BC}\,=\,227.7 N

What if we don’t know the mass? How would we calculate the maximum mass that can be hung?