Determine the tension developed in each wire used to support the 50-kg chandelier.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
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?