Determine the tension developed in each wire


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

Determine the tension developed in each wire

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

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

Determine the tension developed in each wire

We can now write our equations of equilibrium.

\rightarrow ^+\sum \text{F}_\text{x}\,=\,0

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.

T_{DC}\,=\,\dfrac{T_{DB}\text{cos}\,(45^0)}{\text{cos}\,(30^0)}

(Simplify)

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

 

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

0.816T_{DB}\text{sin}\,(30^0)\,+\,T_{DB}\text{sin}\,(45^0)\,-\,490.5\,=\,0

(Solve for T_{DB})

T_{DB}\,=\,439.9 N

 

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

T_{DC}\,=\,(0.816)(439.9)

T_{DC}\,=\,358.9 N

 

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

Determine the tension developed in each wire used to

(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}.

 

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

T_{BA}\text{sin}\,(30^0)\,-\,439.9\text{sin}\,(45^0)\,=\,0

(Solve for T_{BA})

T_{BA}\,=\,622.1 N

 

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

\rightarrow ^+\sum \text{F}_\text{x}\,=\,0

439.9\text{cos}\,(45^0)\,+\,T_{BC}\,-\,622.1\text{cos}\,(30^0)\,=\,0

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

Solve for T_{BC}

T_{BC}\,=\,227.7 N

 

Final Answers:

T_{DB}\,=\,439.9 N

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?
 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-31.

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