If the tension developed in each of the four wires 5


If the tension developed in each of the four wires is not allowed to exceed 600 N , determine the maximum mass of the chandelier that can be supported.

If the tension developed in each of the four wires

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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Let us first draw a free body diagram and assume that the weight of the chandelier is W.

If the tension developed in each of the four wires

Now, we will 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)\,-\,W\,=\,0 (eq.2)

 

We will now write T_{DC} and T_{DB} in terms of W. To do so, we will isolate 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 back into eq.2.

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

(isolate for T_{DB})

T_{DB}\,=\,0.897W

 

We can now substitute this value back into eq.3 to write T_{DC} in terms of W.

T_{DC}\,=\,(0.816)(0.897W)

T_{DC}\,=\,0.732W

 

Now, we will focus on ring B and draw a free body diagram.

If the tension developed in each of the four wires is not allowed

(Remember, we found T_{DB}\,=\,0.897W)

Again, we will write our equations of equilibrium, however, we will first write our equation of equilibrium for the y-axis forces.

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

T_{BA}\text{sin}\,(30^0)\,-\,0.897W\text{sin}\,(45^0)\,=\,0

(Isolate for T_{BA})

T_{BA}\,=\,1.27W

 

Now, write an equation of equilibrium for x-axis forces. Note that we just found T_{BA}\,=\,1.27W.

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

0.897W\text{cos}\,(45^0)\,+\,T_{BC}\,-\,1.27W\text{cos}\,(30^0)\,=\,0

(Isolate for T_{BC})

T_{BC}\,=\,0.465W

 

We now have all of the tensions in the ropes in terms of W. They are the following:

T_{DB}\,=\,0.897W

T_{DC}\,=\,0.732W

T_{BA}\,=\,1.27W

T_{BC}\,=\,0.465W

 
From these values, we can see that rope BA, or T_{BA} experiences the most tension. The question also states that the maximum tension in each rope is 600 N, thus we can write the following:

T_{BA}\,=\,1.27W

600\,=\,1.27W

(solve for W)

W\,=\,472.4 N

 

We just found the maximum weight of the chandelier that can be hung, but the question asks us for the mass. We can use this following formula to find the mass:

W=mg

(Where W is weight, m is mass and g is the force of gravity)

m\,=\,\dfrac{W}{g}

m\,=\,\dfrac{472.4}{9.81}

m\,=\,48.1 kg

 

Final answer:

Maximum mass of the chandelier that can be hung = 48.1 kg

 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-32.

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