The ball D has a mass of 20 kg 1


The ball D has a mass of 20 kg. If a force of F = 100 N is applied horizontally to the ring at A, determine the largest dimension d the force in cable AC is zero.

The ball D has a mass of 20 kg

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 draw a free body diagram around ring A as follows:

The ball D has a mass of 20 kg

(Note that cable AC is not shown in the free body diagram because the force in the cable is zero.)

Let us write our equations of equilibrium.

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

100\,-\,F_{AB}\text{cos}\theta\,=\,0

(Isolate for F_{AB})

F_{AB}\,=\,\dfrac{100}{\text{cos}\theta} (eq.1)
 
 
+\uparrow \sum \text{F}_\text{y}\,=\,0

F_{AB}\text{sin}\theta\,-\,196.2\,=\,0 (eq.2)

 

Substitute the isolated value of F_{AB} from eq.1 into eq.2.

\left(\dfrac{100}{\text{cos}\theta}\right)\text{sin}\theta\,=\,196.2

(Simplify)

\dfrac{\text{sin}\theta}{\text{cos}\theta}\,=\,1.962

(remember that \dfrac{\text{sin}\theta}{\text{cos}\theta}\,=\,\text{tan}\theta)

\text{tan}\theta\,=\,1.962

(solve for \theta)

\theta\,=\,\text{tan}^{-1}(1.962)

\theta\,=\,63^0

 

To figure out d, we can use trigonometry. We can write the following:

\text{tan}\,(63^0)\,=\,\dfrac{(1.5\,+\,d)}{2}

(If you are unclear about this step, remember that \text{tan}\theta\,=\,\dfrac{\text{opposite}}{\text{adjacent}}. Refer to the diagram again to see how we got the values for opposite and adjacent.)

(solve for d)

d\,=\,2.42 m

 

If, you wanted, you can also figure out F_{AB} by substituting the value of \theta we found back into eq.1.

F_{AB}\,=\,\dfrac{100}{\text{cos}\,(63^0)}

F_{AB}\,=\,220 N

 

Final Answer:

d\,=\,2.42 m

 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 3, question 3-42.

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