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.
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:
(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.
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.
(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:
(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}\,=\,220 N
Final Answer:
Thanks, nice resolution!