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.

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

Solution:

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.

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

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

(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

$d\,=\,2.42$ m