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.

#### Solution:

Show me the final answer↓

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!