To determine the value of force $Q$ for which cable $AD$ is just taut, we need to break down the forces acting at point $A$, where the cables meet. We'll work under the assumption that the system is in equilibrium.

Step 1: Identify the forces involved

The forces acting at point $A$ are:

1. Force $P$ = 900 N acting along a specific direction.
2. Force $Q$ (unknown) applied along a different direction.
3. The tension in cable $AD$ is zero when it is "just taut."
4. Tension forces in the other cables (such as $AB$ and $AC$).

Step 2: Resolve forces into components

To find the value of $Q$, we will resolve the forces acting on point $A$ into their horizontal and vertical components. Assuming a 2D plane (for simplicity), the equilibrium condition requires that the sum of the forces in both the horizontal and vertical directions is zero.

We can represent the forces as vectors and set up the following system of equations:

• The sum of the forces in the $x$-direction must be zero.
• The sum of the forces in the $y$-direction must be zero.

Step 3: Set up equilibrium equations

Let $\theta_P$ and $\theta_Q$ be the angles at which forces $P$ and $Q$ act, relative to the horizontal axis. The horizontal and vertical equilibrium equations are:

$\sum F_x = 0 \quad \Rightarrow \quad P \cos(\theta_P) + Q \cos(\theta_Q) = 0$ $\sum F_y = 0 \quad \Rightarrow \quad P \sin(\theta_P) + Q \sin(\theta_Q) = 0$

Step 4: Solve for $Q$

To solve for $Q$, we need the specific geometry or angles $\theta_P$ and $\theta_Q$. Could you provide more details about the directions or angles at which $P$ and $Q$ are applied, as well as the configuration of the cables?

Once we have this information, we can solve for $Q$ by using trigonometric relationships and the conditions of equilibrium.

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Would you like to provide more details on the geometry or angles? If not, I can guide you through common assumptions based on typical configurations.

Here are 5 related questions to deepen understanding:

1. How do we resolve forces in 3D if the problem extends beyond 2D?
2. What happens if the system is not in equilibrium? How would we handle acceleration?
3. How do angles of force application influence the tautness of cables?
4. How does friction or material stiffness in cables affect the solution?
5. How can we apply vector cross products to force problems?

Tip: Always start by resolving forces into components along the axes for clear equilibrium equations.