To solve for the tension $T$ in the left-hand cable, we need to break down the forces acting on the system into horizontal and vertical components. Here's the approach step-by-step:

Step 1: Identify the forces

• The weight $W = 663 \, \text{N}$ is acting vertically downward.
• The right-hand cable has a tension of $810 \, \text{N}$ and makes an angle of $26^\circ$ with the ceiling.
• The left-hand cable has a tension $T$, which makes an angle $\theta$ with the wall (we will solve for this later).

Step 2: Break forces into components

We need to break the forces into their horizontal and vertical components and use the fact that the system is in equilibrium. In equilibrium:

• The sum of horizontal forces is zero: $\sum F_x = 0$
• The sum of vertical forces is zero: $\sum F_y = 0$

Vertical components:

• The vertical component of the tension in the right cable is: $T_{\text{right}, y} = 810 \, \text{N} \cdot \sin(26^\circ)$
• The vertical component of the tension in the left cable is: $T_{\text{left}, y} = T \cdot \sin(\theta)$
• The total vertical force must balance the weight: $T_{\text{left}, y} + T_{\text{right}, y} = W = 663 \, \text{N}$ So, $T \cdot \sin(\theta) + 810 \, \text{N} \cdot \sin(26^\circ) = 663 \, \text{N}$

Horizontal components:

• The horizontal component of the tension in the right cable is: $T_{\text{right}, x} = 810 \, \text{N} \cdot \cos(26^\circ)$
• The horizontal component of the tension in the left cable is: $T_{\text{left}, x} = T \cdot \cos(\theta)$ Since there is no horizontal movement: $T_{\text{left}, x} = T_{\text{right}, x}$ Therefore: $T \cdot \cos(\theta) = 810 \, \text{N} \cdot \cos(26^\circ)$

Step 3: Solve the system of equations

We now have two equations:

1. $T \cdot \sin(\theta) + 810 \cdot \sin(26^\circ) = 663$
2. $T \cdot \cos(\theta) = 810 \cdot \cos(26^\circ)$

Let me calculate these values and solve for $T$.The tension $T$ in the left-hand cable is $810 \, \text{N}$.

Would you like further details on how this result was obtained or have any additional questions?

Here are five follow-up questions to expand your understanding:

1. How do horizontal and vertical force components ensure equilibrium in this scenario?
2. What happens to the tension $T$ if the angle $\theta$ increases?
3. How would you calculate the angle $\theta$ if the problem provided it explicitly?
4. Could you apply this method to a similar setup but with different angles or tensions?
5. How would the equilibrium change if the weight of the object increased?

Tip: Always decompose forces into horizontal and vertical components when solving equilibrium problems involving angled forces.