To determine the tensions in the two cables holding the 200 N weight, let's call the tensions $T_1$ and $T_2$ for the left and right cables, respectively. Here’s how we can approach the problem:

1. Identify forces and angles:

   • The weight $W = 200 \, \text{N}$ acts downward.
   • The angles with the horizontal are 40° for $T_1$ and 30° for $T_2$.

2. Resolve forces into components:

   • The sum of the vertical components of $T_1$ and $T_2$ should equal the weight (since the weight is the only downward force).
   • The horizontal components of $T_1$ and $T_2$ should cancel each other out (since there is no horizontal movement).

3. Set up equations:

   • Vertical components: $T_1 \sin(40^\circ) + T_2 \sin(30^\circ) = 200$.
   • Horizontal components: $T_1 \cos(40^\circ) = T_2 \cos(30^\circ)$.

4. Solve the equations: From these two equations, we can solve for $T_1$ and $T_2$.

Let's calculate this step-by-step.

Calculations

1. Horizontal Equation: $T_1 \cos(40^\circ) = T_2 \cos(30^\circ)$ Solving for $T_1$: $T_1 = T_2 \frac{\cos(30^\circ)}{\cos(40^\circ)}$

2. Vertical Equation: Substitute $T_1$ from the horizontal equation into the vertical equation: $T_2 \frac{\cos(30^\circ)}{\cos(40^\circ)} \sin(40^\circ) + T_2 \sin(30^\circ) = 200$ Solving this will give us $T_2$, which can then be used to find $T_1$.

I’ll proceed with these calculations to find the values of $T_1$ and $T_2$.The tensions in the two cables are as follows:

• $T_1 \approx 184.32 \, \text{N}$ (left cable with 40° angle)
• $T_2 \approx 163.04 \, \text{N}$ (right cable with 30° angle)

These values satisfy both the vertical and horizontal equilibrium conditions for the system.

Would you like a detailed breakdown of the calculations? Let me know if you have further questions.

Here are some related questions to deepen your understanding:

1. How would the tensions change if the weight were increased to 300 N?
2. What would happen to the tensions if both angles were equal, say 35° each?
3. How does changing the angle of one of the cables affect the distribution of tension?
4. Could you solve for tensions using a different approach, such as vector analysis?
5. How would adding a third cable affect the equilibrium conditions?

Tip: Always check equilibrium conditions (both horizontal and vertical) when solving tension problems to ensure accuracy.