To solve this problem, we’ll balance the forces in both the $x$- and $y$-directions and solve for the tension $T_1$.

Given Information

• Weight of the traffic light, $W = 139 \, \text{N}$
• Angle of $T_1$ with the horizontal: $40^\circ$
• Angle of $T_2$ with the horizontal: $60^\circ$

Part (a): Writing Equilibrium Equations

Since the system is in equilibrium, the sum of the forces in both the $x$- and $y$-directions should be zero.

1. In the $x$-direction: $T_1 \cos(40^\circ) = T_2 \cos(60^\circ)$

2. In the $y$-direction: $T_1 \sin(40^\circ) + T_2 \sin(60^\circ) = W$ Substituting $W = 139 \, \text{N}$, we get: $T_1 \sin(40^\circ) + T_2 \sin(60^\circ) = 139$

Part (b): Solving the Simultaneous Equations

Now, let’s solve these equations to find $T_1$.

1. From the $x$-direction equation: $T_2 = \frac{T_1 \cos(40^\circ)}{\cos(60^\circ)}$

2. Substitute $T_2$ from this equation into the $y$-direction equation: $T_1 \sin(40^\circ) + \frac{T_1 \cos(40^\circ) \sin(60^\circ)}{\cos(60^\circ)} = 139$

3. Simplify and solve for $T_1$.

Let's calculate the values using trigonometric approximations and substitutions for the angles.

Approximate Trigonometric Values:

• $\sin(40^\circ) \approx 0.6428$
• $\cos(40^\circ) \approx 0.7660$
• $\sin(60^\circ) \approx 0.8660$
• $\cos(60^\circ) = 0.5$

Substitute these into the equation:

$T_1 \times 0.6428 + \frac{T_1 \times 0.7660 \times 0.8660}{0.5} = 139$

Calculating $T_1$ from this equation will yield the tension in $T_1$.

Let me proceed with the computation.The tension $T_1$ in the cable is approximately $70.57 \, \text{N}$.

Would you like more details on the steps or have any questions?

Here are some related questions that could further deepen your understanding:

1. How would the tensions change if the weight of the traffic light increased?
2. What effect would changing the angles of the cables have on the tension values?
3. How can we derive the trigonometric relationships used for the angles here?
4. What are some real-life scenarios where understanding force equilibrium is essential?
5. How would the system behave if one of the cables was replaced with a stronger or weaker material?

Tip: When solving equilibrium problems, always start by resolving forces into their components and writing down the balance equations before solving.