This is a physics problem involving forces in equilibrium. The box weighing $w = 30.0 \, \text{N}$ is hanging from a vertical rope, with two additional ropes attached to the ceiling at angles $\theta_1 = 27^\circ$ and $\theta_2 = 60^\circ$. The system is in static equilibrium, which means the net force acting on the system is zero in both the vertical and horizontal directions.

To solve for the tensions $T_1$, $T_2$, and $T_3$ in the ropes, we can break the forces down into their components:

Vertical forces:

• The vertical forces must balance out the weight of the box.
• The vertical component of tension $T_1$ is $T_1 \sin(\theta_1)$, and the vertical component of tension $T_2$ is $T_2 \sin(\theta_2)$.
• The sum of these vertical components must equal the weight $T_3 = w$, which directly acts downwards.

Thus: $T_1 \sin(\theta_1) + T_2 \sin(\theta_2) = w$ Substituting $w = 30.0 \, \text{N}$, $\theta_1 = 27^\circ$, and $\theta_2 = 60^\circ$: $T_1 \sin(27^\circ) + T_2 \sin(60^\circ) = 30.0 \, \text{N}$

Horizontal forces:

• The horizontal components of the tensions must cancel each other out.
• The horizontal component of tension $T_1$ is $T_1 \cos(\theta_1)$, and the horizontal component of tension $T_2$ is $T_2 \cos(\theta_2)$.

Thus: $T_1 \cos(\theta_1) = T_2 \cos(\theta_2)$

We now have two equations with two unknowns, $T_1$ and $T_2$, which can be solved simultaneously.

Let me calculate these values for you.The tensions in the two ropes are:

• $T_1 = 15.02 \, \text{N}$
• $T_2 = 26.77 \, \text{N}$

Since the system is in equilibrium, these tensions ensure that both the horizontal and vertical force components balance out.

Would you like further details or have any questions? Here are some related questions for deeper understanding:

1. How does changing the angles $\theta_1$ and $\theta_2$ affect the tension in the ropes?
2. If the weight of the box increased, how would the tensions in $T_1$ and $T_2$ change?
3. Can you explain the role of equilibrium in solving this problem?
4. What assumptions are we making about the mass of the ropes and the pulleys?
5. How would this problem change if there were friction at the ceiling attachments?

Tip: When solving for forces in equilibrium, always break down the forces into horizontal and vertical components and solve using simultaneous equations.