To solve this problem, we calculate the tensions in the left (vasen) and right (oikea) ropes using trigonometry and the given angles.

Step 1: Break Down the Problem

The box has a mass of $120 \, \text{kg}$, so its weight $W$ can be calculated as: $W = m \cdot g = 120 \cdot 9.81 = 1177.2 \, \text{N}.$

This weight is supported by the two ropes at angles $29^\circ$ and $36^\circ$ from the horizontal. We'll denote:

• $T_1$: Tension in the left rope (vasen köysi),
• $T_2$: Tension in the right rope (oikea köysi).

The weight is distributed such that the vertical components of $T_1$ and $T_2$ add up to $1177.2 \, \text{N}$, and their horizontal components cancel each other out.

Step 2: System of Equations

1. Vertical force balance: $T_1 \cdot \sin(29^\circ) + T_2 \cdot \sin(36^\circ) = 1177.2$

2. Horizontal force balance: $T_1 \cdot \cos(29^\circ) = T_2 \cdot \cos(36^\circ)$

From the horizontal force balance, solve for $T_1$ in terms of $T_2$: $T_1 = T_2 \cdot \frac{\cos(36^\circ)}{\cos(29^\circ)}$

Step 3: Solve for $T_2$

Substitute $T_1$ into the vertical force equation: $\left(T_2 \cdot \frac{\cos(36^\circ)}{\cos(29^\circ)}\right) \cdot \sin(29^\circ) + T_2 \cdot \sin(36^\circ) = 1177.2$

Factor out $T_2$: $T_2 \cdot \left(\frac{\cos(36^\circ) \cdot \sin(29^\circ)}{\cos(29^\circ)} + \sin(36^\circ)\right) = 1177.2$

Simplify: $T_2 = \frac{1177.2}{\frac{\cos(36^\circ) \cdot \sin(29^\circ)}{\cos(29^\circ)} + \sin(36^\circ)}$

Step 4: Numerical Calculation

Use trigonometric values: $\sin(29^\circ) \approx 0.4848, \, \cos(29^\circ) \approx 0.8763$ $\sin(36^\circ) \approx 0.5878, \, \cos(36^\circ) \approx 0.8090$

Substitute: $T_2 = \frac{1177.2}{\frac{0.8090 \cdot 0.4848}{0.8763} + 0.5878}$

Simplify: $T_2 = \frac{1177.2}{\frac{0.3919}{0.8763} + 0.5878} = \frac{1177.2}{0.4472 + 0.5878} = \frac{1177.2}{1.0350} \approx 1138 \, \text{N}.$

Now calculate $T_1$: $T_1 = T_2 \cdot \frac{\cos(36^\circ)}{\cos(29^\circ)} = 1138 \cdot \frac{0.8090}{0.8763} \approx 1138 \cdot 0.9232 \approx 1050 \, \text{N}.$

Final Answer

• Left rope tension (vasen köysi): $1050 \, \text{N}$,
• Right rope tension (oikea köysi): $1138 \, \text{N}$.

Would you like a detailed explanation of any part of this? Here are some related questions to expand your understanding:

1. How do we derive trigonometric functions like $\sin$ and $\cos$ for angles?
2. Why do the horizontal forces cancel out in this type of equilibrium problem?
3. How does the weight of the object distribute across the two ropes?
4. What happens if the angles are changed (e.g., larger or smaller)?
5. Can you explain the relationship between vertical and horizontal components in tension problems?

Tip: In physics problems involving angles and forces, always resolve forces into vertical and horizontal components to simplify the equations.