The problem appears to involve calculating the tension in two ropes (T₁ and T₂) supporting a girl who weighs 392 N, with angles 50° and 40° from the vertical. Let me analyze the forces and solve the problem.

Steps to Solve:

1. Force Analysis:

   • The girl's weight acts downward (392 N).
   • Tensions $T_1$ and $T_2$ act along the ropes at angles $\theta_1 = 50^\circ$ and $\theta_2 = 40^\circ$ from the vertical, respectively.

2. Equilibrium Conditions: The forces must balance both horizontally and vertically:

   • Vertical Force Balance: $T_1 \cos(50^\circ) + T_2 \cos(40^\circ) = 392$
   • Horizontal Force Balance: $T_1 \sin(50^\circ) = T_2 \sin(40^\circ)$

3. Solve the System of Equations:

   • Use the horizontal force equation to express one tension in terms of the other.
   • Substitute into the vertical force equation to solve for the tensions.

Let me compute these.import numpy as np

Known values

weight = 392 # N (girl's weight) theta1 = np.radians(50) # Convert degrees to radians theta2 = np.radians(40) # Convert degrees to radians

Equations for the system

Horizontal force balance: T1sin(theta1) = T2sin(theta2)

Solve for T1 in terms of T2

T1_in_terms_of_T2 = lambda T2: T2 * (np.sin(theta2) / np.sin(theta1))

Vertical force balance: T1cos(theta1) + T2cos(theta2) = weight

def solve_for_tensions(weight, theta1, theta2): # Substitute T1 into the vertical equation T2 = weight / (np.cos(theta2) + np.cos(theta1) * (np.sin(theta2) / np.sin(theta1))) T1 = T1_in_terms_of_T2(T2) return T1, T2

Calculate tensions

T1, T2 = solve_for_tensions(weight, theta1, theta2) T1, T2(251.9727429971233, 300.28942170263934)(251.9727429971233, 300.28942170263934)successThe tensions in the ropes are as follows:

• Tension $T_1$: 252.0 N (approximately)
• Tension $T_2$: 300.3 N (approximately)

These values satisfy the equilibrium conditions of the system.

Would you like further details on the calculations or any additional related questions?

Related Questions:

1. How do we derive the equilibrium equations in such problems?
2. What would happen to the tensions if the angles were changed?
3. How would the problem change if the girl's weight was different?
4. Could we determine the forces if one of the ropes was slack?
5. What is the importance of resolving forces into components in such problems?

Tip: Always draw a clear force diagram to visualize and simplify the analysis of forces.