Lecture on Hanging Traffic Light Physics

Problem Overview

• Scenario: Traffic light supported by three cables with angles of 25°, 40°.
• Parameters:
  • Maximum support by any cable: 450 Newtons
  • Mass of traffic light: 40 kg

Objective

• Determine if all cables can support the traffic light by finding tension in each cable.

Solution Approach

Steps

1. Draw Free Body Diagram (FBD):
   • Show traffic light with downward weight (mg) and upward tension.
   • Weight (mg) = 392 Newtons (mass ( m = 40 )kg, ( g = 9.8 )m/s²).
2. Analyze Forces at the Junction:
   • Tensions act in different directions with angles.
   • Decompose tensions into components.
   • Use trigonometry to resolve components.

Equilibrium Conditions

• Traffic light is in static equilibrium.
  • Horizontal (x-axis): ( \sum F_x = 0 )
  • Vertical (y-axis): ( \sum F_y = 0 )

Equations Derived

Horizontal Component Equation

• ( T_1 \cos 35° = T_2 \cos 40° )
  • Solve for ( T_1 ): ( T_1 = \frac{T_2 \cos 40°}{\cos 35°} )

Vertical Component Equation

• ( T_2 \sin 40° + T_1 \sin 35° = mg )
  • Substitute value of ( T_1 ) from horizontal equation.

Solving the Equations

1. Substitute ( T_1 ) in vertical equation:
   • ( T_2 \sin 40° + \frac{T_2 \cos 40° \sin 35°}{\cos 35°} = 392 )
   • Simplify and solve for ( T_2 ).
2. Calculate ( T_2 ):
   • Value found: 332 Newtons
3. Calculate ( T_1 ) using ( T_2 ):
   • Value found: 310.5 Newtons

Conclusion

• Tensions in cables: ( T_3 = 392 )N, ( T_2 = 332 )N, ( T_1 = 310.5 )N
• All tensions are less than 450 Newtons.
• Cables can support traffic light without issue.

Final Remarks

• Cables remain intact as tension is well below the maximum threshold.
• Traffic light remains stable.