Overview
This lecture explains how to calculate the tension in each cable supporting a hanging traffic light using vector resolution and static equilibrium principles.
Problem Setup
- A 40 kg traffic light is suspended by three cables at angles of 25°, 40°, and another determined by geometry.
- Each cable can withstand a maximum tension of 450 N.
- Need to find the tension in each cable and check if they are within safety limits.
Free Body Diagram & Force Resolution
- Weight of the light (W) acts downward: ( W = mg = 40 \times 9.8 = 392,N ).
- Tensions in cables are resolved into x (horizontal) and y (vertical) components.
- For each cable:
- Tension components use sine/cosine based on their angle with the axis: cosine for components adjacent to the angle, sine for opposite.
Equilibrium Equations
- System is in static equilibrium, so total force in both x and y directions are zero.
- X-axis: ( T_1 \cos 35^\circ = T_2 \cos 40^\circ )
- Y-axis: ( T_1 \sin 35^\circ + T_2 \sin 40^\circ = mg )
Solving for Tensions
- Substitute values and solve the two equations:
- ( T_2 = 332,N )
- ( T_1 = 310.5,N )
- ( T_3 = mg = 392,N )
Safety Check
- All tensions (310.5 N, 332 N, 392 N) are less than the cable limit of 450 N.
- The cables can safely support the traffic light.
Key Terms & Definitions
- Static Equilibrium — State where an object is at rest, and all forces and torques are balanced.
- Tension — The pulling force transmitted by a string, cable, or similar object.
- Vector Resolution — Splitting a vector into components, typically along x and y axes.
Action Items / Next Steps
- Practice drawing free body diagrams and resolving forces into components.
- Solve similar equilibrium problems for additional practice.