Demonstration of how wrapping a rope around a post affects tension required to prevent slipping
Comparison using 1, 2, and 4 turns around the post
Constant coefficient of static friction
Goal: Calculate tension required on one end of the rope to prevent slipping when a force is applied on the other end (1,000 Newtons)
Key Concepts
Equation used: T₁ = F / (e^(μ * θ))
T₁: Tension on the less stressed side of the rope
F: Force applied on the other side (1,000 Newtons)
μ: Coefficient of static friction (0.4)
θ: Total angle of wraps around the post in radians
E: Mathematical constant (approximately 2.718)
Calculations & Results
1 Turn
Formula: T₁ = 1,000 / (e^(0.4 * 2π))
Calculation:
T₁ = 81 Newtons
2 Turns
Formula: T₁ = 1,000 / (e^(0.4 * 4π))
Calculation:
T₁ ≈ 6.56 Newtons
Observation:
Doubling the turns drastically reduces tension required from 81 N to 6.56 N
4 Turns
Formula: T₁ = 1,000 / (e^(0.4 * 8π))
Calculation:
T₁ ≈ 0.043 Newtons
Observation:
Quadrupling turns reduces tension required to a very small value
Key Insights
Increasing the number of turns has a similar effect on reducing tension as increasing the coefficient of static friction.
Both the coefficient of static friction and the number of turns are in the exponent in the tension equation, leading to an exponential decrease in required tension.
This relationship between tension, friction, and number of turns demonstrates the mechanical advantage gained from additional rope turns in systems of friction.