Rope Friction Around Pole

The tension force in a rope grows exponentially with the number of turns the rope makes around a pole.

What it shows:

Many people have probably observed that, by wrapping a rope around a post, a person can hold in check a much larger force than would ordinarily be possible. In this experiment a flexible thick rope is wound around a horizontal pipe. Due to the interaction of the frictional forces and tension, there can be a considerable difference in tension between the two ends of the rope. In the demonstration, one end of the rope supports a (heavy) load and the other end is held by (small) "holding" force. The demonstration models the principle by which a capstan works.

How it works:

The Euler-Eytelwein equation(1) relates the tension of the two ends of the rope: T2 = T1 eμθ, where T2 is the tension in the rope due to the load it's supporting, T1 is the tension necessary to hold the load without slipping, μ is the coefficient of friction between the rope and the pipe, and θ is the total angle (measured in radians) made by all the windings of the rope (one full winding is 2∏ radians). The tension force increases exponentially with the coefficient of friction and the number of turns around the pipe. Note that the diameter of the pipe does not come into play. However, the pipe diameter cannot be too small because a significant amount of force would be lost in bending the rope around the pipe, especially if the rope is a little stiff.

65 pounds 65 pounds 100 pounds

Setting it up:

Secure a 2"-diameter steel pipe to the edge of the lecture bench with two C-clamps, as shown in the photographs. Use the 3/8"-diameter nylon rope. Our 65-lb wrecking ball makes an impressive heavy load. Tie one end of the rope to the ball and wrap several turns of the rope around the pipe. Upon slowly unwrapping the rope, you'll discover that 4½ turns is enough to hold the ball with only the weight of the loose rope supplying the "holding" force. That length of loose rope weighs about 0.11 lb. With θ=9∏, this suggests that μ≈0.23 (not an unreasonable number). With 2½ turns you can easily hold the ball with just your fingers. The 3rd photo shows a 100-lb dumbbell held with just 4½ turns of rope and the fingers supplying the tension. Using μ≈0.23, we estimate a 0.15 lb finger holding force.


Exponential relationships are always astounding and this is no exception. Many devices based on belt or wrap friction are used in rappelling, rock climbing (so-called top-roping in which one can hold (belay) a heavy person to prevent a fall), sailing, and rigging of equipment. The ubiquitus V-belt and pulley is another example, cleverly enhanced by the V shape. The relation between tightside and slackside tension for a V-belt is similar:

T2 = T1 eμθ/sin(β/2), where β is the angle of the V in the pulley (note that for β=180 degrees, we are back to the flat belt relationship).


(1) The equation is derived in most mechanical engineering textbooks. For example: R. Becker, Introduction to Theoretical Mechanics, (McGraw-Hill, NY, 1954) p 45-46. L. Goodman and W. Warner, Statics and Dynamics, (Wadsworth, Belmont CA, 1964) pp 308 - 314.  Also see D. Morin, Introduction to Classical Mechanics, (Cambridge University Press, NY, 2007) pp 26-27.