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The following is an example from David Morin's Introduction to Classical Mechanics.

A rope wraps an angle $\theta$ around a pole. You grab one end and pull with a tension $T_0$. The other end is attached to a large object, say, a boat. If the coefficient of static friction between the rope and the pole is $\mu$, what is the largest force the rope can exert on the boat, if the rope is not to slip around the pole?

Solution
Consider a small piece of the rope that subtends an angle $d\theta$. Let the tension in this piece be $T$ (which varies slightly over the small length). As shown in the figure, the pole exerts a small outward normal force, $Nd\theta$ , on the piece. This normal force exists to balance the “inward” components of the tensions at the ends. These inward components have magnitude $T \sin(d\theta/2)$. Therefore, $Nd\theta = 2T \sin(d\theta/2)$.
The solution then continues to calculate the inequality $T\leq T_0e^{\mu\theta}$.

enter image description here
I do not know how to get the magnitude for the inward components as $T sin(d\theta/2)$.

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1 Answer 1

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Since the axial action $T$ is aligned with the axis of the rope at every point, and since the infinitesimal element of the rope describe a arc of circumference of angle $d \theta$, the angle between the directions of the axial action $T$ at the free ends of the elementary rope is $d \theta$.

Now, draw the bisector of angle $d \theta$. This is your radial direction. The projection of each $T$ along this direction is $T \sin(d\theta/2)$.

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  • $\begingroup$ Is $T$ not in the tangential direction to the rope or along the direction of the rope around the pole? By axial action, do you mean $T$ is along the direction of the pole? $\endgroup$
    – 1500kook12
    Sep 6, 2022 at 19:05
  • $\begingroup$ $T$ is along the direction of the rope, but this direction changes, since the rope is rolled up around the pole. $\endgroup$
    – basics
    Sep 6, 2022 at 19:24
  • $\begingroup$ as an example, in your picture the rope describes a small arc of a circle of angle $d \theta$ $\endgroup$
    – basics
    Sep 6, 2022 at 19:26
  • $\begingroup$ To confirm I'm getting it right, should I use the theorem that the angle between the tangents and the angles subtended by the radii of the arc are supplementary? where the tangents would be $T$ and angle subtended by the radii would be $d\theta$ $\endgroup$
    – 1500kook12
    Sep 8, 2022 at 13:06
  • $\begingroup$ Or that the angle between the two directions indicated by the 2 arrows T, is the same as $d \theta$ $\endgroup$
    – basics
    Sep 8, 2022 at 14:02

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