# I want to know how to arrive at the inward components as $T \sin(d\theta /2)$?

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 $$T\leq T_0e^{\mu\theta}$$.

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

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 $$T \sin(d\theta/2)$$.
• 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? Sep 6, 2022 at 19:05
• $T$ is along the direction of the rope, but this direction changes, since the rope is rolled up around the pole. Sep 6, 2022 at 19:24
• as an example, in your picture the rope describes a small arc of a circle of angle $d \theta$ Sep 6, 2022 at 19:26
• 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 $d\theta$ Sep 8, 2022 at 13:06
• Or that the angle between the two directions indicated by the 2 arrows T, is the same as $d \theta$ Sep 8, 2022 at 14:02