Static Equilibrium problem

Question

I would like to ask about a part of a question I saw in an IG textbook, the question starts as "A smooth bead is threaded on a light inextensible string, the ends of the string are attached to the ceilling, the bead is acted on by a horizontal force F and the bead is in equilibrium..... ", the point is that the answer assumes that the tension on both sides of the string are equal, why is this true?

Answer 1

There are (possibly hidden) assumptions in the problem statement, namely that there is no frictional force between the bead and the string and the part of the string in contact with the bead has zero weight. So although the string may bend around its region of contact with the bead, the tension remains equal throughout that piece of the string; there is no tangential component of any other force (from the bead or from gravity) at any point along that part of the string's path.

If we allow the bead to exert a significant frictional force on the string then we are no longer justified in assuming the tension is equal on both sides. Likewise if the part of the string in contact with the bead has significant weight.

The descriptions of the bead and string as "smooth" and "light" are presumably indicators that we are supposed to assume negligible friction and negligible weight of that portion of the string.

Answer 2

In the figure, point B shows the bead location threaded to the massless smooth string ABC.

In black is represented the bead weight, In blue the exerted horizontal force and in red the resultant force in equilibrium with the tension forces in green. The tension forces which are equal in modulus, have their resultant at the bisector dotted line.