Proving T=H+q*h in a Catenary Spanned Cable

[dirk_mec1]

Homework Statement

A cable is spanned between two points at the left touching the ground and at the right smoothly spanned over a large roller. Assume that there's no friction present. The tension left at the bottom is H and the tension T at the top is such that T>H. The cable has a weight per unit length q and is spanned over an height h and is in static equibrilium.

Prove that:

T = H +q*h

http://img220.imageshack.us/img220/537/96775335.png

Homework Equations

The Attempt at a Solution

I thought that the roller would give reaction forces perpendicular to the cable but other students told me that it works like a pulley and since there's the assumption that there's no friction I can disregard these forces, right? But I still don't know how to prove that formula.

 

[LawrenceC]

Hint:

Calculus and Analytic Geometry by Thomas...

 

[dirk_mec1]

Can you guide my in the right direction with the info of the book? Or give me the right chapter of the book?

 

[LawrenceC]

Chapter 10, third edition. The title of the chapter is Hyperbolic Functions.

 

[dirk_mec1]

Got it but it probably doesn't work if it's a beam instead of a cable, right? What happens with the expression if it is a beam?

 

[LawrenceC]

How do you "smoothly span" a beam over a roller? A beam is a different animal.

 

[dirk_mec1]

I mean no friction suppose that at the ends there's no shear force or bending moment (or at least negliglible) is the formula still valid?

 

[LawrenceC]

The reason why the cable pulls on both ends is because it cannot support itself. It has no resistance to bending stress - actually a cable does to a slight extent. If a beam could be bent over the frictionless pulleys such that it is horizontal where it touches them, it would remain there if undisturbed.

Think of a wheelbarrow with a load in it and a good set of bearings on its wheel. If you raise it, it does not want to run away from you. Yet, it is tilted upward.