Force to move a catenary curve

[Smilan]

Homework Statement

I have a string attached on two ends to uneven supports. The upper support is fixed and the lower support can be moved only horizontally. I need to determine the force with which I need to pull the string (horizontally) to obtain a given sag in the string.

Relevant Equations

Attached as an image

Where D is sag, H is horizontal tension
–w is cable weight per unit length
–h is elevation difference between supports
–S is span (horizontal difference between supports)

This is the image of the problem:

My main question is a conceptual one, is the pulling force I'm looking for simply the horizontal tension in all of the string (Th in the image)? Or do I need to find the total tension at the support that I am pulling (T1 in the image)? My intuition tells me it should only be the horizontal tension that matters for what I am looking for, but at the same time it seems weird to me that the vertical tension would not matter.

 

[TSny]

My main question is a conceptual one, is the pulling force I'm looking for simply the horizontal tension in all of the string (Th in the image)? Or do I need to find the total tension at the support that I am pulling (T1 in the image)? My intuition tells me it should only be the horizontal tension that matters for what I am looking for, but at the same time it seems weird to me that the vertical tension would not matter.

I think you're right that your applied force would equal the horizontal component of the tension in the string at the lower support.

Since the lower support is restricted to move only horizontally, you could imagine that the right end of the string is attached to a peg that slides without friction in a horizontal slot. When the peg is at rest in a certain position, the net force on the peg must be zero. You can consider your force as applied to the peg. So, your horizontal force on the peg must balance the horizontal component of the string's tension on the peg.

Can you see how the vertical component of the string's tension on the peg is balanced?

 

[Lnewqban]

You could also compare your situation with a piston-connecting rod-crankshaft of an internal combustion engine.

The only forces doing work are the ones free to move in certain direction (forces F and F_T in the attached diagram).

Movement is restricted in any other direction, reason for which the other forces (forces F_P and F_N in the attached diagram) do not contribute to the mechanical advantage of the mechanism (ratio of output force to input force), but only to increased friction.