Sag in wire due to a weight (given the Young's modulus)

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Discussion Overview

The discussion revolves around calculating the sag in a mild steel wire when a weight is suspended from its midpoint. Participants explore the theoretical framework and equations relevant to the problem, considering the wire's properties and the effects of tension and elasticity.

Discussion Character

  • Homework-related
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents a formula for sag in a metal rod and attempts to apply it to a wire, noting the challenge of relating sag to the increase in length of the wire.
  • Another participant points out that the sag formula is based on assumptions suitable for rods with rectangular or square cross-sections, which may not apply to a typical steel wire that operates under tension rather than bending.
  • Some participants suggest analyzing the problem using a free body diagram to understand the forces acting on the weight and the wire, emphasizing the need to consider tension and its relationship to sag.
  • There is a suggestion to treat the wire as having zero stiffness and to draw diagrams to illustrate the forces and geometry involved in the problem.

Areas of Agreement / Disagreement

Participants express differing views on the appropriate approach to calculate sag, with some advocating for a tension-based analysis while others reference the original sag formula. The discussion remains unresolved, with no consensus on the best method to apply.

Contextual Notes

Participants note the limitations of the sag formula when applied to a wire, highlighting the need for careful consideration of the wire's properties and the assumptions made in the analysis.

ezioauditore
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Homework Statement


A mild steel wire of length 1 m and cross-sectional area 0.5*10^-2 cm^2 is stretched within elastic limit horizontally between two pillars.A mass of 100 g is suspended from mid-point.Depression at mid-point?

Homework Equations



Sag in metal rod is=WL^3/(4BD^3Y) where W=weight, l=length,b=breadth,d=depth,Y=Young's modulus.But here its a wire.

The Attempt at a Solution


I tried to calculate the k of the wire since its stretched within its elasticity limit and found out the increase in length of the wire.But could not relate the sag to increase in length.
 
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ezioauditore said:

Homework Statement


A mild steel wire of length 1 m and cross-sectional area 0.5*10^-2 cm^2 is stretched within elastic limit horizontally between two pillars.A mass of 100 g is suspended from mid-point.Depression at mid-point?

Homework Equations



Sag in metal rod is=WL^3/(4BD^3Y) where W=weight, l=length,b=breadth,d=depth,Y=Young's modulus.But here its a wire.

The Attempt at a Solution


I tried to calculate the k of the wire since its stretched within its elasticity limit and found out the increase in length of the wire.But could not relate the sag to increase in length.
Your sag formula for the metal rod appears to be developed for a rod with a rectangular or square cross section, which is not the cross section of a typical steel wire.
The wire is not going to be in bending; it supports the load by remaining in tension.

You should analyze this problem from first principles by drawing a free body diagram showing the weight suspended between two supporting points. Without getting into catenaries and stuff, you can assume each part of the wire suspending the weight is straight. You want the tensile stress in each part of the wire to be less than the elastic limit, whatever that number is.
 
Show your attempt with all your working.
I think it's fair to assume the wire has zero stiffness, treat it as a string not a beam (or rod).
Draw a free body diagram of the weight and show how tension varies with angle (note how tension goes to infinity as theta goes to zero). Extension/sag depends on tension and tension depends on sag...
 
+1

I would make TWO drawings..

1) A free body diagram showing the forces acting on the weight. This should lead you to equations - hint it's a statics problem.
2) A diagram showing the wire straight and deflected by the weight. Some geometry will give you an equation for the change in length of the wire(s).

Eventually you will have enough equations to solve it all.
 

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