# Equilibrium: Minimum Cable Length

• ledphones
In summary: For part b, stress = force/area. You can calculate the force from the tension (1300N) and the angle theta. Then use the stress-strain relationship to find the change in length.
ledphones

## Homework Statement

The above figure shows a uniform iron beam of mass 253 kg and length L = 3 m. The cable holding the beam in place can take a tension of 1300 N before it breaks. Use the Young's Modulus for steel to be 200e9 N/m2. (You may ignore the small mass of the cable in this calculation.) a)What minimum length of the cable?
Lengthmin= m

HELP: How will the torque placed on the beam by the cable depend on the angle between the cable and the beam??
b)Assume the cable is made of steel and has a diameter of 1". How much will it stretch before it breaks?
Δmax = cm

HELP: Young's modulus for steel is 200X109N/m2.

## The Attempt at a Solution

for a)

I set the forces in the x up like :
mgcos(phi)-Fcos(theta)=0

y:
Fsin(phi)+Tsin(theta)-W=0

torque
-Wdsin(180)+TLsin(theta)=0

where phi is the angle of the wall the normal force and theta is the angle of the cable and the beam. i don't know where x comes into play

for part b i think it would be like this

stress=1050/(1.27^2pi)

stress/20000000=strain

change in L will be strain*the L from part a

#### Attachments

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torque
-Wdsin(180)+TLsin(theta)=0
I don't follow the first term. It should be the weight times the distance from the pivot to the center of mass, which is L/2. There is no angle involved since the weight acts perpendicular to the beam. I agree with your second term. Solve for the angle theta, from which you can calculate the length of the cable.

## 1. What is equilibrium in the context of minimum cable length?

Equilibrium in the context of minimum cable length refers to the state in which the forces acting on a cable are balanced, resulting in no net force and therefore no movement or deformation of the cable.

## 2. Why is minimum cable length important?

Minimum cable length is important because it ensures that the cable is able to carry the required load without breaking or sagging excessively. It also minimizes the amount of material needed for the cable, reducing costs and potential environmental impact.

## 3. How is minimum cable length calculated?

Minimum cable length is calculated using the principle of virtual work, which involves setting up a system of equations and solving for the unknown cable length. This can be done analytically or through computer simulations.

## 4. What factors affect the minimum cable length?

The minimum cable length is affected by several factors, including the load placed on the cable, the properties of the cable material, and the support structures and conditions. Changes in any of these factors can alter the equilibrium of the cable and therefore impact the minimum length.

## 5. How does changing the support structure affect the minimum cable length?

Changing the support structure, such as adding or removing anchors or changing the angle of the cable, can affect the minimum cable length by altering the forces acting on the cable. This can result in a different equilibrium and therefore a different minimum length.

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