A steel cable 3.00 cm^2 in cross-sectional area has a mass of 2.40 kg per meter of length. If 50

A steel cable 3.00 cm^2 in cross-sectional area has a mass...

A steel cable 3.00 $\mathrm{cm}^{2}$ in cross-sectional area has a mass of 2.40 $\mathrm{kg}$ per meter of length. If 500 $\mathrm{m}$ of the cable is hung over a vertical cliff, how much does the cable stretch under its own weight? Take $Y_{\text { steel }}=2.00 \times 10^{11} \mathrm{N} / \mathrm{m}^{2}$ .

Key Concepts

Young's Modulus

Young's Modulus is a fundamental property of materials that quantifies their stiffness. It defines the ratio of tensile stress (force per unit area) to tensile strain (relative deformation) within the elastic limit of the material, determining how much a material will stretch or compress under a given load.

Stress and Strain

Stress is the force applied per unit area while strain is the resulting relative change in length. The relationship between them, governed by Hooke's Law in the elastic regime, is linear, and the constant of proportionality is Young’s modulus, which indicates how much deformation occurs under a specific load.

Distributed Load and Integration

When a structural element is subjected to its own weight, the load is continuously distributed along its length. To determine the overall extension, one must integrate the contributions to elongation from each infinitesimal segment of the cable, which involves considering the variable tension that develops due to the cumulative weight acting on each segment.

Linear Density

Linear density is the mass per unit length of an object and plays a critical role in analyzing its behavior under self-weight. It allows the weight of any segment of the material to be calculated by multiplying the linear density by the gravitational acceleration and the segment's length, which is essential for setting up and solving integration problems in elasticity.

Transcript

00:01 So we're told a steel cable that is, let's see here, three and a half, three cubic centimeters, three square centimeters is a cross -sectional area.

00:15 It has a mass per unit length of 2 .4 kilograms per meter.

00:21 And there's 500 meters of it and it's hung over a vertical cliff and how much has a cable stretch under its own weight? and it still saw the yannis modulus is 200 gigapascals.

00:39 So we have all our information here from the problem.

00:42 And so what we need to do is we basically need to look at a little differential element of, you know, these cables at some distance from the top.

00:53 And so that differential element, we can see if we do a force balance on it, okay, we can see that we have some force here and some force here and then we have the weight of it.

01:05 And so the weight, it has, we have a differential dm.

01:11 I should probably write this a little bit different.

01:13 Probably write it as gdm to make it a little more clear that it's just, we have a differential mass here.

01:23 And so we have, we know if we do a force balance here, we get the the force down here, the d .f, the slight change in force.

01:31 Must be equal to minus g, dm.

01:35 And dm is, i'm gonna, i used here the actual mass density, density per unit volume.

01:45 So then actually if you do that, then a's cancel out.

01:48 But anyway, so this is the, this is the, this right here, a times the x is the volume of this and then this is the volume density.

01:58 Now if we integrate that, we can see that the force in the, in the, the cable, it's a maximum at the top, and then when it gets to the bottom, it's going to be a minimum.

02:11 So when x equals l, this should be zero, because this is, this is just g -a row times l, the mass, the, basically just the weight of the cable...

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