What is the maximum length of a metal cable that can hang vertically supported from one end of the cable? The Young's modulus of this metal is 2.10 × 10¹¹ N/m², its tensile strength (the maximum tensile stress it can support without breaking) is 7.40 × 10⁸N/m², and its density is 7.60 × 10³ kg/m³. A) 9.94 km B) 4.22 km C) 456 km D) 1.75 km E) 24.8 km

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The maximum length of the cable is reached when the stress at the top end of the cable is equal to the tensile strength. The stress at the top end of the cable is given by:$$\sigma = \frac{mg}{A}$$where:* $\sigma$ is the stress* $m$ is the mass of the cable* $g$ is the acceleration due to gravity* $A$ is the cross-sectional area of the cableThe mass of the cable is given by:$$m = ho V$$where:* $ho$ is the density of the cable* $V$ is the volume of the cableThe volume of the cable is given by:$$V = Al$$where:* $l$ is the length of the cableSubstituting these equations into the equation for stress, we get:$$\sigma = \frac{ho g l}{A}$$Setting this equal to the tensile strength, we get:$$\frac{ho g l}{A} = \sigma_{UTS}$$Solving for $l$, we get:$$l = \frac{\sigma_{UTS} A}{ho g}$$Substituting the given values, we get:$$l = \frac{(7.40 	imes 10^8 	ext{ N/m}^2)(1.00 	imes 10^{-6} 	ext{ m}^2)}{(7.60 	imes 10^3 	ext{ kg/m}^3)(9.81 	ext{ m/s}^2)} = 9.94 	ext{ km}$$

What Is the Maximum Length of a Metal Cable That