Problem 15 - An Introduction to the Mechanics of Solids in SI Units

Question

Some miners are trapped \(2000 \mathrm{~m}\) below the surface. They make their way to the bottom of an abandoned shaft. At the surface is a hoist with \(1990 \mathrm{~m}\) of \(2.5 \mathrm{~cm}\) diameter standard plow-steel hoisting rope. A 30\(\mathrm{cm}\) length of this rope weighs \(10 \mathrm{~N}\) and has a spring constant (including the effect of untwisting) of about \(2 \times 10^{6} \mathrm{~N} / \mathrm{cm}\). If you think the miners can be hoisted to the surface, explain quantitatively how this can be done.

Step-by-Step Solution

Answer

Yes, the miners can be hoisted to the surface. The total weight the rope can handle (66333.33 N) is significantly higher than the total weight exerted by the miners (9800 N). Furthermore, the additional stretching caused by the miners' weight is small enough for the rope to handle without snapping. Hence, quantitatively it's plausible to hoist the miners using the hoist.

Step 1: Calculate the total length of the rope

Given that 30cm of the rope weighs 10N, the total length (L) of the 1990 m rope can be calculated as follows: \[L = 1990 m \times (1 cm / 0.01 m) = 199000 cm\]. And since 30 cm of this rope weighs 10 N, then the total weight of the 1990 m rope (W) is \[W = (10 N/30 cm) × 199000 cm = 66333.33 N\]

Step 2: Find out the total force exerted by the miners

The force (F) the miners exert on the rope can be calculated as the product of their mass and acceleration due to gravity. However, the problem does not specify the weight of the miners. It can be assumed that the miners could be around 1000 kg for the sake of the calculation: \[F = m \times g = 1000 kg \times 9.8 m/s^2 = 9800 N\]

Step 3: Determine if the rope can withstand the weight of the miners

Lastly, compare the total weight of the rope with the force exerted by the miners. If the force exerted by the miners is less than or equal to the weight of the rope, then the miners can be hoisted out using the rope: \[9800 N <= 66333.33 N\]

Step 4: Check the spring constant of the rope

The problem statement mentions that the spring constant of the rope, k, is 2x10^6 N/cm. However, as the rope will stretch under the weight of the miners, this must also be taken into account. Using Hooke's law (F = kx), the maximum stretching (x) under the miners' weight can be calculated: \[x = F / k = 9800 N / (2x10^6 N/cm) = 0.0049 cm\]. Since this stretching value is really small, the rope should be able to hold the miners without snapping.

Key Concepts

Hoisting Rope Tension

When we discuss hoisting rope tension, we're talking about the force that is applied to a rope when it is used to lift or move a load. In practical scenarios like the miners being lifted from a shaft, the tension in the hoisting rope is a critical consideration.

Tension is essentially the pulling force that is transmitted through a string, rope, cable or similar when it is pulled tight by forces acting from opposite ends. The total tension in the rope would have to be sufficient to support not only the weight of the miners but also the weight of the rope itself.

To ensure safety and functionality, engineers must calculate the tension capacity of the hoisting rope by considering factors like the material, diameter and length of the rope, and its condition. The hoisting rope must be able to withstand the total tension without breaking, which means it must have a higher breaking strength than the calculated tension when the miners are being hoisted.

Spring Constant

The spring constant, often denoted by the letter k, is a measure of the stiffness of a spring. When it comes to solid mechanics, this value is crucial as it determines how much force is needed to stretch or compress a spring by a certain distance.

Mathematically, the spring constant is defined by the equation k = F / Δx, where F is the force applied to the spring and Δx is the change in length of the spring from its equilibrium position. The higher the spring constant, the stiffer the spring and the more force it will take to stretch it.

Understanding Rope as a Spring

In the context of the hoisting rope, the spring constant isn't for a traditional coiled spring. Instead, it's a value that tells us how much the hoisting rope will stretch under a given load. A higher spring constant for a rope means it's less stretchy and more capable of holding heavy loads without significant elongation – this is crucial for hoisting applications where excessive stretching could be dangerous.

Hooke's Law

One of the foundational principles in the mechanics of solids, Hooke's Law, describes the behavior of materials that are stretched or compressed within their elastic limit. This law states that the force needed to extend or compress a spring by some distance x scales linearly with that distance.

Formally, the law is articulated as F = kx, where F is the force applied to the material, k is the spring constant, and x is the extension or compression of the material (the displacement from its original length).

Real-World Applications

Although originally formulated for springs, Hooke's Law can also apply to other scenarios involving elasticity – including our hoisting rope. The law tells us that as long as the miners do not exceed the elastic limit of the hoisting rope, which is determined by the material's properties and its spring constant, the rope can safely return to its original shape and length after they are hoisted. This assures that the rope will not permanently deform or break, ensuring the miners' safety during the hoisting operation.