Problem 52 - An Introduction to the Mechanics of Solids in SI Units

Question

A steel cable hangs under its own weight. The diameter of the cable is not constant but varies in a manner that makes the tensile stress at all points along the cable the same. Derive the differential equation that describes the variation of cable diameter with position along the cable. Solve the equation and find the expression for \(\mathrm{d}_{1}\) in terms of \(\mathrm{d}_{2}, \mathrm{~L}, \sigma_{0}\) and \(\gamma(\) the weight per unit volume of the cable).

Step-by-Step Solution

Answer

\( d_{1} = d_{2} - L \)

Step 1: Derivation of the differential equation

In the steady state, the tension at any point in the cable must equal the weight of the cable that hangs below it. If the cable hangs vertically and \( x \) measures the distance from the top, the tension \( T \) at a position \( x \) along the cable is given by \( T = \gamma A(L - x) \), with \( A \) denoting cross-sectional area, \( L \) the total length, and \( \gamma \) the cable's weight per unit volume. Then the cable's diameter \( d \) at point \( x \) can be expressed as \( d = (4A/π)^{1/2} \). Because the tensile stress \( σ \) at all points is constant, it's given by \( σ = T/A = \sigma_0 \), where \( \sigma_0 \) is the constat tensile stress. This results in the equation \( \gamma d (L-x) = 4 \sigma_0^{2} = \textrm{const.} \).

Step 2: Solving the differential equation

Differentiating the equation with respect to \( x \), we obtain \( \gamma d - \gamma x \frac{dd}{dx} = 0 \). Rearranging and dividing both sides by \( d \), we get \( dx = - \frac{d}{dd} \). Integrating over the cable length \( L \), the integral \( \int_{0}^{L} dx = - \int_{d_{1}}^{d_{2}} \frac{d}{d'} dd' \), with the diameters \( d_{1}, d_{2} \) at the top and bottom respectively.

Step 3: Determining the expression for \( d_{1} \)

Now, let's solve the integral \( \int_{d_{1}}^{d_{2}} \frac{d}{d'} dd' = -L \) for \( d_{1} \). Integrating, we get \( \left[-d' + d_{2}\right]_{d_{1}}^{d_{2} = -L \). Solving for \( d_{1} \), we obtain \( d_{1} = d_{2} - L \).

Key Concepts

Differential Equations

Differential equations play a crucial role in describing physical phenomena. They are mathematical equations involving derivatives, which represent how a function changes over time or space.
In the context of mechanics of solids, differential equations can describe how physical properties like stress or displacement vary within a material under certain conditions.
When dealing with the steel cable, we derive a differential equation to track how the diameter varies along the cable's length, ensuring the tensile stress remains constant. We start by setting up an equation based on the tension balance at any point in the cable, accounting for its varying weight and diameter.

• The tension at a point is proportional to the weight of the cable hanging below.
• The cross-sectional area changes along the cable, affecting the tension force.

By differentiating this relationship, we arrive at a form that allows us to calculate how the cable diameter should vary. Solving the resulting differential equation provides the basis for knowing how physical characteristics of the cable must change to maintain uniform stress.

Tensile Stress

Tensile stress is the force exerted per unit area within materials, such as cables or ropes, when they are stretched.
It is crucial in mechanics because it helps determine whether a material will withstand applied forces without deforming or breaking.
In the problem of the steel cable, tensile stress is kept constant at all points along the length, which means there's a consistent force-per-area through the cable's varying diameter.

• Tensile stress is calculated using the formula: \( \sigma = \frac{T}{A} \).
• Here, \( T \) represents the tension force, and \( A \) the cross-sectional area.

This constant tensile stress setup is an engineering consideration to prevent weak points, which can lead to structural failure. By managing stress uniformly, the cable can handle heavy loads more effectively. Understanding how to maintain this uniformity through changes in diameter helps in designing safer and more efficient cable systems.

Cable Diameter Variation

The diameter of a cable can affect its mechanical properties significantly. Varying the diameter alters the cable's ability to handle stress, as well as its overall structural integrity.
For the steel cable hanging under its own weight, the diameter must change along its length to meet the requirement of constant tensile stress.
The diameter at any point relates directly to the cross-sectional area, and thus, influences the tension force. If the diameter varied too much without consideration for stress, it could lead to material failure at weaker spots.

• The diameter function is derived using differential equations, maintaining stress uniformity.
• It defines a tapering profile for the cable, ensuring stress is evenly distributed along its length.

Ultimately, solving the differential equation reveals the precise way diameter should change to balance the mechanical demands. By obtaining expressions for the endpoints of the diameter, such as \( d_1 \) in relation to \( d_2 \), engineers can design cables that withstand gravity without disproportionate stress.