Question

A rod of cross-sectional area \(a\) and length \(L\) will stretch by an amount \(e\) when subject to a tensile load of \(P,\) Fig. \(11-27 .\) The modulus of elasticity is given by $$E=\frac{P L}{a e}$$ Solve this equation for \(a.\)

Short answer

\( a = \frac{PL}{Ee} \)

Step-by-step solution

Identify the Given Variables

Recognize the variables that are provided in the problem: modulus of elasticity (E), tensile load (P), length (L), and elongation (e).

Isolate the variable a

Start with the given formula for the modulus of elasticity and rearrange it to solve for the cross-sectional area (a). The given formula is: \( E = \frac{PL}{ae} \). Multiply both sides by the cross-sectional area (a) and then divide by the modulus of elasticity (E) to isolate (a) on one side of the equation.

Solve for a

The rearranged equation after isolating (a) is: \( a = \frac{PL}{Ee} \). This formula allows us to solve for the cross-sectional area (a) directly in terms of the other variables.

Key concepts

Understanding Cross-Sectional Area in Materials Science

In the realm of physics and engineering, the cross-sectional area is a vital concept, particularly when dealing with the mechanical properties of materials. Imagine slicing a rod or wire perpendicular to its length; the area of this cut is the cross-sectional area (a). This value is not just a measure of size, but rather a crucial element in the calculation of other physical properties, including the material's resistance to forces such as tension, compression, and bending.

In our example, involving a rod stretched by a tensile load, the cross-sectional area (a) directly influences the rod's ability to withstand forces without breaking. It's much like comparing a thin thread to a thick rope; a larger area can distribute the applied force more effectively, enhancing the material's tensile strength. The cross-sectional area also plays a key role in the calculation of stress, which is the force applied per unit area. Understanding its significance allows us to make informed choices about the materials and cross-sections needed for specific engineering applications and designs.

Exploring the Impact of Tensile Load

The term tensile load refers to a force that attempts to elongate a material by pulling it apart. It is a stress that can be either beneficial or detrimental, depending on the application. When you hang off a pull-up bar, for example, your weight creates a tensile load that stretches the bar slightly; this is similar to the tensile load (P) applied to our hypothetical rod.

An important aspect when working with tensile loads is the concept of tensile strength, the maximum load a material can withstand while being stretched before failing. Designing materials and structures involves a delicate balance – they must handle the applied loads without exceeding their tensile strength, ensuring safety and reliability. We calculate the amount of tensile load a material can withstand by understanding its physical properties, like the modulus of elasticity and the cross-sectional area, allowing for the safe application to structures such as bridges, buildings, and even everyday objects like utensils or cable wires.

Elongation: Measuring Material Deformity

When a material is subject to an external force, it deforms in some measure—a phenomenon referred to as elongation. This is the measure of how much a material extends under stress, recorded as the change in length (e) from its original shape. Imagine gently pulling on a rubber band; as you stretch, it elongates. The same principle applies to the rod in our example when it's subjected to a tensile load.

The ability of a material to elongate and its limitations are of great importance in material science and engineering, dictating how materials will behave when subjected to forces. Different materials have different capacities for elongation: metals might stretch only a small amount before snapping back or yielding, while plastics and rubbers can typically stretch much more. Elongation is also an indicator of ductility—the more a material can elongate before breaking, the more ductile it is. Understanding the elongation properties helps in designing materials that are both strong and flexible, providing guidelines for their uses in various industries, from construction to electronics.