Question

A solid copper cube has an edge length of \(85.5 \mathrm{~cm}\). How much pressure must be applied to the cube to reduce the edge length to \(85.0 \mathrm{~cm} ?\) The bulk modulus of the copper is \(140 \mathrm{GPa}\).

Short answer

The pressure that must be applied to the cube to reduce the edge length is determined by substituting the values into our derived formula in step 5. This gives us the required pressure value in GigaPascals (GPa).

Step-by-step solution

Understand the terms

In this step, realize that the bulk modulus (B) formula can be written as follows: B = - ∆P/∆V/V. Here, ∆P represents the change in pressure, ∆V represents the change in volume, and V represents the original volume.

Calculate initial and final volumes

Given that the initial cube edge length is 85.5 cm and the final edge length is 85 cm, let's find the initial and final volumes of the cube. The volume of a cube can be found using the formula V = a^3, where a is the edge length. So, the initial volume V1 = 85.5^3 cm^3 and the final volume V2 = 85.0^3 cm^3.

Calculate volume strain

Next, let's calculate the volume strain, which is the change in volume divided by the original volume. The change in volume (∆V) is the final volume minus the initial volume, (V2 - V1). thus, volume strain = ∆V/V = (V2 - V1)/V1.

Rearrange bulk modulus formula to solve for pressure change

Recall the bulk modulus formula B = - ∆P/∆V/V. From this, filter out ∆P (the change in pressure) to find out the unknown pressure to be applied to the cube to cause the change in volume. We get ∆P = -B * ∆V/V.

Substitute the values

Replace B, ∆V/V with their values from earlier steps. ∆P = -140GPa * ((85.0 cm^3 - 85.5 cm^3)/85.5 cm^3). The change in pressure, ∆P is the pressure that must be applied to the cube to reduce the edge length. Hence we find the answer.

Key concepts

Physics Problem Solving

When approaching physics problems, especially those involving material properties, it's essential to have a structured method. Let's dissect the problem at hand – determining the pressure to compress a copper cube using the bulk modulus concept. This requires an understanding of the formula and variables involved. In physics problem solving, we start by identifying the knowns: initial and final dimensions, and the material's bulk modulus. Next, the relevant equations are applied – in this case, the bulk modulus definition and the formula for the volume of a cube.

The physics problem solving approach is about breaking down complex problems into smaller, more manageable parts and applying mathematical relationships between these parts. This not only makes the problem easier to handle but also helps us to understand the principles governing the behavior of the system in question.

Mechanical Properties of Materials

Understanding the mechanical properties of materials is central to physics and engineering. These properties define how materials respond to external stresses and strains. One such property is the bulk modulus, which measures a material's resistance to uniform compression. It is a fundamental parameter that gives us essential information about the material's elasticity, an intrinsic property revealing how much a material will deform when a force is applied.

The mechanical properties of materials like copper - which is the focus of our problem - is often studied in terms of stress-strain relationships. Copper's bulk modulus is given, which provides a direct measure of its compressibility. Knowledge of copper's mechanical properties is not only crucial for solving homework problems but also for practical applications, such as designing structures or technologies that utilize the material's resistance to change in volume under pressure.

Elastic Properties of Materials

Elastic properties of materials is a realm of study concerning how materials deform and return to their original shape when external forces are applied and removed. These properties are important because they determine a material's ability to store mechanical energy. The bulk modulus discussed earlier is part of a larger group of elastic moduli, which also includes Young's modulus and the shear modulus, each describing different types of elastic behavior. In our bulk modulus problem, we deal with an aspect of material elasticity regarding volume change.

Materials like copper exhibit elastic behavior up to a certain limit, characterized by a linear stress-strain relationship, known as Hooke's law. When working with exercises in this field, the focus is on reversible deformations where after pressure removal, the material may return to its original shape. This ability defines how materials can be utilized in various mechanical systems where this elastic property is a necessity, like in springs or cushioning materials.