Problem 90 - University Physics with Modern Physics

Question

Bulk Stress Due to a Temperature Increase. (a) Prove that, if an object under pressure has its temperature raised but is not allowed to expand, the increase in pressure is $$\Delta p=B \beta \Delta T$$ where the bulk modulus \(B\) and the average coefficient of volume expansion \(\beta\) are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from \(20.0^{\circ} \mathrm{C}\) to \(35.0^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

Pressure increase \( \Delta p = 8.64 \times 10^7 \ \mathrm{Pa} \) to keep steel from expanding.

Step 1: Understanding Bulk Stress and Thermal Expansion

When the temperature of a material increases, it tends to expand. If the expansion is not allowed (e.g., a steel block tightly restricted in a container), additional pressure is required to counter the expansion force, causing bulk stress.

Step 2: Relating Bulk Modulus and Change in Pressure

The Bulk Modulus, denoted as \( B \), is defined as \( B = -\frac{\Delta V/V}{\Delta p} \), where \( \Delta V \) is the change in volume, \( V \) is the initial volume, and \( \Delta p \) is the change in pressure. For no change in volume (\( \Delta V = 0 \)), the formula simplifies to \( \Delta p = B \beta \Delta T \) where \( \beta \) is the coefficient of volume expansion and \( \Delta T \) is the change in temperature.

Step 3: Calculating the Pressure Change for Steel

Use the proven relationship \( \Delta p = B \beta \Delta T \). For steel, \( B \approx 1.6 \times 10^{11} \ \mathrm{Pa} \) and \( \beta \approx 36 \times 10^{-6} \ 1/^{\circ}\mathrm{C} \). Substitute these values along with \( \Delta T = 35.0^{\circ}C - 20.0^{\circ}C = 15.0^{\circ}C \) into the equation: \( \Delta p = 1.6 \times 10^{11} \times 36 \times 10^{-6} \times 15 \).

Step 4: Solving for Necessary Pressure

Carry out the multiplication: \( \Delta p = 1.6 \times 10^{11} \times 36 \times 10^{-6} \times 15 = 8.64 \times 10^7 \ \mathrm{Pa} \). This is the pressure needed to prevent a steel block from expanding when heated from \( 20.0^{\circ} \mathrm{C} \) to \( 35.0^{\circ} \mathrm{C} \).

Key Concepts

bulk modulus

The bulk modulus, denoted by the symbol \( B \), is a fundamental property of materials that measures their resistance to uniform compression. It is a type of modulus of elasticity and is particularly important when considering how a material reacts when an external pressure is applied evenly across all its surfaces.
The bulk modulus is defined by the relationship \( B = -\frac{\Delta V}{V} \) over \( \Delta p \), where \( \Delta V \) is the change in volume, \( V \) is the initial volume, and \( \Delta p \) represents the change in pressure. In simpler terms, it describes how much an object will compress under a given amount of external pressure. A higher bulk modulus indicates a less compressible material, meaning it requires more pressure to cause a change in volume.
Understanding bulk modulus is essential when you want to keep an object's volume constant despite increasing temperature. In the context of this exercise, it is used to calculate the additional pressure needed to counteract the natural expansion tendency of materials when they are heated.

thermal expansion

Thermal expansion occurs because most materials increase in size when heated. This expansion happens because heat causes molecules to move more vigorously, thus requiring more space.
For many solids, this expansion is linear to begin with but can become more complex. In most cases considered in physics, when the temperature rises, the length, area, and volume of an object increase. The concept of thermal expansion is crucial in engineering to ensure that different components fit together and function correctly under varying temperature conditions.
In the provided exercise, thermal expansion plays a central role because we're exploring how an increase in temperature causes an increase in volume, unless additional pressure—the topic of focus—is applied to prevent it.

pressure change

Pressure change in the context of this exercise refers to the alteration in force applied per unit area on a surface to counteract unwanted volume expansion due to temperature increase.
When a material is heated and expansion is restricted, as in the scenario given, additional pressure is applied to maintain its original volume. This pressure change can be calculated using the relationship \( \Delta p = B \beta \Delta T \). Here, \( B \) is the bulk modulus, \( \beta \) is the average coefficient of volume expansion, and \( \Delta T \) is the change in temperature.
The ability to calculate this pressure change is particularly useful in applications such as designing systems that can withstand varying environmental conditions without structural failure.

average coefficient of volume expansion

The average coefficient of volume expansion, symbolized as \( \beta \), is a material property that quantifies how much a material's volume increases per degree change in temperature. It serves as a measure of a material's tendency to expand when heated.
This coefficient is crucial for predicting how much an object will expand in a given temperature range and becomes particularly important when calculating changes in pressure needed to prevent expansion.
In this exercise, \( \beta \) is used in conjunction with the bulk modulus and temperature change to determine the pressure change using the formula \( \Delta p = B \beta \Delta T \). Knowing the average coefficient of volume expansion helps in designing construction materials and various mechanical devices that need to withstand high temperatures without changing dimensions.