Problem 32
Question
The bulk modulus of water is \(2.1 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\). The pressure required to increase the density of water by \(0.1 \%\) is : (a) \(2.1 \times 10^{3} \mathrm{~N} / \mathrm{m}^{2}\) (b) \(2.1 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2}\) (c) \(2.1 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\) (d) \(2.1 \times 10^{2} \mathrm{~N} / \mathrm{m}^{2}\)
Step-by-Step Solution
Verified Answer
The pressure required is \(2.1 \times 10^{6} \text{ N/m}^2\), so the answer is (b).
1Step 1: Identify the Relationship Between Pressure and Density
The bulk modulus (B) is defined as the ratio of the change in pressure (ΔP) to the fractional change in volume (- rac{ΔV}{V}) or density. It is given as:\[ B = - V \cdot \frac{ΔP}{ΔV} \]Since volume and density are inversely proportional, we can also write it in terms of density change as:\[ B = \rho \cdot \frac{ΔP}{Δ\rho} \]
2Step 2: Set Up the Equation Using Given Data
According to the problem, the bulk modulus \( B = 2.1 \times 10^{9} \text{ N/m}^2 \). The density of water is to increase by \( 0.1\% \), which is \( \frac{Δ\rho}{\rho} = 0.001 \). Use the relationship:\[ 2.1 \times 10^{9} = \frac{ΔP}{0.001} \]
3Step 3: Solve for Change in Pressure
Rearrange the equation to solve for \( ΔP \):\[ ΔP = (2.1 \times 10^{9}) \times 0.001 \]Calculate:\[ ΔP = 2.1 \times 10^{6} \text{ N/m}^2 \]
4Step 4: Conclusion and Selection of Correct Option
The pressure required to increase the density of water by \(0.1\%\) is \(2.1 \times 10^{6} \text{ N/m}^2\). Thus, the correct answer is option (b).
Key Concepts
Understanding PressureThe Role of DensityExploring Volume ChangeUnderstanding Density Change
Understanding Pressure
Pressure is a fascinating concept. It describes the amount of force exerted per unit area. In a fluid, pressure acts equally in all directions. This makes it crucial for understanding how substances respond to forces.
The unit of pressure is Pascal (Pa), equivalent to one Newton per square meter. Pressure can influence various physical properties, such as volume and density. When you apply pressure to a material, its atoms or molecules are pushed closer together. This can lead to changes in density and volume, which we'll explore more in subsequent sections.
The unit of pressure is Pascal (Pa), equivalent to one Newton per square meter. Pressure can influence various physical properties, such as volume and density. When you apply pressure to a material, its atoms or molecules are pushed closer together. This can lead to changes in density and volume, which we'll explore more in subsequent sections.
The Role of Density
Density is the mass of a substance per unit of volume. Represented by the Greek letter \( \rho \), density is a key factor in determining how materials interact with external pressures.
When the density of a material increases, its mass is being packed into a smaller volume. Conversely, less dense materials often seem lighter for their size. Changes in density often occur in response to external factors such as pressure or temperature.
When the density of a material increases, its mass is being packed into a smaller volume. Conversely, less dense materials often seem lighter for their size. Changes in density often occur in response to external factors such as pressure or temperature.
- In this exercise, we are considering water's density increasing by 0.1%.
- This small change can influence the pressure needed to achieve it, as seen in the previous solutions.
Exploring Volume Change
Volume change occurs when a substance is compressed or expanded. It is inversely related to density change for constant mass. If you compress something, the volume decreases, and the density increases.
Volume change is crucial when discussing bulk modulus. The bulk modulus \( B \) is a measure of a material's resistance to uniform compression. It's calculated as the ratio of pressure change \( \Delta P \) to the fractional volume decrease \( \frac{\Delta V}{V} \).
For instance, in the exercise given, water's volume would decrease as the density increases due to applied pressure. Appreciating this concept helps us comprehend material behavior under stress.
Volume change is crucial when discussing bulk modulus. The bulk modulus \( B \) is a measure of a material's resistance to uniform compression. It's calculated as the ratio of pressure change \( \Delta P \) to the fractional volume decrease \( \frac{\Delta V}{V} \).
For instance, in the exercise given, water's volume would decrease as the density increases due to applied pressure. Appreciating this concept helps us comprehend material behavior under stress.
Understanding Density Change
Density change is important in physics and engineering. It's often involved when materials are subjected to different environmental conditions like pressure variations. A 0.1% increase in water's density might seem small, but it results from significant pressure application.
To calculate this in terms of bulk modulus \( B \), we use the relationship \( B = \rho \cdot \frac{\Delta P}{\Delta \rho} \). This links density change with pressure and the material's bulk modulus, allowing us to quantify how much pressure is needed to alter the density.
To calculate this in terms of bulk modulus \( B \), we use the relationship \( B = \rho \cdot \frac{\Delta P}{\Delta \rho} \). This links density change with pressure and the material's bulk modulus, allowing us to quantify how much pressure is needed to alter the density.
- In our problem, the required pressure for a 0.1% density increase in water was found to be \( 2.1 \times 10^{6} \text{ N/m}^2 \).
- This demonstrates how sensitive materials can be to pressure changes, even for small density alterations.
Other exercises in this chapter
Problem 28
An equilateral triangle \(A B C\) is formed by joining three rods of equal length and \(D\) is the mid-point of \(A B\). The coefficient of linear expansion for
View solution Problem 29
In an anisotropic medium, the coefficients of linear expansion of a solid are \(\alpha_{1}, \alpha_{2}\) and \(\alpha_{3}\) in three mutually perpendicular dire
View solution Problem 33
At a temperature \(t^{\circ} \mathrm{C}\), a liquid is completely filled in a spherical shell of copper. If \(\Delta T\) increases temperature of the liquid and
View solution Problem 35
Using the following, data, at what temperature will the wood just sink in benzene? Density of wood at \(0^{\circ} \mathrm{C}=8.8 \times 10^{2} \mathrm{~kg} / \m
View solution