Calculating the Density of Salt Water

Oct 13, 2025 by windailys.com 63 views

Introduction

In this problem, we are given a cylindrical swimming pool with a radius of 2 m and a depth of 1.3 m. The pool is filled completely with salt water of specific gravity 1.03. We are also given the atmospheric pressure as $1.013 \times 10^5 \, \text{Pa}$ . Our goal is to calculate the density of the salt water in the pool.

Calculating the Density of Salt Water

To calculate the density of the salt water, we need to use the formula for density, which is given by:

\rho = \frac{m}{V}

where $\rho$ is the density, $m$ is the mass, and $V$ is the volume.

However, we are not given the mass of the salt water directly. Instead, we are given the specific gravity, which is the ratio of the density of the salt water to the density of fresh water. The density of fresh water is approximately $1000 \, \text{kg/m}^3$ .

Calculating the Volume of the Pool

To calculate the volume of the pool, we need to use the formula for the volume of a cylinder, which is given by:

V = \pi r^2 h

where $V$ is the volume, $r$ is the radius, and $h$ is the height.

In this case, the radius of the pool is 2 m and the depth is 1.3 m. Plugging these values into the formula, we get:

V = \pi (2 \, \text{m})^2 (1.3 \, \text{m})

V = \pi (4 \, \text{m}^2) (1.3 \, \text{m})

V = 16.26 \, \text{m}^3

Calculating the Mass of the Salt Water

Now that we have the volume of the pool, we can calculate the mass of the salt water using the formula for density:

\rho = \frac{m}{V}

Rearranging this formula to solve for mass, we get:

m = \rho V

We are given the specific gravity of the salt water, which is 1.03. This means that the density of the salt water is 1.03 times the density of fresh water. The density of fresh water is approximately $1000 \, \text{kg/m}^3$ , so the density of the salt water is:

\rho = 1.03 \times 1000 \, \text{kg/m}^3

\rho = 1030 \, \text{kg/m}^3

Now we can plug in the values for the volume and the density to calculate the mass:

m = 1030 \, \text{kg/m}^3 \times 16.26 \, \text{m}^3

m = 16831.8 \, \text{kg}

Calculating the Pressure at the Bottom of the Pool

The pressure at the bottom of the pool is due to the weight of the water above it, as well as the atmospheric pressure. We can calculate the pressure due to the weight of the water using the formula:

P = \rho g h

where $P$ is the pressure, $\rho$ is the density of the water, $g$ is the acceleration due to gravity, and $h$ is the height of the water.

In this case, the density of the water is 1030 kg/m^3, the acceleration due to gravity is 9.8 m/s^2, and the height of the water is 1.3 m. Plugging these values into the formula, we get:

P = 1030 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 \times 1.3 \, \text{m}

P = 13351.4 \, \text{Pa}

The atmospheric pressure is given as $1.013 \times 10^5 \, \text{Pa}$ . To find the total pressure at the bottom of the pool, we add the pressure due to the weight of the water to the atmospheric pressure:

P_{\text{total}} = P + P_{\text{atmospheric}}

P_{\text{total}} = 13351.4 \, \text{Pa} + 1.013 \times 10^5 \, \text{Pa}

P_{\text{total}} = 1.013 \times 10^5 \, \text{Pa} + 13351.4 \, \text{Pa}

P_{\text{total}} = 1.01652 \times 10^5 \, \text{Pa}

Conclusion

In this problem, we calculated the density of the salt water in a cylindrical swimming pool. We used the formula for density and the specific gravity of the salt water to calculate the density. We also calculated the volume of the pool and the mass of the salt water. Finally, we calculated the pressure at the bottom of the pool due to the weight of the water and the atmospheric pressure.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers. Cengage Learning.

Note

Q&A: Frequently Asked Questions

Q: What is the formula for calculating the density of a substance? A: The formula for calculating the density of a substance is:

\rho = \frac{m}{V}

where $\rho$ is the density, $m$ is the mass, and $V$ is the volume.

Q: How do I calculate the volume of a cylinder? A: To calculate the volume of a cylinder, you can use the formula:

V = \pi r^2 h

where $V$ is the volume, $r$ is the radius, and $h$ is the height.

Q: What is the specific gravity of salt water? A: The specific gravity of salt water is 1.03, which means that the density of salt water is 1.03 times the density of fresh water.

Q: How do I calculate the mass of a substance given its density and volume? A: To calculate the mass of a substance given its density and volume, you can use the formula:

m = \rho V

Q: What is the pressure at the bottom of a pool due to the weight of the water? A: The pressure at the bottom of a pool due to the weight of the water can be calculated using the formula:

P = \rho g h

where $P$ is the pressure, $\rho$ is the density of the water, $g$ is the acceleration due to gravity, and $h$ is the height of the water.

Q: How do I calculate the total pressure at the bottom of a pool? A: To calculate the total pressure at the bottom of a pool, you need to add the pressure due to the weight of the water to the atmospheric pressure.

Q: What is the atmospheric pressure? A: The atmospheric pressure is the pressure exerted by the atmosphere on the surface of the Earth. It is typically given as $1.013 \times 10^5 \, \text{Pa}$ .

Q: How do I calculate the density of a substance given its specific gravity? A: To calculate the density of a substance given its specific gravity, you can use the formula:

\rho = \text{specific gravity} \times \rho_{\text{fresh water}}

where $\rho$ is the density, specific gravity is the ratio of the density of the substance to the density of fresh water, and $\rho_{\text{fresh water}}$ is the density of fresh water.

Q: What is the significance of the density of a substance? A: The density of a substance is an important property that can affect its behavior in various situations. For example, the density of a substance can affect its buoyancy in water, its ability to conduct heat, and its reactivity with other substances.

Q: How do I apply the concepts learned in this article to real-world problems? A: The concepts learned in this article can be applied to various real-world problems, such as designing swimming pools, calculating the pressure at the bottom of a pool, and understanding the behavior of substances in different environments.

Conclusion

In this article, we have discussed the concept of density and its importance in various situations. We have also provided a step-by-step guide on how to calculate the density of a substance given its specific gravity and volume. Additionally, we have answered frequently asked questions related to the topic. We hope that this article has provided you with a better understanding of the concept of density and its applications in real-world problems.