Calculating Water Level Rise When a Cube is Immersed in a Rectangular Vessel

In this article, we will explore the basic principles of fluid mechanics and specifically how to calculate the rise in water level when a cube is immersed in a rectangular vessel containing water. This concept is fundamental to understanding various real-world applications, from everyday tasks to more complex engineering problems.

Introduction to the Problem

Consider a cube that has an edge length of 9 cm. This cube is completely immersed in a rectangular vessel that contains water. The dimensions of the base of the vessel are 15 cm in length and 12 cm in width. Our goal is to calculate the rise in the water level due to the immersion of the cube.

Step-by-Step Calculation

Step 1: Calculate the Volume of the Cube

The volume of a cube is given by the formula:

V edge^3

Given the edge length of the cube is 9 cm, we calculate the volume as:

V 9^3 729 cm^3

Step 2: Calculate the Area of the Base of the Rectangular Vessel

The area of the base of the rectangular vessel is calculated using the formula:

A length times; width

Given the length and width are 15 cm and 12 cm respectively, we find:

A 15 cm times; 12 cm 180 cm^2

Step 3: Calculate the Rise in Water Level

The rise in the water level can be calculated using the formula:

h V / A

Substituting the values we calculated:

h 729 cm^3 / 180 cm^2 4.05 cm

Conclusion

The rise in the water level when the cube is completely submerged is 4.05 cm. This calculation is based on the principle that the volume of the cube that is displaced is equal to the volume of water displaced, which in turn raises the water level in the rectangular vessel.

Alternative Calculations

There are several ways to approach this problem, and here are a few additional methods to derive the same conclusion:

Alternative Method 1: Direct Calculation of Water Displaced by Cube

Volume of water displaced by the cube:

Volume of Cube 9^3 729 cm^3

Bottom area of the rectangular vessel:

15 cm times; 12 cm 180 cm^2

Rise in water level:

h 729 cm^3 / 180 cm^2 4.05 cm

Alternative Method 2: Cross-Sectional Approach

The cross-sectional area of the standing water column in the rectangular vessel is:

15 cm times; 12 cm 180 cm^2

When the cube is immersed in water in the vessel, the water level rises by:

729 cm^3 / 180 cm^2 4.05 cm

Conclusion

Regardless of the method used, the rise in the water level when the cube is completely submerged is consistently found to be 4.05 cm, confirming the initial calculation.

Key Concepts to Remember

Volume of the Cube: Calculated using the formula V edge^3. Area of the Base of the Vessel: Calculated using the formula A length times; width. Rise in Water Level: Calculated using the formula h V / A.

FAQs

Q: What is the principle behind this calculation? A: The principle is the displacement of water by an object of known volume, as described by Archimedes' principle. Q: Can this calculation be applied to other shapes besides a cube? A: Yes, the same principle applies to any solid object. The volume of the object is used to calculate the displacement and subsequent rise in the water level. Q: How can this concept be practically applied? A: This concept is used in various fields, such as hydrology, agriculture, and civil engineering, to measure the volume of fluids or solids.

Conclusion Summary

In conclusion, the rise in water level when a cube of 9 cm edge is completely immersed in a rectangular vessel is 4.05 cm. This practical example helps us understand the fundamental relationship between the volume of an object and the effect it has on the displacement of water, which is a critical concept in fluid mechanics.