Calculating Force To Lift A Cube From Aquarium Bottom A Physics Problem

Jul 7, 2025 by Admin 72 views

Calculation of Force Required to Lift a Cube from the Bottom of an Aquarium

Introduction

In this article, we will delve into the physics behind calculating the force required to detach a polished cube from the bottom of an aquarium. This is a classic problem that combines concepts from fluid mechanics, pressure, and buoyancy. We will explore the various forces acting on the cube, including the gravitational force (weight), the buoyant force exerted by the water, and the force due to hydrostatic pressure. Understanding these forces is crucial to determining the net force required to lift the cube.

This problem is not just a theoretical exercise; it has practical applications in various fields, such as marine engineering, underwater construction, and even aquarium maintenance. For instance, engineers need to understand these principles when designing structures that are submerged in water, and aquarium enthusiasts may encounter similar scenarios when rearranging decorations or equipment in their tanks. By breaking down the problem step by step, we will provide a clear and comprehensive solution that is accessible to both students and professionals alike.

Our focus will be on a specific scenario: a polished cube with a side length of 10 cm and a mass of 10 kg resting at the bottom of an aquarium filled with water to a depth of 50 cm. The atmospheric pressure is given as 10⁵ Pascals. We will calculate the force needed to lift this cube and compare it with other relevant forces to provide a complete understanding of the situation. This article aims to provide a detailed and insightful analysis of the problem, making it easier for readers to grasp the underlying principles and apply them to similar situations.

Problem Statement

Let's clearly define the problem we aim to solve. We have a polished cube with a side length (*s*) of 10 cm (0.1 m) and a mass (*m*) of 10 kg resting at the bottom of an aquarium. The aquarium is filled with water to a depth (*h*) of 50 cm (0.5 m). The atmospheric pressure (*P₀*) is 10⁵ Pascals. Our objective is to calculate the force (*F*) required to detach the cube from the bottom of the aquarium.

To solve this, we need to consider several factors. First, the cube experiences a gravitational force (*Fg*), which is its weight, pulling it downwards. Second, the water exerts a buoyant force (*Fb*) on the cube, pushing it upwards. Third, the hydrostatic pressure of the water at the bottom of the aquarium creates a force pressing the cube against the bottom. We need to account for all these forces to determine the net force required to lift the cube.

The problem also requires us to compare this lifting force with other relevant forces. This comparison will help us understand the relative magnitudes of the forces involved and provide a more intuitive sense of the situation. For example, we might compare the lifting force with the weight of the cube or the force due to the hydrostatic pressure. Such comparisons can highlight the significance of each force and how they interact to affect the cube's behavior.

By providing a clear problem statement, we set the stage for a detailed analysis and calculation. The subsequent sections will break down the problem into manageable steps, calculating each force individually before combining them to find the final answer. This structured approach will ensure a thorough and accurate solution.

Forces Acting on the Cube

To determine the force required to lift the cube, we must first identify and calculate all the forces acting upon it. There are three primary forces to consider:

Gravitational Force (Weight) (*Fg*): This is the force due to gravity pulling the cube downwards. It is calculated using the formula:*Fg = m * g*, where *m* is the mass of the cube and *g* is the acceleration due to gravity (approximately 9.81 m/s²).
Buoyant Force (*Fb*): This is the upward force exerted by the water on the cube. It is equal to the weight of the water displaced by the cube. The buoyant force is calculated using Archimedes' principle:*Fb = ρ * V * g*, where *ρ* is the density of water (approximately 1000 kg/m³), *V* is the volume of the cube, and *g* is the acceleration due to gravity.
Force due to Hydrostatic Pressure (*Fp*): This force is caused by the pressure of the water at the bottom of the aquarium pressing on the cube. The hydrostatic pressure (*P*) at a depth (*h*) in a fluid is given by:*P = P₀ + ρ * g * h*, where *P₀* is the atmospheric pressure, *ρ* is the density of water, *g* is the acceleration due to gravity, and *h* is the depth of the water. The force due to this pressure acting on the bottom surface of the cube (*A*) is:*Fp = P * A*. This force acts downwards, pressing the cube against the bottom of the aquarium.

By understanding these three forces, we can begin to quantify their magnitudes and directions. This is a crucial step in determining the net force required to lift the cube. In the following sections, we will calculate each of these forces in detail, using the given parameters of the problem.

Calculations

Now, let's calculate each of the forces acting on the cube using the formulas and values provided. This step-by-step calculation will help us understand the magnitude of each force and their overall impact on the cube.

1. Gravitational Force (Fg)

The gravitational force, or weight, of the cube is calculated using the formula:*Fg = m * g*, where *m* is the mass of the cube (10 kg) and *g* is the acceleration due to gravity (9.81 m/s²).

*Fg = 10 * 9.81 = 98.1 * Newtons*

So, the gravitational force pulling the cube downwards is 98.1 Newtons.

2. Buoyant Force (Fb)

The buoyant force is calculated using Archimedes' principle:*Fb = ρ * V * g*, where *ρ* is the density of water (1000 kg/m³), *V* is the volume of the cube, and *g* is the acceleration due to gravity (9.81 m/s²).

The volume of the cube (*V*) is calculated as:*V = s³*, where *s* is the side length of the cube (0.1 m).

*V = (0.1)³ = 0.001 * m³*

Now we can calculate the buoyant force:

*Fb = 1000 * 0.001 * 9.81 = 9.81 * Newtons*

Thus, the buoyant force pushing the cube upwards is 9.81 Newtons.

3. Force due to Hydrostatic Pressure (Fp)

The hydrostatic pressure (*P*) at the bottom of the aquarium is given by:*P = P₀ + ρ * g * h*, where *P₀* is the atmospheric pressure (10⁵ Pa), *ρ* is the density of water (1000 kg/m³), *g* is the acceleration due to gravity (9.81 m/s²), and *h* is the depth of the water (0.5 m).

*P = 10⁵ + (1000 * 9.81 * 0.5) = 10⁴ + 4905 = 104905 * Pascals*

The area (*A*) of the bottom surface of the cube is:

*A = s² = (0.1)² = 0.01 * m²*

The force due to hydrostatic pressure (*Fp*) is then:

*Fp = P * A = 104905 * 0.01 = 1049.05 * Newtons*

So, the force due to hydrostatic pressure pressing the cube against the bottom is 1049.05 Newtons.

Force Required to Detach the Cube

Now that we have calculated the individual forces acting on the cube, we can determine the force required to detach it from the bottom of the aquarium. This involves understanding how these forces interact and the net effect they have on the cube.

The forces acting on the cube are as follows:

Gravitational Force (*Fg*): 98.1 N (downwards)
Buoyant Force (*Fb*): 9.81 N (upwards)
Force due to Hydrostatic Pressure (*Fp*): 1049.05 N (downwards)

The gravitational force pulls the cube downwards, while the buoyant force pushes it upwards. The hydrostatic pressure also presses the cube downwards against the bottom of the aquarium. To lift the cube, we need to overcome the combined effect of the gravitational force and the hydrostatic pressure, while also considering the opposing buoyant force.

The net downward force (*F_down*) is the sum of the gravitational force and the hydrostatic pressure force:

*F_down = Fg + Fp = 98.1 + 1049.05 = 1147.15 * Newtons*

The net upward force is the buoyant force (*Fb*) which is 9.81 N.

The force required to detach the cube (*F_detach*) must counteract the net downward force minus the buoyant force:

*F_detach = F_down - Fb = 1147.15 - 9.81 = 1137.34 * Newtons*

Therefore, the force required to detach the cube from the bottom of the aquarium is approximately 1137.34 Newtons. This calculation accounts for the weight of the cube, the upward buoyant force, and the significant force exerted by the hydrostatic pressure at the bottom of the aquarium.

Comparison with Other Forces

To better understand the magnitude of the force required to detach the cube, it is helpful to compare it with other relevant forces. This comparison will provide a more intuitive sense of the forces at play and their relative significance.

1. Comparison with the Weight of the Cube

The weight of the cube (*Fg*) is 98.1 Newtons. The force required to detach the cube (1137.34 N) is significantly larger than its weight. This difference highlights the impact of the hydrostatic pressure, which is the primary force pressing the cube against the bottom of the aquarium. The hydrostatic pressure alone contributes a force of 1049.05 N, which is more than ten times the weight of the cube.

2. Comparison with the Buoyant Force

The buoyant force (*Fb*) is 9.81 Newtons, which is relatively small compared to both the weight of the cube and the detachment force. The buoyant force partially counteracts the gravitational force, but its effect is minimal in this scenario due to the high hydrostatic pressure.

3. Significance of Hydrostatic Pressure

The hydrostatic pressure exerts a force of 1049.05 Newtons, which is the dominant factor in determining the detachment force. This force arises from the weight of the water column above the cube and the atmospheric pressure acting on the water surface. The depth of the water (50 cm) is sufficient to create a substantial pressure at the bottom of the aquarium, resulting in a significant force pressing the cube against the bottom.

4. Practical Implications

The high force required to detach the cube has practical implications. For example, if one were to try to lift the cube manually, it would require a considerable effort due to the hydrostatic pressure. This is a common consideration in underwater construction and maintenance, where objects at significant depths can be difficult to move due to the water pressure.

By comparing the detachment force with other forces, we gain a comprehensive understanding of the physical dynamics involved. The hydrostatic pressure is the key factor in this scenario, emphasizing the importance of considering fluid pressure in submerged systems.

Conclusion

In this article, we have thoroughly analyzed the forces acting on a polished cube resting at the bottom of an aquarium and calculated the force required to detach it. We identified three primary forces: the gravitational force (weight), the buoyant force, and the force due to hydrostatic pressure. Through detailed calculations, we found that the gravitational force is 98.1 N, the buoyant force is 9.81 N, and the force due to hydrostatic pressure is a substantial 1049.05 N.

The calculation of the detachment force involved understanding the interplay of these forces. The net downward force, combining gravity and hydrostatic pressure, amounted to 1147.15 N. By subtracting the buoyant force, we determined that the force required to detach the cube from the bottom of the aquarium is approximately 1137.34 Newtons. This value is significantly higher than the cube's weight, primarily due to the considerable hydrostatic pressure at the 50 cm depth.

Our comparison with other forces underscored the dominance of hydrostatic pressure in this scenario. The hydrostatic pressure force is more than ten times the weight of the cube, highlighting the significant impact of fluid pressure at depth. This observation has practical implications in various fields, such as underwater engineering, where understanding and managing hydrostatic forces are crucial for the design and operation of submerged structures.

This analysis not only provides a quantitative answer to the problem but also offers a deeper understanding of the underlying physical principles. By breaking down the problem into manageable steps and comparing the results with other relevant forces, we have provided a comprehensive and insightful solution. This approach can be applied to similar problems involving submerged objects and fluid dynamics, making it a valuable exercise for students and professionals alike. The key takeaway is the significant role of hydrostatic pressure in submerged systems, which must be carefully considered in any practical application.