Calculating The Force To Lift A Cube From The Bottom Of An Aquarium

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Introduction

In this article, we will delve into the physics behind calculating the force required to lift a submerged object from the bottom of a liquid-filled container. Specifically, we will analyze the scenario of a polished cube lying at the bottom of an aquarium. This problem involves understanding the interplay of various forces, including the force of gravity, buoyant force, and the pressure exerted by the surrounding fluid. By carefully considering these factors, we can determine the additional force needed to overcome these opposing forces and lift the cube.

We will explore the concepts of hydrostatic pressure, buoyancy, and the Archimedes' principle to arrive at a comprehensive solution. This exercise not only enhances our understanding of fluid mechanics but also provides a practical application of these principles in a real-world scenario. Let's embark on this fascinating journey of physics and problem-solving.

Problem Statement

Imagine a scenario where a polished cube, with a side length of 10 cm and a mass of 10 kg, rests at the bottom of an aquarium. The aquarium has a smooth, horizontal, and flat bottom. The water depth in the aquarium is maintained at 50 cm, and the atmospheric pressure is 10⁾ Pascals. Our task is to determine the force required to detach this cube from the bottom of the aquarium. Furthermore, we will compare this force with other relevant forces to provide a comprehensive understanding of the situation.

Defining the Parameters

To begin, let's define the key parameters of our problem:

  • Side length of the cube (a): 10 cm = 0.1 m
  • Mass of the cube (m): 10 kg
  • Water depth (h): 50 cm = 0.5 m
  • Atmospheric pressure (P₀): 10⁾ Pa
  • Density of water (ρ): Approximately 1000 kg/mÂł
  • Acceleration due to gravity (g): Approximately 9.81 m/s²

These parameters form the foundation of our calculations. By understanding their roles and relationships, we can accurately determine the force needed to lift the cube.

Key Concepts and Principles

1. Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at a given point due to the weight of the fluid above it. This pressure increases with depth and is a crucial factor in determining the forces acting on the submerged cube. The hydrostatic pressure (P) at a depth (h) in a fluid of density (ρ) is given by:

P = P₀ + ρgh

Where:

*   P₀ is the atmospheric pressure
*   ρ is the density of the fluid
*   g is the acceleration due to gravity
*   h is the depth from the surface of the fluid

2. Buoyant Force (Archimedes' Principle)

The buoyant force is an upward force exerted by a fluid that opposes the weight of an immersed object. This force is equal to the weight of the fluid displaced by the object, a principle known as Archimedes' Principle. The buoyant force (Fb) can be calculated as:

Fb = ρVg

Where:

*   ρ is the density of the fluid
*   V is the volume of the fluid displaced (which is equal to the volume of the submerged object)
*   g is the acceleration due to gravity

3. Force Due to Pressure Difference

When an object is resting on the bottom of a container filled with fluid, the pressure exerted by the fluid on the bottom surface of the object is greater than the pressure exerted on the top surface. This pressure difference creates an upward force that contributes to the force required to lift the object. This force (Fpressure) can be calculated as:

Fpressure = (P_bottom - P_top) * A

Where:

*   P_bottom is the pressure at the bottom surface of the cube
*   P_top is the pressure at the top surface of the cube
*   A is the area of the bottom (or top) surface of the cube

4. Weight of the Cube

The weight of the cube (W) is the force exerted on it by gravity and is calculated as:

W = mg

Where:

*   m is the mass of the cube
*   g is the acceleration due to gravity

Calculating the Required Force

To determine the force needed to lift the cube, we need to consider all the forces acting on it. These include the weight of the cube (W), the buoyant force (Fb), and the force due to the pressure difference (Fpressure).

1. Calculate the Weight of the Cube (W)

Using the formula W = mg, we can calculate the weight of the cube:

W = 10 kg * 9.81 m/s² = 98.1 N

2. Calculate the Volume of the Cube (V)

The volume of the cube can be calculated using the formula V = aÂł, where a is the side length:

V = (0.1 m)Âł = 0.001 mÂł

3. Calculate the Buoyant Force (Fb)

Using the formula Fb = ρVg, we can calculate the buoyant force:

Fb = 1000 kg/m³ * 0.001 m³ * 9.81 m/s² = 9.81 N

4. Calculate the Pressure at the Top and Bottom Surfaces of the Cube

  • Pressure at the bottom surface (P_bottom): The bottom surface is at a depth of 0.5 m, so:

    P_bottom = P₀ + ρgh = 10⁵ Pa + (1000 kg/m³ * 9.81 m/s² * 0.5 m) = 10⁵ Pa + 4905 Pa = 104905 Pa

  • Pressure at the top surface (P_top): The top surface is at a depth of 0.4 m (0.5 m - 0.1 m), so:

    P_top = P₀ + ρgh = 10⁵ Pa + (1000 kg/m³ * 9.81 m/s² * 0.4 m) = 10⁵ Pa + 3924 Pa = 103924 Pa

5. Calculate the Force Due to Pressure Difference (Fpressure)

The area of the cube's surface (A) is a² = (0.1 m)² = 0.01 m². Using the formula Fpressure = (P_bottom - P_top) * A, we get:

Fpressure = (104905 Pa - 103924 Pa) * 0.01 m² = 981 Pa * 0.01 m² = 9.81 N

6. Calculate the Total Upward Force

The total upward force is the sum of the buoyant force and the force due to the pressure difference:

Total Upward Force = Fb + Fpressure = 9.81 N + 9.81 N = 19.62 N

7. Calculate the Force Required to Lift the Cube (F_lift)

The force required to lift the cube is the difference between the weight of the cube and the total upward force:

F_lift = W - (Fb + Fpressure) = 98.1 N - 19.62 N = 78.48 N

Therefore, the force required to detach the cube from the bottom of the aquarium is approximately 78.48 N.

Comparison and Discussion

To put this force into perspective, let's compare it with the weight of the cube and the buoyant force.

  • The weight of the cube is 98.1 N.
  • The buoyant force is 9.81 N.
  • The force due to the pressure difference is 9.81 N.
  • The force required to lift the cube is 78.48 N.

We can see that the buoyant force and the force due to the pressure difference significantly reduce the force required to lift the cube. The force required to lift the cube is less than its actual weight due to these upward forces.

The force due to the pressure difference arises from the fact that the water pressure at the bottom of the cube is higher than at the top. This pressure difference creates an upward force that effectively counteracts a portion of the cube's weight. This phenomenon is a direct consequence of hydrostatic pressure and is crucial in understanding fluid behavior in submerged environments.

Factors Affecting the Force

Several factors can influence the force required to lift the cube:

  1. Density of the Fluid: A denser fluid would exert a greater buoyant force, reducing the lifting force required.
  2. Depth of Submersion: The deeper the cube is submerged, the greater the pressure difference, and consequently, the greater the force due to pressure difference.
  3. Volume of the Cube: A larger cube would displace more water, resulting in a greater buoyant force.
  4. Mass of the Cube: A heavier cube would require a greater lifting force.

Practical Implications

Understanding the principles behind calculating the force required to lift a submerged object has various practical applications in fields such as:

  • Marine Engineering: Designing underwater structures and equipment.
  • Submersible Operations: Calculating buoyancy and stability for submarines and remotely operated vehicles (ROVs).
  • Diving: Understanding the forces acting on divers and equipment at different depths.
  • Naval Architecture: Designing ships that can efficiently displace water and maintain stability.

Conclusion

In conclusion, the force required to detach the polished cube from the bottom of the aquarium is approximately 78.48 N. This force is significantly less than the weight of the cube due to the combined effects of the buoyant force and the force resulting from the pressure difference in the water. By understanding the principles of hydrostatic pressure, buoyancy, and Archimedes' Principle, we can accurately calculate the forces acting on submerged objects and apply this knowledge in various practical scenarios.

This exercise demonstrates the importance of considering all relevant forces when analyzing situations involving fluids and submerged objects. The interplay between gravity, buoyancy, and pressure differences is crucial in understanding the behavior of objects in fluid environments.