Problem 42 - Fundamentals of Physics

Question

A flotation device is in the shape of a right cylinder, with a height of \(0.500 \mathrm{~m}\) and a face area of \(4.00 \mathrm{~m}^{2}\) on top and bottom, and its density is \(0.400\) times that of fresh water. It is initially held fully submerged in fresh water, with its top face at the water surface. Then it is allowed to ascend gradually until it begins to float. How much work does the buoyant force do on the device during the ascent?

Step-by-Step Solution

Answer

The buoyant force does 9810 J of work on the device.

Step 1: Understand the Given Problem

We have a cylindrical object fully submerged in water. The object has a height of 0.5 m, a top and bottom surface area of 4.00 m², and a density equal to 0.4 of the density of fresh water. We need to find the work done by the buoyant force as the cylinder ascends to float.

Step 2: Calculate Volume and Mass of the Cylinder

The volume of the cylinder is calculated using the formula \( V = ext{Area} \times ext{Height} \), which gives \( V = 4.00 ext{ m}^2 \times 0.500 ext{ m} = 2.00 ext{ m}^3 \). The mass of the cylinder can be found from its density: \( m = \text{Density} \times V = 0.4 \rho_{\text{water}} \times 2.00 \). If \( \rho_{\text{water}} = 1000 ext{ kg/m}^3 \), then \( m = 0.4 \times 1000 \times 2.00 = 800 \text{ kg} \).

Step 3: Calculate Buoyant Force

The buoyant force is equal to the weight of the displaced fluid, given by \( F_b = \rho_{\text{water}} \times V \times g \), where \( g = 9.81 ext{ m/s}^2 \) is the acceleration due to gravity. Thus, \( F_b = 1000 \times 2.00 \times 9.81 = 19620 \text{ N} \).

Step 4: Determine Work Done During Ascent

As the object rises to the surface, the buoyant force does work in lifting the cylinder. The work done by the buoyant force is given by the formula \( W = F_b \times d \), where \( d \) is the distance the center of gravity of the object moves, which is the height of the cylinder. Thus, \( W = 19620 \times 0.500 = 9810 \text{ J} \).

Step 5: Conclusion

The work done by the buoyant force is equal to the buoyant force times the height that the object travels during ascent until it starts floating. Here, the distance is equal to its height of 0.5 m because it starts fully submerged and ends with its top at the water surface.

Key Concepts

Flotation

Flotation is a fascinating concept related to objects submerged in or floating on fluids. When an object is placed in a fluid, it experiences an upward force known as the buoyant force. The object's ability to stay afloat, or its flotation, depends on this force.

For an object to float, the buoyant force must balance the gravitational force pulling it downward. In simpler words, if the weight of the fluid displaced by an object is greater than the object’s weight, it will float. This principle is governed by Archimedes' principle.

In the context of the exercise, once the cylinder is free to move, the buoyant force works against gravity, pushing the cylinder upwards until it begins to float. At this point, the upward buoyant force equals the gravitational force, leading to equilibrium and flotation.

Cylinder Volume

Understanding the volume of a cylinder is crucial when considering its interaction with fluids. The volume of a cylinder is determined by its base area and height. Mathematically, it can be expressed as:
\[ V = ext{Area} imes ext{Height} \]

This formula gives us the internal space it occupies, which is essential for calculating how much fluid the cylinder displaces. More displaced water leads to more buoyant force.

In the given problem, the cylinder's volume is calculated using its top and bottom face area of 4 square meters and its height of 0.5 meters. Therefore, the volume comes out to be 2 cubic meters.

The volume not only helps in determining the buoyant force but also plays a role in understanding how deeply the object initially submerges and how it behaves upon being released.

Work Done

Work done is an essential concept in physics, especially when analyzing forces and motion. It is defined as the product of force and the distance over which it acts. In the case of the cylinder in water, we're interested in how much work the buoyant force does as the cylinder moves.

The equation for work done is: \[ W = F imes d \] Where \( F \) represents the force, and \( d \) represents the distance the object moves in the direction of the force.

In the exercise, the work done by the buoyant force is calculated by multiplying the buoyant force by the distance, which is the height of the cylinder. This tells us the energy transferred by the buoyant force as it lifts the cylinder upwards.

• The work done is crucial for understanding how forces affect motion.
• It's an important measure of energy transfer within a system.

Density of Fluids

Density is a characteristic of matter that relates mass to volume, expressed as \( \rho = \frac{m}{V} \). For fluids, density determines how it interacts with submerged or floating matter.

The density of a fluid affects the buoyant force exerted on an object. Higher fluid density results in greater buoyant force.

In the problem, the fluid is freshwater, typically with a density of 1000 kg/m³. The device's density is 0.4 times this, making it less dense than the water, which is why it eventually floats after being released.

Understanding density allows us:

• To predict buoyancy
• To determine the mass of fluids displaced
• To assess whether an object will float or sink

These insights are crucial for applications involving ships, watercrafts, and even in designing flotation devices.