Problem 65 - College Physics

Question

In seawater, a life preserver with a volume of 0.0400 \(\mathrm{m}^{3}\) will support a 75.0 \(\mathrm{kg}\) person (average density 980 \(\mathrm{kg} / \mathrm{m}^{3} )\) with 20\(\%\) of the person's volume above water when the life preserver is fully submerged. What is the density of the material composing the life preserver?

Step-by-Step Solution

Answer

The density of the life preserver material is 719.5 kg/m³.

Step 1: Calculate the Volume of the Person

The volume of the person can be found using their mass and density. The formula to find volume is \( V = \frac{m}{\rho} \). So, the volume of the person \( V_{person} = \frac{75.0}{980} = 0.0765 \, \mathrm{m}^3 \).

Step 2: Determine Submerged Volume of the Person

Since 20% of the person's volume is above water, the submerged volume is 80% of the total volume. Thus, \( V_{submerged, person} = 0.8 \times 0.0765 = 0.0612 \, \mathrm{m}^3 \).

Step 3: Apply Buoyancy Principle

For the system to float, the buoyant force must equal the gravitational force on the person plus the life preserver. The buoyant force is due to the total submerged volume of the life preserver and person \( V_{submerged, total} = 0.0400 + 0.0612 = 0.1012 \, \mathrm{m}^3 \).

Step 4: Calculate Mass of Displaced Seawater

The density of seawater is about 1025 \( \mathrm{kg/m^3} \). The mass of the displaced seawater is \( m_{seawater} = 1025 \times 0.1012 = 103.78 \, \mathrm{kg} \).

Step 5: Determine Mass of Life Preserver

Since the buoyant force equals the weight of the person plus the life preserver, we find the mass of the life preserver by subtracting the mass of the person from the displaced water's mass: \( m_{preserver} = 103.78 - 75 = 28.78 \, \mathrm{kg} \).

Step 6: Calculate Density of Life Preserver Material

Density is mass divided by volume. The density of the life preserver \( \rho_{preserver} = \frac{m_{preserver}}{V_{preserver}} = \frac{28.78}{0.0400} = 719.5 \, \mathrm{kg/m^3} \).

Key Concepts

Density calculations

Density is a crucial concept in physics and helps us understand how mass is distributed in a given volume. To calculate density (\( \rho \)), you need to use the formula:

• \( \rho = \frac{m}{V} \)

where \( m \) is the mass and \( V \) is the volume.
In our example, a life preserver must have a calculated density to ensure it assists a person in floating. By finding the mass of the life preserver and dividing it by its volume, we can calculate its density.
Understanding density also helps explain why certain objects float or sink in fluids.
In simpler terms, if an object's density is less than the fluid it is in, it will float. Here, the life preserver must have a lower density than seawater for it to support the person on water successfully.

Buoyant force

Buoyant force is the upward push a fluid exerts on an object submerged in it. This is the reason objects feel lighter in water.
The buoyant force can be understood by looking at how much fluid an object displaces.
The key to buoyant force is that it equals the weight of the fluid that the object displaces, which is fundamental in determining if an object will float or sink. When you place the life preserver and the person in water, they displace a certain volume of seawater. The weight of this displaced seawater equals the buoyant force applied by the water.
So, when the total weight of the person and the life preserver equals the buoyant force, the system floats.
This balance leads to 80% of the person's volume being submerged while the rest remains above the water, maintaining buoyancy.

Archimedes' Principle

Archimedes' Principle is a vital principle in fluid mechanics that helps us understand buoyancy. It states that an object wholly or partially submerged in a fluid experiences an upward force known as the buoyant force. This force is equal to the weight of the fluid displaced by the object.
For example, when you immerse a life preserver and a person in water, they push aside a certain volume of that water.
According to Archimedes' Principle:

• The weight of the displaced water is equal to the buoyant force acting on the submerged object.

In our case, the displacement of 0.1012 \( \mathrm{m}^3 \) of seawater results in a buoyant force that precisely balances the combined weight of the person and the life preserver.
By understanding this principle, we can predict and analyze the floating and sinking behavior of various objects when placed in a fluid.