Problem 51 - College Physics

Question

An aquarium is filled with a liquid. A cork cube, \(10.0 \mathrm{~cm}\) on a side, is pushed and held at rest completely submerged in the liquid. It takes a force of \(7.84 \mathrm{~N}\) to hold it under the liquid. If the density of cork is \(200 \mathrm{~kg} / \mathrm{m}^{3}\), find the density of the liquid.

Step-by-Step Solution

Answer

The density of the liquid is approximately 800 kg/m³.

Step 1: Understand the Problem

We have a submerged cork cube in an aquarium filled with a liquid. We need to find the density of the liquid. We know the force required to keep the cube submerged, the cube's size, and its density.

Step 2: Calculate the Volume of the Cube

The side of the cube is given as \( 10.0 \text{ cm} \). Convert this to meters: \( 0.10 \text{ m} \). The volume \( V \) of the cube is then \( V = \text{side}^3 = (0.10 \text{ m})^3 = 0.001 \text{ m}^3 \).

Step 3: Determine the Buoyant Force

The buoyant force \( F_b \) acting on the cube is equal to the force needed to hold it submerged, which is given as \( 7.84 \text{ N} \). Archimedes' principle states that the buoyant force is equal to the weight of the liquid displaced by the cube.

Step 4: Apply Archimedes' Principle

According to Archimedes' principle: \( F_b = \rho_l \cdot V \cdot g \), where \( \rho_l \) is the density of the liquid, \( V \) is the volume of the displaced liquid (the same as the volume of the cube), and \( g \) is the acceleration due to gravity \( 9.81 \text{ m/s}^2 \).

Step 5: Solve for the Density of the Liquid

Rearrange the formula to solve for \( \rho_l \):\[ \rho_l = \frac{F_b}{V \, g} = \frac{7.84 \text{ N}}{0.001 \text{ m}^3 \times 9.81 \text{ m/s}^2} \approx 800 \text{ kg/m}^3 \].

Step 6: Analyze the Results

The density of the liquid is found to be approximately 800 kg/m³. This result indicates the liquid is denser than cork (200 kg/m³), which makes sense as cork floats in most liquids.

Key Concepts

Buoyant Force

When an object is submerged in a liquid, it experiences an upward force called the buoyant force.
This force is a result of the pressure difference between the top and bottom of the object.
Archimedes' Principle gives us the key to understanding buoyant force:

• It states that the buoyant force is equal to the weight of the liquid displaced by the object.

For a cork cube held underwater, the buoyant force helps determine how much force is needed to keep it submerged.
The cube displaces a volume of liquid equal to its own volume, and this displacement causes the buoyant force.
In our example, the buoyant force is equal to the force required to hold the cork cube submerged, which is 7.84 N.

Density Calculation

Density is a property of matter defined as mass per unit volume. It helps determine how substances interact, like floating or sinking.
The formula for density \( \rho \) is:

• \( \rho = \frac{m}{V} \)

Where \( m \) is mass and \( V \) is volume.
In the exercise, to find the density of the liquid, Archimedes' principle provides a useful insight:

• The buoyant force \( F_b \) equals the weight of the displaced liquid, calculated using \( \rho_l \cdot V \cdot g \).
• Rearranging the formula gives the liquid density \( \rho_l = \frac{F_b}{V \, g} \).

Given the buoyant force, volume of the cube, and gravity, it's straightforward to calculate the density.

Volume of Cube

A cube's volume is found by raising its side length to the power of three.
This is a straightforward calculation because of the cube's regular shape.
For example, if each side of a cube is 10 cm, the volume \( V \) in cubic meters is:

• First, convert the side length to meters: \( 10.0 \text{ cm} = 0.10 \text{ m} \)
• Then, calculate the volume: \( V = (0.10 \text{ m})^3 \)
• Resulting in \( V = 0.001 \text{ m}^3 \)

Calculating volume accurately is essential when dealing with buoyant forces and density calculations, as seen in the exercise. Understanding volume in practical applications helps explain why certain objects float or sink.