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Chapter 9: Problem 51

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.

Short Answer

Expert verified
The density of the liquid is approximately 800 kg/m鲁.

Step by step solution

01

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.
02

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 \( V = \text{side}^3 = (0.10 \text{ m})^3 = 0.001 \text{ m}^3 \).
03

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.
04

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 \( 9.81 \text{ m/s}^2 \).
05

Solve for the Density of the Liquid

Rearrange the formula to solve for \( \rho_l \):\[ \rho_l = \frac{F_b}{V \, g} = \frac{7.84 \text{ N}}{0.001 \text{ m}^3 \times 9.81 \text{ m/s}^2} \approx 800 \text{ kg/m}^3 \].
06

Analyze the Results

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

(a) Given a piece of metal with a light string attached, a scale, and a container of water in which the piece of metal can be submersed, how could you find the volume of the piece without using the variation in the water level? (b) An object has a weight of \(0.882 \mathrm{~N}\). It is suspended from a scale, which reads \(0.735 \mathrm{~N}\) when the piece is submerged in water. What are the volume and density of the piece of metal?

When railroad tracks are installed, gaps are left between the rails. (a) Should a greater gap be used if the rails are installed on (1) a cold day or (2) a hot day? Or (3) does the temperature not make any difference? Why? (b) Each steel rail is \(8.0 \mathrm{~m}\) long and has a cross-sectional area of \(0.0025 \mathrm{~m}^{2}\). On a hot day, each rail thermally expands as much as \(3.0 \times 10^{-3} \mathrm{~m}\) If there were no gaps between the rails, what would be the force on the ends of each rail?

The gauge pressure in both tires of a bicycle is \(690 \mathrm{kPa}\). If the bicycle and the rider have a combined mass of \(90.0 \mathrm{~kg}\), what is the area of contact of each tire with the ground? (Assume that each tire supports half the total weight of the bicycle.)

The speed of blood in a major artery of diameter \(1.0 \mathrm{~cm}\) is \(4.5 \mathrm{~cm} / \mathrm{s}\). (a) What is the flow rate in the artery? (b) If the capillary system has a total cross-sectional area of \(2500 \mathrm{~cm}^{2}\), the average speed of blood through the capillaries is what percentage of that through the major artery? (c) Why must blood flow at low speed through the capillaries?

Here is a demonstration Pascal used to show the importance of a fluid's pressure on the fluid's depth (vFig. 9.36): An oak barrel with a lid of area \(0.20 \mathrm{~m}^{2}\) is filled with water. A long, thin tube of cross- sectional area \(5.0 \times 10^{-5} \mathrm{~m}^{2}\) is inserted into a hole at the center of the lid, and water is poured into the tube. When the water reaches \(12 \mathrm{~m}\) high, the barrel bursts. (a) What was the weight of the water in the tube? (b) What was the pressure of the water on the lid of the barrel? (c) What was the net force on the lid due to the water pressure?
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