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Chapter 12: Problem 27

An ore sample weighs 17.50 N in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 11.20 N. Find the total volume and the density of the sample.

Short Answer

Expert verified
The volume is approximately \(6.42 \times 10^{-4} \, \text{m}^3\) and the density is approximately \(2779 \, \text{kg/m}^3\).

Step by step solution

01

Understand the Problem

We are given that the ore sample weighs 17.50 N in air and that the tension in the cord when the sample is immersed in water is 11.20 N. We need to find the volume and density of the ore sample.
02

Calculate Buoyant Force

The buoyant force can be found using the difference in the weight of the sample in air and the tension in the cord when immersed in water. The formula is:\[ F_b = W_{air} - T_{water} \]Substitute the given values:\[ F_b = 17.50 \, \text{N} - 11.20 \, \text{N} = 6.30 \, \text{N} \]
03

Use Archimedes' Principle to Find Volume

According to Archimedes' principle, the buoyant force is equal to the weight of the water displaced by the sample. The weight of the displaced water can be expressed as:\[ F_b = \rho_{water} \cdot V \cdot g \]Assuming the density of water, \(\rho_{water} = 1000 \, \text{kg/m}^3\) and \(g = 9.81 \, \text{m/s}^2\), we can solve for the volume \(V\):\[ V = \frac{F_b}{\rho_{water} \cdot g} = \frac{6.30}{1000 \cdot 9.81} \, \text{m}^3 \]\[ V \approx 6.42 \times 10^{-4} \, \text{m}^3 \]
04

Calculate Density of the Sample

The density of the sample can be calculated using its mass and the volume found in the previous step. First, find the mass:\[ m = \frac{W_{air}}{g} = \frac{17.50}{9.81} \, \text{kg} \]\[ m \approx 1.784 \text{ kg} \]Now calculate the density \(\rho\) using the formula:\[ \rho = \frac{m}{V} = \frac{1.784}{6.42 \times 10^{-4}} \, \text{kg/m}^3 \]\[ \rho \approx 2779 \, \text{kg/m}^3 \]

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Key Concepts

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

Understanding Buoyant Force
Buoyant force is a core concept in physics related to floating and sinking objects in fluids. Archimedes' Principle helps us understand this phenomenon. It states that any object submerged in a fluid experiences an upward, or buoyant, force equal to the weight of the fluid displaced by the object.
In practical terms, like in the example of the ore sample, you can calculate the buoyant force by subtracting the tension in the cord when the sample is immersed in water from its weight in air:
  • Weight in air: 17.50 N
  • Tension when immersed: 11.20 N
So, the buoyant force is found using the formula:\[ F_b = W_{air} - T_{water} = 17.50 \, \text{N} - 11.20 \, \text{N} = 6.30 \, \text{N} \]This buoyant force helps determine other factors like the volume of the displaced fluid, assisting in calculations about the object's density.
Mastering Density Calculation
Think of density as how much stuff is packed into a given space. It鈥檚 pivotal in identifying material properties. Mathematically, density \(\rho\) is the object's mass divided by its volume:\[ \rho = \frac{m}{V} \]
In the case of the ore sample, using the given gravitational force, the object's mass \(m\) is found as follows:\[ m = \frac{W_{air}}{g} = \frac{17.50}{9.81} \, \text{kg} \approx 1.784 \, \text{kg} \]
After the volume is determined through buoyant force calculations, the density is calculated:\[ \rho = \frac{1.784}{6.42 \times 10^{-4}} \, \text{kg/m}^3 \approx 2779 \, \text{kg/m}^3 \]
This value shows us how dense the material of our ore sample is compared to other substances.
Exploring Volume Calculation
The volume of an object measures the space it occupies. In our exercise, we used Archimedes' Principle to find the volume of water displaced when the ore sample is submerged. This displaced volume is equal to the volume of the ore itself.
The formula to derive the volume \(V\) using the buoyant force \(F_b\) is:\[ V = \frac{F_b}{\rho_{water} \cdot g} \]
Where:
  • \(\rho_{water}\) is the density of water, typically \(1000 \, \text{kg/m}^3\)
  • \(g\) is the acceleration due to gravity, \(9.81 \, \text{m/s}^2\)
  • \(F_b\) is the buoyant force, \(6.30 \, \text{N}\)
Let's put these values into action:\[ V = \frac{6.30}{1000 \times 9.81} \, \text{m}^3 \approx 6.42 \times 10^{-4} \, \text{m}^3 \]
This calculation provides a precise measurement of the sample's volume, crucial for subsequent density calculations.

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Most popular questions from this chapter

When an open-faced boat has a mass of 5750 kg, including its cargo and passengers, it floats with the water just up to the top of its gunwales (sides) on a freshwater lake. (a) What is the volume of this boat? (b) The captain decides that it is too dangerous to float with his boat on the verge of sinking, so he decides to throw some cargo overboard so that 20\(\%\) of the boat's volume will be above water. How much mass should he throw out?

There is a maximum depth at which a diver can breathe through a snorkel tube (Fig. E12.17) because as the depth increases, so does the pressure difference, which tends to collapse the diver's lungs. Since the snorkel connects the air in the lungs to the atmosphere at the surface, the pressure inside the lungs is atmospheric pressure. What is the external-internal pressure difference when the diver's lungs are at a depth of 6.1 \(\mathrm{m}\) (about 20 \(\mathrm{ft}\) )? Assume that the diver is in freshwater. (A scuba diver breathing from compressed air tanks can operate at greater depths than can a snorkeler, since the pressure of the air inside the scuba diver's lungs increases to match the external pressure of the water.)

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 45.0 -kg woman to be able to stand on it without getting her feet wet?

You are designing a diving bell to withstand the pressure of seawater at a depth of 250 \(\mathrm{m}\) . (a) What is the gauge pressure at this depth? (You can ignore changes in the density of the water with depth. \((\mathrm{b})\) At this depth, what is the net force due to the water outside and the air inside the bell on a circular glass window 30.0 \(\mathrm{cm}\) in diameter if the pressure inside the diving bell equals the pressure at the surface of the water? (You can ignore the small variation of pressure over the surface of the window.)

A 950 -kg cylindrical can buoy floats vertically in salt water. The diameter of the buoy is 0.900 \(\mathrm{m} .\) Calculate the additional distance the buoy will sink when a 70.0 -kg man stands on top of it.
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