Problem 32 - College Physics

Question

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

Step-by-Step Solution

Answer

The volume of the sample is approximately 0.000643 m³ and its density is about 2777 kg/m³.

Step 1: Understand the Problem

We need to find the total volume and density of an ore sample. Initially, the sample weighs 17.50 \( \mathrm{N} \) in air and has a tension of 11.20 \( \mathrm{N} \) when immersed in water. The difference in weight is due to the buoyant force, which can be used to find the volume of the sample using Archimedes' principle.

Step 2: Calculate the Buoyant Force

The buoyant force \( F_b \) equals the difference in weight of the sample in air and in water. Thus, \( F_b = 17.50 \mathrm{N} - 11.20 \mathrm{N} = 6.30 \mathrm{N} \).

Step 3: Use Archimedes' Principle to Find the Volume

According to Archimedes' principle, the buoyant force is equal to the weight of the displaced water. The weight of the displaced water is the volume of the sample times the density of water times gravity: \( F_b = V \times \rho_{water} \times g \). Given \( \rho_{water} = 1000 \mathrm{kg/m^3} \) and \( g = 9.8 \mathrm{m/s^2} \), we find the volume \( V \) by rearranging: \( V = \frac{F_b}{\rho_{water} \times g} \). Substituting in values: \( V = \frac{6.30 \mathrm{N}}{1000 \mathrm{kg/m^3} \times 9.8 \mathrm{m/s^2}} \approx 0.000643 \mathrm{m^3} \).

Step 4: Calculate the Density of the Sample

The density \( \rho \) of the sample is given by \( \rho = \frac{\text{Mass}}{\text{Volume}} \). The mass can be found from the weight of the sample in air divided by gravity: \( m = \frac{W}{g} = \frac{17.50 \mathrm{N}}{9.8 \mathrm{m/s^2}} \approx 1.7857 \mathrm{kg} \). Now substitute this mass and the volume into the density formula: \( \rho = \frac{1.7857 \mathrm{kg}}{0.000643 \mathrm{m^3}} \approx 2777 \mathrm{kg/m^3} \).

Key Concepts

Buoyant Force

The concept of buoyant force is rooted deeply in Archimedes’ Principle, which states that any object fully or partially submerged in a fluid experiences an upward force, or buoyant force, equal to the weight of the fluid displaced by the object. In simpler terms, when you put something in water, it pushes water out of the way and, in turn, the water pushes back against the object, trying to float it.

In our exercise, the ore sample weighed 17.50 N in air and the tension in the cord was 11.20 N when immersed in water. The buoyant force is the difference between these two weights.

• Weight in air: 17.50 N
• Weight in water: 11.20 N
• Buoyant Force: 6.30 N

The buoyant force here is calculated as: \( F_b = 17.50 \, \mathrm{N} - 11.20 \, \mathrm{N} = 6.30 \, \mathrm{N} \).

This force represents the weight of the water displaced by the sample, which is key to determining the next steps: calculating the volume and density of the sample.

Density Calculation

Density is a measure of how much mass is packed into a unit volume of a substance. It's calculated by dividing the mass by the volume: \( \rho = \frac{\text{mass}}{\text{volume}} \). Understanding this allows us to figure out how tightly packed the atoms or molecules are within an object.

To find the density of the ore sample, we first need its mass. We can calculate the sample’s mass by dividing the weight in air by the gravitational acceleration: \( m = \frac{17.50 \, \mathrm{N}}{9.8 \, \mathrm{m/s^2}} \approx 1.7857 \, \mathrm{kg} \).

With mass and volume determined, the density calculation becomes straightforward.

• Density \( \rho = \frac{1.7857 \, \mathrm{kg}}{0.000643 \, \mathrm{m^3}} \approx 2777 \, \mathrm{kg/m^3} \)

The calculated density helps in understanding the composition and identification of the material, as each substance has a characteristic density.

Volume Determination

Determining the volume of an object using Archimedes' Principle involves understanding how much water is displaced when the object is submerged. This displaced water is directly correlated to the volume of the object itself, since the buoyant force equals the weight of the displaced water.

The formula relates these concepts: \( F_b = V \times \rho_{water} \times g \), where \( V \) is the volume, \( \rho_{water} \) is the density of water, and \( g \) is the acceleration due to gravity.

Rearranging to solve for \( V \), we have:

• \( V = \frac{F_b}{\rho_{water} \times g} = \frac{6.30 \, \mathrm{N}}{1000 \, \mathrm{kg/m^3} \times 9.8 \, \mathrm{m/s^2}} \approx 0.000643 \, \mathrm{m^3} \)

This volume reflects how much space the ore sample takes up, key in helping us calculate its density. Knowing the volume is crucial for understanding an object's physical properties, applicable in real-world scenarios like material science and engineering.