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Chapter 9: Problem 41

A sample of an unknown material appears to weigh \(300 \mathrm{~N}\) in air and \(200 \mathrm{~N}\) when immersed in alcohol of specific gravity \(0.700\). What are (a) the volume and (b) the density of the material?

Short Answer

Expert verified
The volume of the material is \(0.014 \mathrm{~m^3}\) and the density of the material is \(2186.43 \mathrm{~kg/m^3}\).

Step by step solution

01

Compute the Buoyant Force

Calculate the buoyant force. This can be done by subtracting the weight of the material in alcohol from its weight in air. Therefore, buoyant force (F) = weight in air - weight in alcohol = \(300 N - 200 N = 100 N\).
02

Determine the Volume of the Material

Next, we use the buoyant force to compute the volume of the material. According to Archimedes' principle, buoyant force equals the weight of liquid displaced by the material. Hence, we have F = volume (V) x density of alcohol (Da) x acceleration due to gravity (g). By rearranging this formula, we get V = F / (Da * g). As the specific gravity of alcohol is given as 0.700, this means Da = \(0.700 * 1000 \mathrm{~kg/m^3} = 700 \mathrm{~kg/m^3}\), (since water's density is \(1000 \mathrm{~kg/m^3}\)). And by taking g approximately as \(9.8 \mathrm{~m/s^2}\), we find V = \(100 N / (700 \mathrm{~kg/m^3} * 9.8 \mathrm{~m/s^2}) = 0.014 \mathrm{~m^3}\).
03

Calculate the Density of the Material

Lastly, we calculate the density of the unknown material. This can be done by the relationship density (D) = mass (m) / volume (V). But we have to convert the weight in air to mass first, which can be calculated with mass = weight_in_air / g. So, m = \(300 N / 9.8 \mathrm{~m/s^2} = 30.61 \mathrm{~kg}\). Therefore, the density D = m/V = \(30.61 \mathrm{~kg} / 0.014 \mathrm{~m^3} = 2186.43 \mathrm{~kg/m^3}\).

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Key Concepts

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

Buoyant Force
The concept of buoyant force comes from Archimedes' Principle, which states that any object submerged in a fluid is acted upon by an upward force. This force is equal to the weight of the fluid that the object displaces. It is what makes objects feel lighter in water or any other liquid.

In the exercise given, an unknown material weighs 300 N in air. However, its weight reduces to 200 N when it is submerged in alcohol. This difference is attributed to the buoyant force. Therefore, it is calculated by subtracting the submerged weight from the weight in air. Here, the buoyant force is 100 N (300 N - 200 N).

This force is crucial for calculating the volume of the material using the properties of the liquid in which it is submerged.
Density Calculation
Density is a measure of how much mass is contained in a given volume. Calculating the density of an object involves dividing its mass by its volume. In the context of this exercise, density is an important physical property that helps us identify the material.

To find the mass of the material, we use the weight of the material in air, converted to mass using the equation: \[ \text{mass} = \frac{\text{weight in air}}{g},\]where \( g \approx 9.8 \, \mathrm{m/s^2} \) is the acceleration due to gravity. Thus, the mass is calculated as 30.61 kg (300 N / 9.8 m/s虏).

Once the mass is known, we divide it by the previously calculated volume (0.014 m鲁) to find the density: \[ \text{density} = \frac{\text{mass}}{\text{volume}} = \frac{30.61 \, \mathrm{kg}}{0.014 \, \mathrm{m}^3} = 2186.43 \, \mathrm{kg/m^3}.\]
Specific Gravity
Specific gravity is a ratio of the density of a substance to the density of a reference substance, typically water for liquids and solids. It is a dimensionless number that indicates how heavy a material is relative to water.

In this problem, the specific gravity of alcohol is given as 0.700. This implies that alcohol is 0.700 times as dense as water. Since the standard density of water is 1000 kg/m鲁, the density of alcohol is calculated as: \[ \text{density of alcohol} = 0.700 \times 1000 \, \mathrm{kg/m^3} = 700 \, \mathrm{kg/m^3}. \]

This value of density is used along with the volume displaced by the material to determine the buoyant force, according to Archimedes' Principle.
Volume Displacement
Volume displacement is the volume of fluid that is moved aside when an object is submerged in it. This concept is crucial in applying Archimedes' Principle to calculate the amount of buoyant force and further to find the volume of the submerged object.

According to the exercise, the buoyant force is equal to the weight of the displaced alcohol. Using the formula:\[ \text{buoyant force} = V \times \text{density of alcohol} \times g,\]where the buoyant force is 100 N, the density of alcohol is 700 kg/m鲁, and \( g \approx 9.8 \, \mathrm{m/s^2} \), we rearrange to solve for V:\[ V = \frac{\text{buoyant force}}{\text{density of alcohol} \times g} = \frac{100 \, \text{N}}{700 \, \mathrm{kg/m^3} \times 9.8 \, \mathrm{m/s^2}} = 0.014 \, \mathrm{m^3}.\]

Understanding how volume displacement works allows us to unlock further information about the material's physical properties, such as its density.

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Most popular questions from this chapter

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