Understanding the Relative Density of Objects Immersed in Water
Understanding the Relative Density of Objects Immersed in Water
Often, we come across tables or charts that list the relative density of various materials. However, have you ever calculated the relative density of an object by yourself? In this article, we will walk through a practical example to understand the principles using basic physics principles.
Problem Statement
An object weighs 10.0 Newtons (N) in air and 7.0 N when totally immersed in water. What is the relative density of the object? This calculation will be made in two different ways, using the Archimedes Principle and relative density definition.
Calculating Relative Density Using Archimedes Principle
First, note that the weights given are in Newtons (N). The relative density (density ratio) of an object can be derived by comparing its density to that of water. At 4 degrees Celsius, the density of water is 1 g/cm3. Relative density can be defined as the ratio of the object's density to water's density, or the mass of the object per unit volume of equal mass of water.
Step 1: Determine the Mass and Volume of the Object
Let's start by finding the mass of the object. The object's mass in air is calculated as follows:
Mass of the object (m) 10 N / 10 N/kg 1 kg
The object's upthrust when submerged is given by the difference in its weight in air and its weight in water:
Upthrust weight of the object in air - weight of the object in water 10 N - 7 N 3 N
By Archimedes' Principle, the upthrust (buoyant force) is equal to the weight of the water displaced. Let's denote the volume of the object as ( V ), and the density of water as ( rho_w ). Thus,
Mass of the displaced water 3 N / 10 N/kg 0.3 kg
So, the relative density (or specific gravity) can be calculated as follows:
Relative density mass of the object / mass of the same volume of water 1 kg / 0.3 kg 3.3
Calculating Relative Density Using the Definition
Alternatively, we can calculate the relative density using the definition provided in the problem statement. This involves the use of the Archimedes Principle and some basic physics equations.
Step 2: Derive the Relative Density Using the Definition
The weight of the object in air can be expressed as:
Object weight in air ( V times rho_o times g )
The weight of the object in water can be expressed as:
Object weight in water ( (V - V_w) times rho_o times g )
Where ( V_w ) is the volume of water displaced, ( rho_o ) is the density of the object, and ( g ) is the gravitational acceleration. Using the given values, we can write:
10 N ( V times rho_o times g )
7 N ( (V - V_w) times rho_o times g )
Subtracting the second equation from the first:
10 N - 7 N 3 N ( V times rho_o times g - (V - V_w) times rho_o times g )
3 N ( V times rho_o times g - V times rho_o times g V_w times rho_o times g )
3 N ( V_w times rho_o times g )
Dividing the first equation by the second equation, we get:
(frac{rho_o}{rho_w} frac{10 , text{N/kg}}{3 , text{N/kg}} approx 3.33)
Conclusion
The relative density of the object, or its specific gravity, can be determined using both the Archimedes Principle and the basic definition of relative density. In this case, the relative density of the object is approximately 3.33, which is greater than that of water (1 g/cm3) and less than aluminum (2.7 g/cm3).