Problem 79 - Fundamentals of Physics

Question

An object hangs from a spring balance. The balance registers \(30 \mathrm{~N}\) in air, \(20 \mathrm{~N}\) when this object is immersed in water, and \(24 \mathrm{~N}\) when the object is immersed in another liquid of unknown density. What is the density of that other liquid?

Step-by-Step Solution

Answer

The density of the unknown liquid is 600 kg/m³.

Step 1: Finding the Buoyant Force in Water

The difference in weight between the object in air and in water gives us the buoyant force when the object is immersed in water. Thus, the buoyant force is \(30 \mathrm{~N} - 20 \mathrm{~N} = 10 \mathrm{~N}\).

Step 2: Calculating the Volume of the Object

Since buoyant force equals the weight of the displaced liquid, and the force is provided by the displaced water, which has a density of \(1000 \mathrm{~kg/m^3}\), use the formula \(F_b = \rho_{water} \cdot V \cdot g\). Plugging in the known values, \(10 \mathrm{~N} = 1000 \mathrm{~kg/m^3} \cdot V \cdot 9.8 \mathrm{~m/s^2}\), we solve for \(V\): \(V = \frac{10 \mathrm{~N}}{1000 \mathrm{~kg/m^3} \cdot 9.8 \mathrm{~m/s^2}}\).

Step 3: Finding the Buoyant Force in the Unknown Liquid

The buoyant force in the unknown liquid can be found similarly, \(30 \mathrm{~N} - 24 \mathrm{~N} = 6 \mathrm{~N}\).

Step 4: Determining the Density of the Unknown Liquid

Use the formula \(F_b = \rho_{liquid} \cdot V \cdot g\) to solve for the density of the unknown liquid: \(6 \mathrm{~N} = \rho_{liquid} \cdot V \cdot 9.8 \mathrm{~m/s^2}\). Since \(V\) was calculated in Step 2, substitute it into this equation and solve for \(\rho_{liquid}\). The density of the unknown liquid is found to be \(600 \mathrm{~kg/m^3}\).

Key Concepts

Density

Density plays a crucial role in understanding buoyancy. It is defined as the mass per unit volume of a substance and is a measure of how much matter there is in a given space. Mathematically, density is expressed as \( \rho = \frac{m}{V} \), where \( m \) is mass, and \( V \) is volume. In the exercise, we are given an object immersed in water and an unknown liquid, both providing different readings on the spring balance. This change is due to the density of the fluids involved. When an object is immersed, it displaces a volume of fluid equal to its own submerged volume. The denser the fluid, the greater the mass of the fluid displaced, affecting the buoyancy force experienced by the object. Understanding the concept of density helps to figure out the forces at play and how different fluids alter the apparent weight of submerged objects.

Archimedes' Principle

Archimedes' Principle is key to solving problems involving buoyancy, as seen in this exercise. This principle states that any object submerged in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Simply put, if you know the volume of fluid displaced and the fluid's density, you can find the buoyant force.

Here’s how Archimedes' Principle was applied in the exercise:

• The object submerged in water displaced some amount of water, providing a buoyant force of 10 N, as calculated from the difference in weights (30 N in air and 20 N in water).
• Using water's known density of 1000 kg/m³, this buoyant force helped calculate the volume of the object, which then served in further calculations.

Understanding Archimedes' Principle helps us grasp why the apparent weight of an object changes when submerged and how the displaced fluid's characteristics contribute to buoyancy.

Fluid Mechanics

Fluid mechanics is the study of fluids (liquids and gases) and the forces acting upon them. Buoyancy is one interaction described within fluid mechanics, often encountered in engineering and physics. In this exercise, fluid mechanics principles explain the variation in buoyant force under different fluids.

Fluid mechanics allows us to solve:

• The variation in readings when the object is submerged, due to different fluid densities affecting the buoyant force.
• How to compute unknown liquid density using previously understood behaviors in another fluid (water).

The study of fluid mechanics illuminates why objects float or sink and how different fluids interact with objects in complex environments. It is not just about understanding the laws at play but also applying them to real-world scenarios, as demonstrated in the problem, to solve for unknowns such as the density of unfamiliar fluids.