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 approximately 600 kg/m³.

Step 1: Understand the Problem

The object weighs 30 N in air, 20 N in water, and 24 N in an unknown liquid. We need to find the density of this unknown liquid.

Step 2: Calculate the Buoyant Force in Water

The buoyant force in water is the difference between the object's weight in air and its weight in water. So, the buoyant force is \(30 - 20 = 10\, \mathrm{N}\).

Step 3: Calculate the Volume of the Object

Using the buoyant force in water, calculate the volume of the object. From Archimedes' principle, the buoyant force equals the weight of the displaced fluid: \(F_b = V \times \rho_w \times g\). With \(F_b = 10 \text{ N}\), \(\rho_w = 1000\, \mathrm{kg/m^3}\), and \(g = 9.8 \mathrm{\, m/s^2}\), solve for \(V\).\[ V = \frac{10}{1000 \times 9.8}\approx 0.00102\, \mathrm{m^3}.\]

Step 4: Calculate the Buoyant Force in Unknown Liquid

The buoyant force in the unknown liquid is the difference between the object's weight in air and its apparent weight in the liquid: \(30-24 = 6 \, \mathrm{N}\).

Step 5: Determine the Density of the Unknown Liquid

Using the calculated volume of the object and the buoyant force in the unknown liquid, solve for the density.\[ 6 = 0.00102 \times \rho_{\text{liquid}} \times 9.8\]\[ \rho_{\text{liquid}} = \frac{6}{0.00102 \times 9.8} \approx 600\, \mathrm{kg/m^3}.\]

Key Concepts

Buoyant Force and Its Role in Archimedes' Principle

The buoyant force is a key concept in understanding why objects seem to weigh less in a liquid compared to their weight in air. This force is an upward push exerted by a fluid, countering the weight of an object submerged in it. According to Archimedes' Principle, the buoyant force can be formulated as:

• Buoyant Force = Weight of the Displaced Fluid

If you have ever tried to lift something submerged in water, you might have noticed that it is easier than lifting it in air. This happens because the buoyant force is supporting the object from beneath.
In the exercise, the object weighed 30 N in air and only 20 N when immersed in water, resulting in a 10 N buoyant force. What happens here is that the water displaced by the object weighs 10 N, which is why the object feels 10 N lighter. This same principle allows aircraft carriers, massive ships, and even icebergs to float, despite their substantial weight. By subtracting the smaller weight registers from the initial weight in air, you determine the buoyant force in various liquids.

Density Calculation and Why It's Important

Density is defined as mass per unit volume (\[ \rho = \frac{m}{V} \]). It indicates how much mass is packed into a given volume and is typically expressed in kilograms per cubic meter (\(\text{kg/m}^3\)). Knowing an object's density allows us to understand its mass distribution and how it interacts with different fluids.
In the exercise, we determined the volume of the object first in order to calculate the density of an unknown liquid. By using the buoyant force in water and knowing that 1000 kg/m³ is the density of water, we calculated the volume of the object.
Once you have the volume, you can rearrange the buoyancy formula to solve for the unknown density of another liquid. This involves setting the buoyant force in the new liquid equal to the product of the volume of the object, the unknown density, and the gravitational acceleration (\(9.8 \text{ m/s}^2\)). Knowing how liquid densities compare can manifest in many real-world scenarios like determining if an object will float or sink, or even in determining chemical concentrations in solutions.

Understanding a Spring Balance's Function

A spring balance, also known as a spring scale, is a device used to measure weight by using the tension of a spring. The measurement is based on Hooke's Law, which states:

• The force exerted by a spring is proportional to its extension.

This is expressed mathematically as \(F = k \times x\), where \(F\) is the force in newtons, \(k\) is the spring constant, and \(x\) is the displacement of the spring from its resting position.
In the context of this exercise, the spring balance showed 30 N when the object is in air, and its value decreased when the object was submerged in water (20 N) and in another liquid (24 N). The readings change because the buoyant force reduces the effective weight exerted on the spring scale.
Using a spring balance gives a direct visual or numerical cue for the weight of an object, making it a valuable tool in both experimental and practical measurements of mass and weight, especially in educational and laboratory settings. Its simplicity and effectiveness stem from the material properties of springs and gravitational force interactions.