A vessel of water of total mass \(5 \mathrm{~kg}\) stands on a parcel balance. An iron block of mass \(2.7 \mathrm{~kg}\) and relative density \(7.5\) is suspended by a fine wire from a spring balance and is lowered into the water until it is completely immersed. What are the readings on the two balances?

The buoyant force is the upward force that fluids like water exert on objects that are completely or partially immersed in them.
This force is governed by Archimedes' Principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object.
In this exercise, when the iron block is lowered into the water, it displaces a volume of water equal to its own volume.
The weight of this displaced water gives us the buoyant force.
For instance, if the volume of the iron block is 0.00036 m³, and the density of water is 1000 kg/m³, then using the formula for buoyant force \( F_b = \text{volume} \times \text{density of water} \times g \), the buoyant force is calculated as \( 0.00036 \text{ m}^3 \times 1000 \frac{kg}{m^3} \times 9.8 \frac{m}{s^2} = 3.528 N \).
This understanding is crucial in determining how objects behave when submerged in fluids and affects readings on balances.

Relative density, also known as specific gravity, is a measure of the density of a material compared to the density of water at a specified temperature, typically 4°C.
The relative density of the iron block in this exercise is given as 7.5.
This means the iron is 7.5 times denser than water.
To find the density of iron, we simply multiply the density of water by the relative density: \( \rho_{iron} = 7.5 \times 1000 \frac{kg}{m^3} = 7500 \frac{kg}{m^3} \).
Understanding the relative density helps us find the volume of the iron block by using its mass and density.
For example, the volume of a 2.7 kg iron block is calculated using \( V = \frac{mass}{density} = \frac{2.7 \text{ kg}}{7500 \frac{kg}{m^3}} = 0.00036 \text{ m}^3 \).
This is essential for calculating the buoyant force and understanding buoyancy in different fluids.

A spring balance measures the tension in the wire or string from which an object is suspended.
When the iron block is fully immersed in water, the spring balance no longer reads just the object's weight.
Instead, it reads the weight minus the buoyant force.
To get this reading, we first calculate the actual weight of the iron block: \( W_{iron} = 2.7 \text{ kg} \times 9.8 \frac {m}{s^2} = 26.46 N \).
Then, we subtract the buoyant force from this weight: \( T = 26.46 N - 3.528 N = 22.932 N \).
Hence, the spring balance shows 22.932 N, reflecting the reduced weight due to the upward buoyant force supporting part of the iron block’s weight.
This principle is fundamental in various applications, including underwater weighing and determining the density of objects.

Weight measurement involves determining the gravitational force acting on an object’s mass.
Different types of balances measure this force, including parcel balances and spring balances.
In the example, the parcel balance initially measures the vessel and water's combined weight, which totals 5 kg.
When the iron block is immersed, part of its weight is transferred to the water as the buoyant force but does not change the vessel's actual weight.
The added weight due to the buoyant force manifests as an increase in the parcel balance reading.
To reflect this, we add the mass equivalent to the buoyant force to the initial mass: \( 5 \text{ kg} + 0.36 \text{ kg} = 5.36 \text{ kg} \). This converts to a new weight of \( 5.36 \times 9.8 \frac{m}{s^2} = 52.528 N \).
Such accurate weight measurements are crucial for scientific experiments, shipping, and various industrial applications.