Archimedes purportedly used his principle to verify that the king's crown was pure gold by weighing the crown submerged in water. Suppose the crown's actual weight was \(25.0 \mathrm{N}\). What would be its apparent weight if it were made of (a) pure gold and (b) \(75 \%\) gold and \(25 \%\) silver, by volume? The densities of gold, silver, and water are \(19.3 \mathrm{g} / \mathrm{cm}^{3}, 10.5 \mathrm{g} / \mathrm{cm}^{3},\) and \(1.00 \mathrm{g} / \mathrm{cm}^{3},\) respectively.

When an object is submerged in a fluid, it experiences an upward force called buoyant force. This is the core of Archimedes' Principle, which states that the buoyant force on a submerged object is equal to the weight of the fluid that the object displaces. In the case of the crown, when it is submerged in water, it displaces a volume of water equal to its own volume, and this displaced water has weight. The crown will therefore feel a lighter force acting on it, which is what we refer to as a reduction in the object's apparent weight.

Understanding Through Experimentation

One way to truly understand the concept is to perform a simple experiment. Take a solid object, measure its weight in the air and then measure it again while it's fully submerged in water. The difference you observe is due to the buoyant force.

Density is a measure of how compact the mass in a substance or object is. It is defined as mass per unit volume and is a critical factor in determining whether an object will sink or float in a fluid. In our exercise involving the crown, knowing the density of gold, silver, and water allows us to calculate the volume of the crown, given its mass. This is because the crown's mass, distributed over the volume it occupies, gives us its density, which should be consistent with that of pure gold. Different densities of materials can drastically change the apparent weight of the object when submerged.

The Secret Ingredient

When it comes to density, it's like the secret ingredient that defines the character of a substance. For example, packing more mass into the same volume will increase the density, just like adding more seasoning makes the flavor of a dish more intense.

Apparent weight refers to the weight of an object as measured when it is submerged in a fluid, such as water. The concept of apparent weight is directly tied to the buoyant force. As Archimedes' Principle tells us, the object will feel an upward buoyant force equal to the weight of the fluid it displaces. Thus, the object seems lighter and its weight, as measured while submerged, is reduced. This is what we call the apparent weight.

Real-World Application

Understanding apparent weight is crucial in many fields like shipbuilding, where calculating the buoyancy of a vessel is necessary to ensure it can float and carry a designated load without sinking. Apparent weight helps us grasp how submerged objects interact with the fluid around them, reinforcing concepts like buoyancy and density with tangible consequences.