Understanding Apparent Mass of a Metal Block in Water

Let's break down this problem about finding the apparent mass of a metal block when it's submerged in water. The apparent mass is what the mass seems to be when the object is weighed in a fluid, and it's less than the real mass because of the buoyant force exerted by the fluid.

Given Information:

• Real mass of the metal block (\(m_{real}\)) = 500 g
• Relative density of the metal block (\(RD\)) = 2.5

Finding the Density of the Metal Block

Relative density is the ratio of the density of a substance to the density of a reference substance, usually water for solids and liquids. The density of water (\(\rho_{water}\)) is approximately 1 g/cm³.

The formula for relative density is:

\(\text{Relative Density} = \frac{\text{Density of Substance}}{\text{Density of Water}}\)

So, the density of the metal block (\(\rho_{block}\)) can be calculated as:

\(\rho_{block} = \text{Relative Density} \times \rho_{water}\)

\(\rho_{block} = 2.5 \times 1 \text{ g/cm}^3\)

\(\rho_{block} = 2.5 \text{ g/cm}^3\)

Calculating the Volume of the Metal Block

The volume of the block can be found using its mass and density. The formula for density is Density = Mass / Volume, so Volume = Mass / Density.

\(\text{Volume of block} (V) = \frac{m_{real}}{\rho_{block}}\)

\(V = \frac{500 \text{ g}}{2.5 \text{ g/cm}^3}\)

\(V = 200 \text{ cm}^3\)

Understanding Buoyancy and Apparent Mass

When the metal block is fully immersed in water, it displaces a volume of water equal to its own volume. According to Archimedes' Principle, the buoyant force is equal to the weight of the fluid displaced by the object.

The mass of the water displaced (\(m_{displaced}\)) is:

\(m_{displaced} = \text{Volume of water displaced} \times \text{Density of water}\)

\(m_{displaced} = V \times \rho_{water}\)

\(m_{displaced} = 200 \text{ cm}^3 \times 1 \text{ g/cm}^3\)

\(m_{displaced} = 200 \text{ g}\)

The buoyant force is equivalent to the weight of this displaced water. When an object is immersed in a fluid, its apparent weight is less than its real weight by the amount of the buoyant force.

Apparent weight = Real weight - Buoyant force

Since weight is Mass \(\times\) acceleration due to gravity (\(g\)), and \(g\) is constant for both the object and the displaced fluid, we can relate apparent mass and real mass directly to the mass of the displaced fluid:

Apparent mass (\(m_{apparent}\)) = Real mass (\(m_{real}\)) - Mass of fluid displaced (\(m_{displaced}\))

Calculating the Apparent Mass

Using the values we found:

\(m_{apparent} = m_{real} - m_{displaced}\)

\(m_{apparent} = 500 \text{ g} - 200 \text{ g}\)

\(m_{apparent} = 300 \text{ g}\)

So, the apparent mass of the metal block when fully immersed in water is 300 g.

Summary of Calculations:

Revision Table: Key Concepts for Apparent Mass

Additional Information on Buoyancy and Fluids

Understanding how objects behave in fluids is a fundamental concept in physics, known as fluid mechanics. Buoyancy is a key part of this. Here are some related points:

• Fluid Pressure: Pressure in a fluid increases with depth. This pressure difference between the bottom and top of a submerged object causes the net upward buoyant force.
• Why Objects Float or Sink: An object floats if its density is less than the fluid's density (buoyant force is greater than or equal to its weight). It sinks if its density is greater than the fluid's density (buoyant force is less than its weight). When submerged, the buoyant force on a sinking object is constant regardless of depth (assuming constant fluid density).
• Buoyancy in Different Fluids: The apparent mass change depends on the density of the fluid. An object will feel lighter in a denser fluid (like saltwater or mercury) than in a less dense fluid (like water or air) because the denser fluid provides a greater buoyant force for the same volume displaced.
• Applications: Buoyancy principles are vital in designing ships, submarines, hot air balloons, hydrometers, and understanding phenomena like icebergs floating.