Three sinking blocks have the same mass (1.0 kg) but different densities:

Question

Three sinking blocks have the same mass (1.0 kg) but different densities:

𝜌1 = 1001 kg⁄m3, 𝜌2 = 2000 kg⁄m3 and 𝜌3 = 3000 kg⁄m3. How do you calculate the apparent mass of each block?

Answer 1

To calculate the apparent mass of each block, we need to consider the concept of buoyancy.

The apparent mass of an object submerged in a fluid is the effective or apparent weight of the object, which takes into account the buoyant force exerted on it by the fluid. Buoyant force is the upward force exerted on an object submerged in a fluid, and its magnitude depends on the density of the fluid and the volume of the submerged part of the object.

The formula for calculating the apparent mass of an object submerged in a fluid is:

Apparent Mass = (True Mass) + (Buoyant Force / g)

Where:
- True Mass is the actual mass of the object
- Buoyant Force is the upward force exerted by the fluid on the object
- g is the acceleration due to gravity (approximately 9.8 m/s^2)

To find the buoyant force, we can use Archimedes' principle, which states that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object. The weight of the fluid displaced can be calculated using the formula:

Weight of Fluid Displaced = Volume of Object Submerged * Density of Fluid * g

Now, let's calculate the apparent mass of each block step by step:

Block 1:
- True Mass = 1.0 kg
- Density of Fluid (𝜌) = 1000 kg/m^3 (assuming the fluid is water)
- Volume of Object Submerged = True Mass / Density of Object = 1.0 kg / 1001 kg/m^3 = 0.000999 m^3
- Weight of Fluid Displaced = Volume of Object Submerged * Density of Fluid * g = 0.000999 m^3 * 1000 kg/m^3 * 9.8 m/s^2 = 9.79 N
- Apparent Mass = True Mass + (Buoyant Force / g) = 1.0 kg + (9.79 N / 9.8 m/s^2) = 1.0 kg + 1.0 kg = 2.0 kg

Block 2:
- True Mass = 1.0 kg
- Density of Fluid (𝜌) = 1000 kg/m^3 (assuming the fluid is water)
- Volume of Object Submerged = True Mass / Density of Object = 1.0 kg / 2000 kg/m^3 = 0.0005 m^3
- Weight of Fluid Displaced = Volume of Object Submerged * Density of Fluid * g = 0.0005 m^3 * 1000 kg/m^3 * 9.8 m/s^2 = 4.9 N
- Apparent Mass = True Mass + (Buoyant Force / g) = 1.0 kg + (4.9 N / 9.8 m/s^2) = 1.0 kg + 0.5 kg = 1.5 kg

Block 3:
- True Mass = 1.0 kg
- Density of Fluid (𝜌) = 1000 kg/m^3 (assuming the fluid is water)
- Volume of Object Submerged = True Mass / Density of Object = 1.0 kg / 3000 kg/m^3 = 0.000333 m^3
- Weight of Fluid Displaced = Volume of Object Submerged * Density of Fluid * g = 0.000333 m^3 * 1000 kg/m^3 * 9.8 m/s^2 = 3.26 N
- Apparent Mass = True Mass + (Buoyant Force / g) = 1.0 kg + (3.26 N / 9.8 m/s^2) = 1.0 kg + 0.33 kg = 1.33 kg

Therefore, the apparent masses of the three sinking blocks are:
- Block 1: 2.0 kg
- Block 2: 1.5 kg
- Block 3: 1.33 kg