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Buoyant force is a phenomenon described by Archimedes' Principle. It explains why objects feel lighter when submerged in a fluid. In simple terms, buoyant force is the force exerted by the fluid that opposes an object's weight. This force acts upwards, countering the force of gravity.

Archimedes' Principle states that the upward, or buoyant, force is equal to the weight of the fluid displaced by the object. Imagine holding a ball underwater – you feel the force trying to push it back to the surface. That's the buoyant force at work. The calculation for the buoyant force is straightforward:

• Measure the weight of the object in air ( \( F_{\text{air}} \)).
• Measure the weight of the object when submerged in the fluid ( \( F_{\text{water}} \)).
• Subtract the submerged weight from the air weight ( \( F_b = F_{\text{air}} - F_{\text{water}} \)).

For our example, a sample weighing 17.50 N in air weighs only 11.20 N in water. The buoyant force is 6.30 N, as this is the difference in measured weights. This understanding is crucial for the subsequent steps to determine volume and density.

To calculate density, we need the mass and the volume of the object. Density is simply mass divided by volume. It helps us understand how compact or heavy a substance is for its size.

Firstly, determining the mass of our ore sample involves dividing its weight by gravitational acceleration, calculated through:

• \( m = \frac{F_{\text{air}}}{g} \)

With \( F_{\text{air}} = 17.50 \text{ N} \) and \( g = 9.81 \text{ m/s}^2 \), the mass \( m \) is approximately 1.784 kg. We have already calculated the volume in another step, which is about \( 6.42 \times 10^{-4} ext{ m}^3 \).

Now, to get the density \( \rho \), use:

• \( \rho = \frac{m}{V} \)

So, the density of the sample turns out to be around 2780 kg/m³. This means the sample is quite dense, implying a high amount of mass packed into a specific volume.

Volume determination using buoyancy is a practical approach when working with Archimedes' Principle, especially when dealing with irregularly shaped objects. In this context, volume can be deduced from the buoyant force, as this force reflects the volume of fluid displaced.

The formula connecting buoyant force with volume is:

• \( V = \frac{F_b}{\rho_{\text{water}} \cdot g} \)

Here, \( F_b \) is the buoyant force (6.30 N), \( \rho_{\text{water}} \) is the density of water (1000 kg/m³), and \( g \) is the gravitational acceleration (9.81 m/s²). Plugging in the values gives us the volume of the sample:

• \( V \approx 6.42 \times 10^{-4} \text{ m}^3 \)

This volume calculation is invaluable in scientific applications since it allows determination of other properties like density and can help in further physical analyses of the material properties.