Floating Wood Density Calculation

When an object floats, the buoyant force acting on it equals its weight. This principle, derived from Archimedes' Principle, allows us to determine the object's density relative to the fluid it floats in.

Physics Principle for Floating Objects

For an object to float, the weight of the object must be equal to the weight of the fluid displaced by the submerged part of the object. Mathematically, this is expressed as:

Weight of Object = Buoyant Force

Using the formula Weight = Volume \(\times\) Density \(\times\) gravity (\(g\)), we get:

\(V_{total} \times \rho_{object} \times g = V_{submerged} \times \rho_{fluid} \times g\)

Where:

• \(V_{total}\) is the total volume of the object.
• \(\rho_{object}\) is the density of the object.
• \(V_{submerged}\) is the volume of the object submerged in the fluid.
• \(\rho_{fluid}\) is the density of the fluid.

The gravity term (\(g\)) cancels out from both sides:

\(V_{total} \times \rho_{object} = V_{submerged} \times \rho_{fluid}\)

Calculating Wood Density

We can rearrange the equation to solve for the object's density:

\(\rho_{object} = \frac{V_{submerged}}{V_{total}} \times \rho_{fluid}\)

The question states that 65% of the wood's volume is under water. This means:

\(\frac{V_{submerged}}{V_{total}} = 0.65\)

The fluid is water, and its density (\(\rho_{fluid}\)) is approximately \(1000 \text{ kg/m}^3\). Substituting these values:

\(\rho_{wood} = 0.65 \times 1000 \text{ kg/m}^3\)

\(\rho_{wood} = 650 \text{ kg/m}^3\)

To match the format of the options, we express this in scientific notation:

\(\rho_{wood} = 0.65 \times 10^3 \text{ kg/m}^3\)

This corresponds to option 4.