To determine the fraction of the cork submerged in water when it floats, we first need to understand the principle of buoyancy, which is governed by Archimedes' principle. This principle states that a floating object displaces a volume of fluid equal to the weight of the object.
Let's denote the following:
- \( m \) as the mass of the cork.
- \( \rho_c \) as the density of the cork.
- \( \rho_w \) as the density of water.
- \( V_c \) as the volume of the cork.
- \( V_{sub} \) as the volume of the cork submerged in water.
- \( g \) as the acceleration due to gravity (which will cancel out in the fraction, so it's not necessary to consider its numerical value).
Given:
- Mass of cork, \( m = 20 \) g.
- Density of cork, \( \rho_c = 0.25 \) g/cm³.
- Density of water, \( \rho_w = 1 \) g/cm³ (approximately, since the exact value is about 0.997 g/cm^3, we typically use 1 g/cm³ for simplicity).
First, we determine the volume of the cork using its mass and density:
\[ V_c = \frac{m}{\rho_c} = \frac{20 \text{ g}}{0.25 \text{ g/cm}^3} = 80 \text{ cm}^3 \]
For the cork to float, the weight of the displaced water must equal the weight of the cork. The weight of the water displaced is given by the volume of the submerged cork multiplied by the density of water (and gravity, which will cancel out):
\[ \text{Weight of displaced water} = V_{sub} \times \rho_w \]
The weight of the cork is:
\[ \text{Weight of cork} = m \]
Since the cork floats, these weights are equal:
\[ V_{sub} \times \rho_w = m \]
Substituting \( m \) and \( \rho_w \) into the equation gives:
\[ V_{sub} \times 1 \text{ g/cm}^3 = 20 \text{ g} \]
So,
\[ V_{sub} = 20 \text{ cm}^3 \]
The fraction of the cork submerged is the ratio of the submerged volume to the total volume of the cork:
\[ \text{Fraction submerged} = \frac{V_{sub}}{V_c} = \frac{20 \text{ cm}^3}{80 \text{ cm}^3} = \frac{1}{4} \]
Thus, one-fourth (or 25%) of the cork is immersed when it floats in water.
The mass of piece of cork is 20 g. what fraction of the cork is immersed when it floats in water, given the density of the cork is 0.25 g/cm^3
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