A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 65.0-kg woman to be able to stand on it without getting her feet wet?

검증된 단계별 안내

First, understand that the problem involves buoyancy, which is the ability of an object to float in a fluid. The slab of ice must displace a volume of water equal to the weight of the woman plus the weight of the ice itself.

Use Archimedes' principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. The buoyant force must equal the total weight of the woman and the ice for the slab to float without submerging.

Calculate the weight of the woman using the formula: \( \text{Weight} = m \cdot g \), where \( m = 65.0 \text{ kg} \) is the mass of the woman and \( g = 9.81 \text{ m/s}^2 \) is the acceleration due to gravity.

Determine the density of freshwater, which is approximately \( 1000 \text{ kg/m}^3 \). The volume of water displaced by the ice slab must be equal to the total weight divided by the density of water.

Set up the equation for the volume of the ice slab: \( V = \frac{\text{Weight of woman} + \text{Weight of ice}}{\text{Density of water}} \). Solve for \( V \) to find the minimum volume of the ice slab required to support the woman without submerging.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object submerged in it, counteracting the object's weight. For an object to float, the buoyant force must equal the object's weight. In this scenario, the ice slab must displace a volume of water equal to the combined weight of the slab and the woman to keep her feet dry.

Density

Density is the mass per unit volume of a substance, often expressed in kilograms per cubic meter (kg/m³). The density of ice is less than that of water, which is why ice floats. Understanding the density difference between ice and water is crucial to calculating the volume of ice needed to support additional weight without submerging.

Archimedes' Principle

Archimedes' Principle states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This principle helps determine the minimum volume of the ice slab required to support the woman's weight by ensuring the displaced water's weight equals the total weight of the woman and the ice.