Calculate the true mass (in vacuum) of an aluminum sphere whose apparent mass is 4.0000 kg when weighed in air.

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Understand the problem: The apparent mass of the aluminum sphere is given as 4.0000 kg when weighed in air. This means the buoyant force exerted by the air affects the measurement. To find the true mass (in vacuum), we need to account for this buoyant force.

Recall the formula for the apparent mass: \( m_{apparent} = m_{true} - F_b / g \), where \( F_b \) is the buoyant force and \( g \) is the acceleration due to gravity. Rearrange this to solve for the true mass: \( m_{true} = m_{apparent} + F_b / g \).

Determine the buoyant force: The buoyant force is given by \( F_b = \rho_{air} V g \), where \( \rho_{air} \) is the density of air, \( V \) is the volume of the sphere, and \( g \) is the acceleration due to gravity. To find \( V \), use the relationship \( V = m_{apparent} / \rho_{aluminum} \), where \( \rho_{aluminum} \) is the density of aluminum.

Substitute \( V \) into the buoyant force equation: \( F_b = \rho_{air} (m_{apparent} / \rho_{aluminum}) g \). Simplify this to \( F_b = (\rho_{air} / \rho_{aluminum}) m_{apparent} g \).

Finally, substitute \( F_b \) into the true mass equation: \( m_{true} = m_{apparent} + (\rho_{air} / \rho_{aluminum}) m_{apparent} \). Simplify this expression to calculate the true mass of the aluminum sphere in vacuum.

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주요 개념

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Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is responsible for the apparent loss of weight when an object is weighed in a fluid, such as air or water. The buoyant force depends on the density of the fluid and the volume of the displaced fluid, which is crucial for understanding how the apparent mass differs from the true mass.

Archimedes' Principle

Archimedes' Principle states that an object submerged in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. This principle helps explain why the apparent mass of an object, like the aluminum sphere, is less than its true mass when weighed in air. Understanding this principle is essential for calculating the true mass from the apparent mass.

Density

Density is defined as mass per unit volume and is a key property of materials. It influences how much buoyant force an object experiences when submerged in a fluid. For the aluminum sphere, knowing its density allows for the calculation of its true mass based on the apparent mass measured in air, as the difference is due to the buoyant force acting on it.