Archimedes' principle says that the buoyant force exerted on...

Archimedes' principle says that the buoyant force exerted on an object that is (partially or totally) submerged in water is equal to the weight of the water displaced by the object (see figure). Let $\rho_{w}=1 \mathrm{g} / \mathrm{cm}^{3}=1000 \mathrm{kg} / \mathrm{m}^{3}$ be the density of water and let $\rho$ be the density of an object in water. Let $f=\rho / \rho_{w} .$ If $0<f \leq 1,$ then the object floats with a fraction $f$ of its volume submerged; if $f>1,$ then the object sinks. Consider a cubical box with sides 2 m long floating in water with one-half of its volume submerged ( $\rho=\rho_{w} / 2$ ). Find the force required to fully submerge the box (so its top surface is at the water level). (See the Guided Project Buoyancy and Archimedes' Principle for further explorations of buoyancy problems.)

Archimedes' principle says that the buoyant force exerted on an object that is (partially or totally) submerged in water is equal to the weight of the water displaced by the object (see figure). Let $\rho_{w}=1 \mathrm{g} / \mathrm{cm}^{3}=1000 \mathrm{kg} / \mathrm{m}^{3}$ be the density of water and let $\rho$ be the density of an object in water. Let $f=\rho / \rho_{w} .$ If $0<f \leq 1,$ then the object floats with a fraction $f$ of its volume submerged; if $f>1,$ then the object sinks.
Consider a cubical box with sides 2 m long floating in water with one-half of its volume submerged ( $\rho=\rho_{w} / 2$ ). Find the force required to fully submerge the box (so its top surface is at the water level). (See the Guided Project Buoyancy and Archimedes' Principle for further explorations of buoyancy problems.)

Key Concepts

Archimedes' Principle

Archimedes' Principle states that an object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This principle is essential in understanding how objects float or sink, depending on whether the buoyant force can balance the object’s weight.

Buoyant Force

Buoyant force is the upward force exerted by a fluid that opposes the weight of an object immersed in it. It is the result of differences in pressure exerted by the fluid on different parts of the object and is directly proportional to the volume of displaced fluid.

Density and Volume Displacement

The relationship between an object's density and the density of the fluid determines how much volume must be displaced for the object to float. By comparing the density of the object with the fluid density, one can predict the fraction of the object’s volume that will be submerged in equilibrium.

Floating Equilibrium

Floating equilibrium occurs when the buoyant force acting on an object exactly equals its weight, resulting in the object remaining stationary in the fluid. This balance of forces is used to determine the submerged fraction of the object’s volume in a stable floating state.

Net Force for Submergence

To fully submerge an object that is initially floating, an additional external force must be applied to overcome the excess buoyant force that would otherwise cause the object to rise. This force is calculated as the difference between the buoyant force when the object is fully submerged and the buoyant force at the floating equilibrium.

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