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# Volume forces

## Pressure related to volume forces

A volume force is a force that is a function of the volume.

The pressure in a fluid corresponds to a volume force since the force applies to all the particles in the fluid.

Gravitational and electric forces are also volume forces. They apply to each particle located in the gravitational and the electric fields.

Any non-contact force is in fact a volume force, while contact forces are surface and point forces.

The density of a body/fluid is defined by:$$\rho=\frac{m}{V}$$$m=$mass of the body $(\text{kg})$; $V=$volume of the body $(\text{m}^3)$.

This shows that pressure in fluids is a consequence of gravitational forces:$$P=\rho gh=mg\frac{h}{V}=W\frac{h}{V}$$$W=mg=$weight $(\text{ kg m s}^{-2})$.

Pressure in fluids is also a thermodynamic quantity since it changes with temperature and volume.

## Upthrust and buoyancy

Buoyancy $(\vecphy{B})$ or upthrust is the force exerted by a fluid to oppose the weight of an object floating on or submerged in the fluid. This force is also called the buoyant force.

Archimedes' principle states that the buoyancy of an object immersed in a fluid (not touching the base of a container) is equal in magnitude to the weight of the fluid displaced by the object: \begin{align*}B&=m_{\text{displaced fluid}}g\\&=\rho V_{\text{displaced fluid}}g\end{align*}

A ship on the sea is pulled down by gravitational force (weight), but it is pushed upwards by the buoyant force. The more water the ship displaces, the stronger the buoyancy.

A body will sink or float depending on whether the upthrust is lower or higher than the weight. A body is in equilibrium when its weight is equal in magnitude to the upthrust.

$P=$fluid pressure; $\rho=$density of the fluid, $g=$magnitude of gravitational acceleration, $h=$depth of the point.

## Explaining buoyancy

Buoyancy is the force exerted by a fluid on an object opposing the weight of the object.

The origin of buoyancy is the difference in pressure at different depths. The depth influences the pressure at a given point of an object according to $P=\rho gh$.

The top and bottom of a submerged sphere (located at $h_1$ and $h_2$ respectively) will be subject to different pressures: \begin{align*}P_1=\rho gh_1 \quad& \quad P_2=\rho gh_2\\ h_{1}&\lt h_{2}\\\therefore P_{1}&\lt P_{2}\end{align*}

This difference in pressure results in an upward force on the ball, known as the buoyant force.

The upthrust is due to the difference in pressure at different heights.