State of Matter, Buoyancy, Archimede's principle

States of matter

In physics, a state of matter is a form of existence, such as solid, liquid, and gas. The solid state maintains a fixed volume and shape. The liquid state maintains a fixed volume with adaptive shape. The gaseous state varies in both volume and shape.

Other states of matter that are more exotic and frequently investigated in physics research: plasma, supersolid, superfluid, superconductivity, Bose-Einstein condensate, and fermionic condensate.

Pressure of fluid

In physics, pressure is the force applied perpendicular to a surface of an object per unit area:

$$p=\frac{F_{\perp}}{A}$$

Its SI unit is pascal ($Pa=N/m^2$).

If the applied perpendicular force is the same, the smaller the contact area, the larger the pressure. That is why a nail has a pointy tip to get into the surface, and why a sharpened knife is better than a blunt knife for cutting.

Fluid is either liquid or gas. Fluid pressure is a type of pressure that results from compressive force within a static fluid.

$$p=\rho gh$$

where $\rho$ is the mass density of the fluid, the ratio of its mass and volume ($\rho=\frac{m}{V}$), $g$ is the gravitational acceleration, $h$ is the depth (distance from the point of interest to the surface) of the fluid.

The deeper an object is in the fluid, the more pressure it has. Deep sea organisms need to counter-balance the extremely high pressure from surrounding water. Altitude sickness results from failure to adapt quickly enough to the decrease in air pressure.

Buoyant force

The difference of pressure causes a force on a partially or fully immersed object, called buoyancy. It is a force exerted by a fluid opposing the weight of the object.

Archimedes' principle

Archimedes' principle states that the magnitude of the buoyant force equals to the weight of the displaced fluid (the amount of fluid that would have occupied the space of the object if the object was not there).

We can express its weight as

$$G=mg=\rho Vg$$

Assume the mass density of a fluid is $\rho_f$, when the volume of the fluid is the same as the volume of the object, that is $V_f=V$, then the weight of that amount of the fluid:

$$G_f=m_fg=\rho_fV_fg=\rho_fVg$$

Therefore, the buoyant force when the object is fully immersed in the fluid, buoyant force has the same magnitude as the weight of the displaced fluid:

$$B=G_f=\rho_fVg$$

We know when an object tends to float in the fluid, buoyant force is larger than the gravitational force:

$$B>G\Rightarrow \rho_fVG>\rho Vg\Rightarrow \rho_f>\rho$$

In other words, an object with smaller average density than that of the fluid floats.

Similarly, when an object tends to sink in the fluid, buoyant force is smaller than the gravitational force:

$$B<G\Rightarrow \rho_fVG<\rho Vg\Rightarrow \rho_f<M\rho$$

In other words, an object with larger average density than that of the fluid sinks.

Here is the illustration with free-body diagrams.