**Useful Preliminaries : **

$ \rho = \frac{m}{V} $

$ w = mg = \rho Vg$

$p = \frac{F}{A} $

$p = p_0 + \rho g h $

#### Pressure depends on depth:

In a tank filled with a fluid, the bottom most part of the tank experiences a higher pressure than the top most part of the tank.

This is simply because if you are at the bottom, there are more layers of the fluid above your head “weighing you down” than compared to the top where you have fewer layers.

*Fact: Marina Trench is 1000 times the pressure at the sea level.*

As a result of the pressure depending on depth, any object placed in a fluid experiences a pressure difference between its top and bottom surfaces.

The top is at a lower pressure and the bottom at a higher pressure. Therefore, there is an upward force called the ‘Buoyant Force’ that acts on the object when you submerge it in a fluid.

Anything that you submerge underwater will feel lighter than it actually is because the Buoyant force acts upward on the objects an help you counteract the effect of gravity.

$ F_{net} = mg – F_B$

(i) If $ mg>F_B $, object sinks

(ii) If $ mg<F_B $, object floats

(i) If $ mg=F_B $, object remains stationary. (It is neutrally buoyant)strived(i) If $ mg=F_B $, object remains stationary. (It is neutrally buoyant)

**Note on the Buoyant force:**

- Air is also a fluid and also offers a Buoyant force on any object.
- The Buoyant force is a property of the fluid and has nothing to do with the nature of the object that you submerge. A 1 $ m^3$ of Iron, Styrofoam, Lead, etc all would feel the same Buoyant force.

#### An equation to represent Buoyant force:

So far, all of this has been qualitative. Let’s try to obtain an expression for the Buoyant force that we can work with.

Consider an object submerged in a fluid with density $ \rho_f $ as follows:

$F_B = (P_{high} – P_{low}) A $

$ F_B = \left[ p_0 + \rho_f g (h+d) – (p_0 + \rho_f g h ) \right] A $

where $ \rho_f $ -> Density of the fluid.

$ F_B = \rho_f g (d A ) $

$ F_B = \rho_f g V_{object} $

Notice that the Buoyant force only depends on the volume occupied by the object and not it’s density and as a result all objects with the same volume irrespective of its density experience the same Buoyant force!

We can make this even better by realizing that $ V_{object} = V_{fluid- displaced} $ because when an object submerges in water it pushes away all the fluid that was already there to occupy it for itself.

How much fluid does it need to displace ? For submerged objects it’s exactly how much volume it needs to accommodate the object in the fluid. Therefore we can rewrite the above formula like so:

$ F_B = (\rho_f V_{fluid-displaced}) g $

$F_B = m_f g $ (since $\rho = m/V $)

This is the Archimedes principle. It reads that the Buoyant force experienced by any object is equal to the weight of the fluid displaced.

Eureka!

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