Density, Upthrust & Viscous Drag

• Density (ρ) = mass per unit volume. It tells you how tightly packed the matter in an object is.
• Upthrust is the upward push a fluid gives any object submerged (fully or partly) in it — caused by pressure being greater at greater depth.
• Archimedes' Principle: upthrust = weight of the fluid displaced by the object.
• An object floats when upthrust can equal its weight before it's fully submerged; it sinks when even fully submerged, upthrust never catches up to weight.
• Viscous drag is the "friction" a fluid exerts on an object moving through it — governed for small spheres by Stokes' Law: F = 6πηrv.
• Falling objects reach terminal velocity when weight = upthrust + viscous drag (no more resultant force, so no more acceleration).
• Core Practical 2 uses falling ball bearings in a viscous liquid to experimentally determine the liquid's viscosity, η.

What Density Actually Means

Imagine two boxes of exactly the same size. One is stuffed full of marbles, the other has just a handful rattling around inside. Even though both boxes take up the same space (same volume), the full one is much heavier. That's density in a nutshell — it's a measure of how much mass is squeezed into a given volume.

A balloon and a small lead weight can occupy very different volumes, and yet the lead — despite being physically smaller — is far denser, because its atoms are packed much more tightly and each atom itself is heavier.

Working Out the Volume

Often the volume isn't handed to you directly — you have to calculate it from the object's shape and dimensions first. The three you'll meet most:

   SPHERE:            CUBE:              CYLINDER:
   V = (4/3)πr³        V = d³             V = πr²l

     .-‾‾-.             ┌────┐              ___
    /      \            │    │d            ( r )
   |   •r   |         d │    │              |  |
    \      /            └────┘              | l|
     ‾-..-‾                d                |__|
        

Unit Conversions — the part everyone slips up on