• 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

Why Things Float

Push a beach ball underwater and let go — it shoots back up. That upward shove is upthrust, and it exists because pressure in a fluid increases with depth. The bottom of a submerged object feels more pressure pushing up on it than the top feels pushing down, and that pressure difference creates a net upward force.

Here's the intuition: when you push an object into water, it has to shove the water out of the way — it displaces a volume of water equal to its own submerged volume. Archimedes' Principle says the upthrust you feel is exactly equal to the weight of that displaced water.

Floating vs Sinking

• An object sinks until the weight of fluid it has displaced equals its own weight.
• If that balance point happens before the object is fully submerged → it floats.
• If the object is denser than the fluid, it will still be sinking (accelerating downward) even when fully submerged, because it can never displace enough weight of fluid to match its own weight → it sinks all the way.

              Weight (down, = buoyancy force)
                        |
                        v
                  ┌───────────┐
                  │   SHIP    │
        ~~~~~~~~~~└───────────┘~~~~~~~~~~~
                        ^
                        |
              Buoyancy force (up, = weight of
              displaced water)

     mass of displaced water = mass of ship submerged
        

Calculating Upthrust — the 3-Step Method

1. Find the volume of the submerged part of the object (= volume of fluid displaced).
2. Use ρ = m/V (rearranged: m = ρV) to find the mass of that displaced fluid.
3. Use W = mg to find the weight of the displaced fluid — that weight is the upthrust.

What Is Viscous Drag?

Pour water from a jug and it glugs out easily. Try to pour honey and it crawls out reluctantly. That resistance to flowing — and the resistance a moving object feels as it pushes through the fluid — is viscosity. The frictional force this creates on a moving object is viscous drag.

Viscosity itself (η, the Greek letter "eta") is a property of the fluid — how "thick" it is at a given temperature. Low viscosity fluids (water) pour and flow easily. High viscosity fluids (honey, tomato ketchup) resist flowing. Crucially: the rate of flow of a fluid is inversely proportional to its coefficient of viscosity — thicker fluid, slower flow.

Stokes' Law

Laminar Flow vs Turbulent Flow

As a fluid flows around an object (or the object moves through it), the fluid forms into layers. In laminar flow, every layer moves in the same direction and none of them mix — this happens for slow-moving objects in slow-flowing fluid, and it's the only situation where Stokes' Law applies. In turbulent flow, the layers move in different, chaotic directions and mix together — think of rapids in a river versus a calm stream.

   LAMINAR FLOW                    TURBULENT FLOW
   (layers stay separate,          (layers mix and swirl,
    all same direction)             chaotic directions)

   ──────►  ●  ──────►             ~~~↷  ●  ↶~~~
   ──────►     ──────►             ↷~~     ~~↶
   ──────►  ──────►                  ~~↷↶~~
        

Effect of Temperature on Viscosity

• Liquids get less viscous as temperature increases (heat makes molecules move more freely past each other).
• Gases get more viscous as temperature increases (opposite behaviour to liquids — a classic trick question).

Terminal Velocity of a Falling Sphere

Drop a ball bearing into a tall tube of oil. At first it accelerates downward under gravity. But as it speeds up, viscous drag (and upthrust) grow larger and larger, fighting back against its fall. Eventually these forces balance out weight exactly — at that point there's no resultant force, so no more acceleration. The sphere carries on at a constant speed: terminal velocity.

              ↑ F_d (viscous drag)
              ↑ U (upthrust)
             ( o )   ← sphere falling through fluid
              |
              ↓ W (weight)

     At terminal velocity: W = F_d + U  (forces balanced)
        

Deriving the Terminal Velocity Equation

This derivation is genuinely worth being able to reproduce — it shows up as "derive an expression for..." in exams. Walk through it slowly:

1. At terminal velocity: W_s = W_f + 6πηrv_term (sphere's weight = weight of displaced fluid + viscous drag)
2. Mass of sphere: m_s = ρ_sV = (4/3)πr³ρ_s, so W_s = (4/3)πr³ρ_sg
3. Mass of displaced fluid (same volume as sphere): m_f = ρ_fV = (4/3)πr³ρ_f, so W_f = (4/3)πr³ρ_fg
4. Substitute both into the balance equation: (4/3)πr³ρ_sg = (4/3)πr³ρ_fg + 6πηrv_term
5. Rearrange for v_term, cancelling one factor of r from top and bottom:

Aim & Setup

Aim: drop small spherical ball bearings through a viscous liquid, let them reach terminal velocity, then use the terminal velocity equation (rearranged for η) to calculate the fluid's viscosity.

Equipment

• Long measuring cylinder
• Viscous liquid (e.g. thin oil of known density, or washing-up liquid)
• Stand and clamp
• Metre rule & rubber bands (as distance markers)
• Steel ball bearings of different diameters
• Digital scales & Vernier calipers (to find sphere density)
• Digital stopwatch
• Magnet (to retrieve the balls without draining the tube!)

Method — Step by Step

1. Weigh the balls, measure their radius with Vernier calipers, and calculate their density.
2. Place three rubber bands around the outside of the tube. The highest band must be far enough below the liquid's surface that the ball has definitely reached terminal velocity by the time it passes it (if the ball is still accelerating when it crosses the first marker, move that marker further down). The remaining two bands should be 10–15 cm apart for accurate timing.
3. Release the ball and start the timer as it passes the first (highest) band. Use a lap timer to record the time taken to fall distance d₁ (to the middle band) and d₂ (to the lowest band).
4. Measure and record d₁ (highest to middle band) and d₂ (highest to lowest band) using the metre rule.
5. Repeat at least 3 times for that diameter, then repeat the whole process for each different ball diameter.
6. Retrieve the ball bearings from the bottom using the magnet held against the outside of the cylinder.

Analysis — Deriving η from the Data

At terminal velocity the sphere is in equilibrium: W_s = F_d + U. Substituting the weight, drag, and upthrust expressions in (exactly as in the derivation above) and rearranging for η instead of v_term gives:

Evaluating the Experiment

Safety

• Measuring cylinders are unstable — clamp them at both top and bottom.
• Clean up spillages immediately (viscous liquids are extremely slippery underfoot).
• Avoid getting the fluid in your eyes.

• I can define density and state the formula ρ = m/V with correct units.
• I can calculate the volume of spheres, cubes, and cylinders and convert between mm³, cm³, and m³.
• I can state Archimedes' Principle word-for-word.
• I can explain the 3-step method for calculating upthrust (volume → mass → weight).
• I can explain why an object floats or sinks in terms of displaced fluid weight.
• I can define viscous drag and state Stokes' Law, F = 6πηrv.
• I know the four conditions required for Stokes' Law to apply (small, spherical, slow, laminar).
• I can distinguish laminar flow from turbulent flow.
• I know how viscosity changes with temperature for liquids AND gases.
• I can explain what "terminal velocity" means in terms of balanced forces (W = U + F_d).
• I can derive v_term = 2r²g(ρ_s − ρ_f)/9η from first principles.
• I can describe the full method for Core Practical 2, including how markers are positioned.
• I can identify systematic and random errors in the viscosity practical.
• I can rearrange the terminal velocity equation to solve for η instead of v.