Lesson summary
4.3.5 Archimedes' Principle — AS PHYSICS 9702 notes: State Archimedes' principle and use it for floating objects, weighing in a fluid and density measurement.Archimedes' principle = an object submerged in a fluid at rest has an upward buoyancy force (upthrust) equal to the weight…
- U = mfg = ρfVdispg
- Since m = ρV
- For a fully submerged object Vdisp equals its volume
- Two-step method
- That fluid sits in equilibrium
- The surrounding fluid cannot tell what has replaced the lump
Section 1 of 9The principle
1Archimedes' principle
Archimedes' principle an object submerged in a fluid at rest has an upward buoyancy force (upthrust) equal to the weight of the fluid displaced by the object.
2U
U mfg = ρfVdispg
- mf is the mass of displaced fluid, ρf the fluid density and Vdisp the displaced volume.
3Since m = ρV
- Since m = ρV the upthrust equals the weight of the displaced fluid.
4For a fully submerged object Vdisp equals its volume
- For a fully submerged object Vdisp equals its volume for a floating object it is only the submerged part.
5Two-step method
- Two-step method find the volume of fluid displaced, then use m = ρV and W = mg.
Section 2 of 9Why it holds
1
Replace the object with an equal volume of the fluid itself.
2That fluid sits in equilibrium
- That fluid sits in equilibrium so the surrounding fluid supplies an upward force equal to its weight.
3The surrounding fluid cannot tell what has replaced the lump
- The surrounding fluid cannot tell what has replaced the lump so the forces on the surface are unchanged.
4
It agrees with the pressure-difference derivation U = ρfgV (see 4.3.4).
Section 3 of 9Floating
1
The object sinks until the weight of the fluid displaced equals its own weight.
2An object floats when the upthrust equals its weight
- An object floats when the upthrust equals its weight so it displaces its own mass of fluid.
3Boats float because they displace an amount of water equal to their weight
- Boats float because they displace an amount of water equal to their weight a 50 000 tonne ship displaces 50 000 tonnes of water.
4A ship floats slightly higher in sea water than fresh water
- A ship floats slightly higher in sea water than fresh water the Plimsoll line marks safe loading depths.
5A person getting into a floating boat
- A person getting into a floating boat makes it sink until the extra displaced water weighs as much as the person.
6
The principle applies to gases as well as liquids.
Section 4 of 9Using the principle to find density
1
Weigh the object in air (Wair) and fully in water (Wwater).
2
Upthrust U = Wair − Wwater = ρwgV.
3
Volume V = (Wair − Wwater) ÷ (ρwg).
4
Density of the object ρo = ρw × Wair ÷ (Wair − Wwater).
5Weighing in an unknown liquid
- Weighing in an unknown liquid gives its density, ρliquid = (Wair − Wliquid) ÷ gV.
6A hydrometer sinks until it displaces its own weight of liquid
- A hydrometer sinks until it displaces its own weight of liquid the denser the liquid, the less it sinks.
Section 5 of 9Verifying the principle
1
Hang an object from a newton-meter above a displacement can full of water.
2
Lower the object into the water and note the falling meter reading.
3
Collect the overflow in a beaker.
4
Weigh the beaker before and after, and subtract to get the weight of displaced water.
5
The weight of displaced water equals the drop in meter reading (the upthrust) within experimental error.
Section 6 of 9Types of calculation
Types of calculation
Floating object
- Floating object set weight equal to upthrust and solve for the submerged fraction or the fluid density.
Held fully submerged by a string
- Held fully submerged by a string tension T = W − U, or T = U − W if the object is lighter than the fluid.
Weighing question
- Weighing question use the difference between readings in air and in fluid to find U, then V and density.
A loaded barge moving from sea to fresh water
- A loaded barge moving from sea to fresh water the displaced volume rises in proportion to 1030 ÷ 1000.
Write the equation in symbols before substituting numbers.
Section 7 of 9More from the specification
More from the specification
Hydrostatic pressure and liquid depth
- Hydrostatic pressure and liquid depth hydrostatic pressure in fluids increases with depth,
- producing greater upward force on the bottom of an immersed cylinder.
Atmospheric pressure and unit conversion
- Atmospheric pressure and unit conversion converting atmospheric pressure quantities to pascals ensures accurate calculations of total pressure acting beneath open liquid columns.
Volume and mass density relations
- Volume and mass density relations multiplying fluid density in kilograms per cubic metre by displaced volume yields the mass of displaced liquid.
Statement of Archimedes' principle
- An object wholly or partially immersed in a fluid experiences an upward buoyant force (upthrust) equal to the weight of the fluid displaced by the object.
Hydrostatic origin of upthrust
- Upthrust arises because fluid pressure increases with depth;
- the upward hydrostatic force acting on the bottom surface of an immersed body exceeds the downward force on its top surface.
Condition for equilibrium and floating
- A floating body reaches stable equilibrium when its displaced fluid volume has a weight that exactly balances the total weight of the body.
Manometer
- Device measuring pressure differences with a liquid column.
Hydrostatic pressure
- P = P0 + ρgh.
Atmospheric pressure
- Pressure of the air, about 1.01 × 105 Pa at sea level.
Gravitational field
- Region where mass experiences gravitational force.
Field strength
- G = Fg/m.
Section 8 of 9Key terms
| Term | Meaning |
|---|---|
| Archimedes' principle | The upthrust on an object in a fluid equals the weight of fluid displaced. |
| Upthrust | The upward buoyancy force on an object in a fluid. |
| Displaced volume | The volume of fluid pushed aside by the immersed part of an object. |
| Plimsoll line | A mark on a ship's hull showing the safe loading depth. |
| Hydrometer | A floating instrument used to measure liquid density. |
| Apparent weight | The weight measured when an object is immersed, W − U. |
| Displacement can | A can used to collect water displaced by an object. |
| Neutral buoyancy | The state where upthrust equals weight with the object fully submerged. |
Section 9 of 9Exam technique
1
Saying the upthrust is the weight of the object.
2
Using the total volume of a floating object instead of the submerged volume.
3
Forgetting to subtract the weight of the empty beaker in the experiment.
4
Using the density of the object in the formula for upthrust.
5
Believing the principle only applies to liquids — it applies to gases too.