Lesson 12 of 12 · 19 min
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Must-know facts
22 facts
- 1Pressure P = F/A (normal force); scalar; unit Pa = N m⁻²; dimensions [ML⁻¹T⁻²].
- 21 atm = 1.013 × 10⁵ Pa (1.01 × 10⁵ in the summary); 1 bar = 10⁵ Pa; 1 torr = 1 mm of Hg = 133 Pa.
- 3Density of water at 4 °C = 1.0 × 10³ kg m⁻³; relative density is a pure number.
- 4Pascal's law: equal pressure at equal heights in a fluid at rest; applied pressure is transmitted undiminished to every point.
- 5P = P_a + ρgh; gauge pressure P − P_a = ρgh; container shape does not matter (hydrostatic paradox).
- 6Barometer: P_a = ρgh, about 76 cm of mercury at sea level.
- 7Hydraulic lift: F₂ = F₁A₂/A₁, mechanical advantage A₂/A₁; distances A₁L₁ = A₂L₂.
- 8Syringes 1.0 cm and 3.0 cm diameter: 10 N becomes 90 N; 6.0 cm in gives 0.67 cm out.
- 9Streamlines never cross in steady flow; above a critical speed flow turns turbulent.
- 10Continuity: Av = constant for an incompressible fluid (conservation of mass).
- 11Bernoulli: P + ½ρv² + ρgh = constant along a streamline (energy conservation, ideal fluid, steady flow).
- 12Torricelli: v = √(2gh) from an open tank, the free-fall speed.
- 13Magnus effect: a spinning ball gets lift because air moves faster on one side than the other.
- 14Aircraft example: ΔP = 6.5 × 10³ N m⁻², upper-surface air only about 8% faster.
- 15η = (F/A)/(v/l); unit Pa s = N s m⁻² = poiseuille; dimensions [ML⁻¹T⁻¹].
- 16Liquid viscosity falls with temperature; gas viscosity rises.
- 17Stokes: F = 6πηav; terminal velocity v_t = 2a²(ρ − σ)g/(9η) ∝ a².
- 18Surface tension S = F/l per surface = surface energy per unit area; unit N m⁻¹ = J m⁻²; water 0.0727 N m⁻¹ at 20 °C.
- 19Surface tension usually falls with temperature.
- 20Angle of contact: acute for water-glass (wets), obtuse for mercury-glass (does not wet); S_la cos θ + S_sl = S_sa.
- 21Excess pressure: drop or air bubble in a liquid 2S/r; soap bubble in air 4S/r.
- 22Capillary rise h = 2S cos θ/(ρga); 2.98 cm for water in a 0.05 cm-radius tube; mercury is depressed.
Common traps
Where marks are lost
Treating pressure as a vector because force is a vector.
Using absolute pressure where gauge pressure is asked, or the other way round.
Thinking a wider container gives a larger pressure at the bottom.
Using the diameter ratio instead of its square in a hydraulic lift.
Believing a hydraulic lift gives work for free.
Expecting higher pressure where a pipe narrows.
Applying Bernoulli's equation to turbulent or viscous flow.
Saying the viscosity of air falls when it is heated.
Using 2S/r for a soap bubble in air.
Using the tube's diameter for a in h = 2S cos θ/(ρga).
Formulas
16 to know
Pressure
P = F/A; P = lim ΔF/ΔA as ΔA → 0
Normal force only; scalar.
Density
ρ = m/V
Relative density = ρ/ρ_water(4 °C).
Pressure with depth
P₂ − P₁ = ρgh; P = P_a + ρgh
Gauge pressure = ρgh.
Barometer
P_a = ρ_Hg g h
h ≈ 76 cm at sea level.
Hydraulic lift
F₂ = F₁A₂/A₁; A₁L₁ = A₂L₂
Mechanical advantage A₂/A₁.
Continuity
A₁v₁ = A₂v₂ (Av = constant)
Incompressible steady flow; Av is the volume flow rate.
Bernoulli's equation
P + ½ρv² + ρgh = constant
Along a streamline; ideal fluid.
Speed of efflux
v = √[2gh + 2(P − P_a)/ρ]; open tank v = √(2gh)
Torricelli's law.
Lift on a wing
ΔP × A = weight; ΔP = ½ρ(v₂² − v₁²)
(v₂ − v₁)/v_av = ΔP/(ρv_av²).
Coefficient of viscosity
η = (F/A)/(v/l) = Fl/(vA)
Pa s; [ML⁻¹T⁻¹].
Stokes' law
F = 6πηav
Sphere of radius a at speed v.
Terminal velocity
v_t = 2a²(ρ − σ)g/(9η)
ρ sphere, σ fluid.
Surface tension
S = F/2l (film with two surfaces); S_la = W/2l
N m⁻¹ = J m⁻².
Angle of contact
S_la cos θ + S_sl = S_sa
θ acute: wets; θ obtuse: does not wet.
Excess pressure
drop or cavity: 2S/r; soap bubble: 4S/r
Concave side at higher pressure.
Capillary rise
h = 2S cos θ/(ρga)
a = tube radius; negative h (depression) when θ > 90°.
Key terms
16 terms
- Fluid
- A substance that can flow: a liquid or a gas.
- Pressure
- Normal force per unit area; a scalar measured in pascals.
- Relative density
- Density of a substance divided by the density of water at 4 °C.
- Gauge pressure
- Pressure above atmospheric pressure, P − P_a.
- Hydrostatic paradox
- Differently shaped connected vessels fill to the same level because only depth sets the pressure.
- Torr
- Pressure of 1 mm of mercury, 133 Pa.
- Steady flow
- Flow in which the velocity at each fixed point does not change with time.
- Streamline
- The path of a fluid particle in steady flow; its tangent gives the local velocity.
- Turbulent flow
- Unsteady, irregular flow that sets in above the critical speed.
- Equation of continuity
- Av = constant: conservation of mass for an incompressible fluid.
- Magnus effect
- Sideways or upward force on a spinning ball moving through a fluid.
- Viscosity
- Internal friction between layers of a fluid in relative motion.
- Terminal velocity
- Constant speed of a body falling through a fluid when drag and buoyancy balance its weight.
- Surface tension
- Force per unit length, or energy per unit area, of a liquid's interface.
- Angle of contact
- Angle inside the liquid between the liquid surface and the solid at their line of contact.
- Capillary rise
- The climb of a wetting liquid in a narrow tube, driven by surface tension.