High School Physics
(Dated: July 4, 2005)
∆x Fr
l 7. Surface tension, Laplace law
surface enery
work done ∆W = F · ∆x surface energy increased ∆Us
F = T · , T : surface tension
∴ ∆Us = T · · ∆x = T · ∆A ∴ T = ∆Us
∆A surface energy density Relation betw the pressure difference and the radius of the bubble.
two surfaces, Laplace law.
1. Force approach
Pi P0
(Pi− P0)· πR2 = 2· T · 2πR (bubble) Pi− P0 = 4T
R (bubble with two surfaces)
= 2T
R (drop with only one surface)
= T
R (tube with cylindrical surface)
2. energy approach R→ R + ∆R
V = 4
3πR3 ∴ ∆V = 4πR2· ∆R A = 4πR2 ∴ ∆A = 8πR · ∆R
2
Pi
P0
∆x work done to expand
∆W = (Pi− P0)· A · ∆x
= (Pi− P0)· ∆V
work done = increase of surface energy
∆W = (Pi− P0)· 4πR2∆R
= ∆Us= T · ∆A
= T · 8πR · ∆R (drop)
∴ Pi− P0 = 2T
R (drop)
8. Contact of two bubbles with different size
R1
R2
contact of two bubbles
R1 > R2 →? R1 ↑ , R2 ↓
1. How the bubbles inside the lung work?
surfacant
2. How the strokes in blood vessel happen.
R
State 1
State 2
l l
h
9. surface tension forbids the down flow of the liquid if R ≤ Rth.
energy approach state 1 → state 2
gravitational P.E. ∆UG ↓ area of surface ∆A↑ surface energy ∆Us ↑
if ∆UG+ ∆Us > 0 stable against the transition from state 1 to state 2 or ≤ threshold
3
∆A = 2·
"
1 +
µ4h¶2#12
− 2 ≈ 8h2
∴ ∆Us = 8h2 · T
∆m = 1 4ρ 2h
∴ ∆UG =−1
4ρ 2h· g ·2
3h =−1
6ρg 2h2
∴ ∆UG+ ∆Us ≥ 0 if <
µ48T ρ· g
¶12
= th
for water, T = 72 N/m ∴ th = 1.88 cm
