Classical Mechanics - Fluid

An open-ended U-tube of uniform cross-sectional area contains water (density 1.0 gram/cm³) standing initially 20 cm from the bottom in each arm. An immiscible liquid of density 4.0 gr/cm³ is added to one arm until a layer 5 cm high forms. What is the ratio h₂/h₁ of the heights of the liquid in the two arms?

A. 3/1
B. 5/2
C. 2/1
D. 3/2
E. 1/1

(GR9677 #30)

Solution:

Given:
ρ_water = 1 gr/cm³
ρ_liquid = 4 gr/cm³
h_a = 5 cm

P₁ = P₂
ρ_lgh_a = ρ_wgh_b 
h_b = ρ_lh_a / ρ_w = (4)(5) / (1) = 20

h₁ = h + h_a  
h₂ = h + h_b 
-------------- −
h₁ − h₂ = h_a − h_b  = 5 − 20 = − 15 (Eq.1)

Initially, h₁ + h₂ = 20 + 20 = 40
Finally, h₁ + h₂ = 40 + 5 = 45 (Eq.2)

(Eq.1) → h₁ − h₂ = − 15
(Eq.2) → h₁ + h₂ = 45
------------------------------ +
2h₁ = − 15 + 45 = 30
h₁ = 15

h₂ = 45 − 15 = 30

h₂/h₁ = 30/15 = 2

Answer: C

Classical Mechanics - Fluid

A sphere of mass m is released from rest in a stationary viscous medium. In addition to the gravitational force of magnitude mg, the sphere experiences a retarding force of magnitude bv, where v is the speed of the sphere and b is a constant. Assume that the buoyant force is negligible. Which of the following statements about the sphere is correct?

1. Its kinetic energy decreases due to the retarding force.
2. Its kinetic energy increases to a maximum, then decreases to zero due to the retarding force.
3. Its speed increases to a maximum, then decreases back to a final terminal speed.
4. Its speed increases monotonically, approaching a terminal speed that depends on b but not on m.
5. Its speed increases monotonically, approaching a terminal speed that depends on b and m.

(GR9677 #31)

Solution:

ma = mg − bv

Terminal speed, a = 0

mg = bv
v = mg/b → (E) TRUE

(A) FALSE.
m is released from rest v_i = 0 → KE_i = 0

(B) FALSE.
Terminal speed released from rest v_f  = mg/b ≠ 0 → KE_f  ≠ 0

(C) FALSE.
v_i = 0 to v_f  = mg/b, no v_max

(D) FALSE.
 v_f  = mg/b, depend on both m and b

Answer: E

Fluid - Buoyancy

A balloon is to be filled with helium and used to suspend a mass of 300 kilograms in air. If the mass of the balloon is neglected, which of the following gives the approximate volume of helium required? (The density of air is 1.29 kilograms per cubic meter and the density of helium is 0.18 kilogram per cubic meter.)

A. 50 m³
B. 95 m³
C. 135 m³
D. 270 m³
E. 540 m³

(GR0177 #56)

Solution:

ΣF = 0
ρ_agV =  ρ_hgV + m₀g

(ρ_a -  ρ_h)V = m₀
V = m₀ / (ρ_a - ρ_h) = 300 / (1.29 - 0.18) = 270 m³

Answer: D

Fluid - Force

A stream of water of density ρ, cross-sectional area A and speed v strikes a wall that is perpendicular to the direction of the stream, as shown in the figure. The water then flows sideways across the wall. The force exerted by the stream on the wall is

A. ρv²A
B. ρvA/2
C. ρghA
D. v²A/ρ
E. v²A/2ρ

(GR0177 #57)

Solution:

Dimensional analysis:
Force has unit Newton = kg m/s²
The only choice has the right unit of force is choice A = (kg/m³)(m/s)²(m²) = kg m/s²

Answer: A