46
3 Mechanical Aspects of Biosystems
given mass. Boltzmann showed that when equilibrium is reached, the probability
that a particle will be found to have an energy near in a system of such interacting
particles is proportional to exp (− B T ). The energy of a macromolecule at radius
r in a centrifuge relative to its energy at some larger distance from the center can be
found by the work needed to move that macromolecule from the larger radius to r.
The force (in the rotating frame) on the macromolecule is the ‘weight’ mg minus
the buoyant force, where g = ω 2 r, so =
(mω 2 r − ρ F V ω 2 r)dr. (We can neglect
the force from the Earth, mg, as g >> g.)
Letting ρ M be the density of the macromolecule, we find that the concentration,
C, of the macromolecules at the radius r in terms of the concentration at radius r 0
is given by
C = C o exp
1
2k B T
mω
2
r
2
− r
2
o
(1 − ρ F /ρ M )
.
(3.10)
The concentrations at various radii can be measured with an optical centrifuge.
From the concentration gradient, and the independently measured density of the
macromolecules, one can solve for the mass of an individual macromolecule.
If an object moves sufficiently fast in a fluid, the drag force will no longer be
proportional to the velocity. The transition occurs as the fluid flow changes from
streamline to turbulent flow around the object. 27 If the object must push fluid ahead
as it moves, then the momentum transfer slows the object. The momentum loss of
the object per unit time will be the object’s velocity times the mass of fluid taking
up momentum. The latter is proportional to the volume swept up by the object per
unit time, which is the density of the fluid times the object’s cross-sectional area A
times the object’s velocity. Thus, the net drag force has the form
f drag = −bv −
1
2
C d ρAv
2 .
(3.11)
For a spherical ball, the quadratic drag coefficient can be approximated by C d ≈
24/R e for R e < 60 and C d ≈ 0.4 for 60 < R e < 2 × 10 5 , where the unitless
‘Reynolds’ number R e = 2ρav/η for a fluid of viscosity η moving around a ball
with radius a.
3.4 Forces on Materials
And thus Nature will be very conformable to herself and very simple, performing all the
great motions of the heavenly bodies by the attraction of gravity which intercedes those
27 As we will describe in the chapter on fluid dynamics, this transition can be characterized by the
value of the Reynolds number, which is R e = 2ρav/η for a spherical body. As we will discuss,
turbulence occurs for large values of the Reynolds number.
3 Mechanical Aspects of Biosystems
given mass. Boltzmann showed that when equilibrium is reached, the probability
that a particle will be found to have an energy near in a system of such interacting
particles is proportional to exp (− B T ). The energy of a macromolecule at radius
r in a centrifuge relative to its energy at some larger distance from the center can be
found by the work needed to move that macromolecule from the larger radius to r.
The force (in the rotating frame) on the macromolecule is the ‘weight’ mg minus
the buoyant force, where g = ω 2 r, so =
(mω 2 r − ρ F V ω 2 r)dr. (We can neglect
the force from the Earth, mg, as g >> g.)
Letting ρ M be the density of the macromolecule, we find that the concentration,
C, of the macromolecules at the radius r in terms of the concentration at radius r 0
is given by
C = C o exp
1
2k B T
mω
2
r
2
− r
2
o
(1 − ρ F /ρ M )
.
(3.10)
The concentrations at various radii can be measured with an optical centrifuge.
From the concentration gradient, and the independently measured density of the
macromolecules, one can solve for the mass of an individual macromolecule.
If an object moves sufficiently fast in a fluid, the drag force will no longer be
proportional to the velocity. The transition occurs as the fluid flow changes from
streamline to turbulent flow around the object. 27 If the object must push fluid ahead
as it moves, then the momentum transfer slows the object. The momentum loss of
the object per unit time will be the object’s velocity times the mass of fluid taking
up momentum. The latter is proportional to the volume swept up by the object per
unit time, which is the density of the fluid times the object’s cross-sectional area A
times the object’s velocity. Thus, the net drag force has the form
f drag = −bv −
1
2
C d ρAv
2 .
(3.11)
For a spherical ball, the quadratic drag coefficient can be approximated by C d ≈
24/R e for R e < 60 and C d ≈ 0.4 for 60 < R e < 2 × 10 5 , where the unitless
‘Reynolds’ number R e = 2ρav/η for a fluid of viscosity η moving around a ball
with radius a.
3.4 Forces on Materials
And thus Nature will be very conformable to herself and very simple, performing all the
great motions of the heavenly bodies by the attraction of gravity which intercedes those
27 As we will describe in the chapter on fluid dynamics, this transition can be characterized by the
value of the Reynolds number, which is R e = 2ρav/η for a spherical body. As we will discuss,
turbulence occurs for large values of the Reynolds number.