36
3 Mechanical Aspects of Biosystems
The displacements r i to each mass m i are calculated from the center of mass. The
symbol δ represents the ‘identity tensor’, defined so that when ‘dotted’ into a vector,
it turns that vector into itself. The factored expression in the braces depends only on
the geometric distribution of the masses, and is called the ‘moment of inertia tensor’.
For systems whose masses are distributed about the axis symmetrically, the angular
momentum L expression simplifies to I ω, wherein the moment of inertia for the
system becomes a number given by I ≡
m i ρ 2
i , where ρ i is the distance from the
axis of the mass m i . 16
An ice skater spinning on the tip of her skate has some frictional force acting
on her skate, but with a relatively small lever arm (no bigger than the radius of the
small circle the skate carves in the ice when she spins). This means the external
torque on her will be relatively small, and the value of her angular momentum,
L = I ω =
mρ 2
ω, will not change much over several rotations. If she pulls in
her arms, some mass in her arms will have a smaller distance from the axis, making
her moment of inertia smaller. From the conservation of angular momentum we
conclude that as she pulls in her arms, her angular speed must increase, keeping I ω
constant. 17
Examples of spinning objects in biology include the tail of certain bacteria,
ballerinas, and centrifuges to separate organics.
3.2.3 Conserved Quantities
Isaac Newton did not use energy conservation in solving for the motion of masses,
since, in the problems he posed, the forces were known or could be found. 18
However, as we will describe, energy conservation is deeply seated in Nature and
has wide practical purpose, even for dissipative processes if heat and microscopic
kinetic motion are recognized.
In a brilliant paper in 1918, Emmy Nöther 19 showed that for every continuous
symmetry of a system, there will be a corresponding conserved quantity, i.e. a
function of the dynamical variables which will not change in time. With Nöther’s
theorem, momentum conservation is a consequence of the deeper idea of symmetry
under translations. If the dynamics of a system does not change when the whole
16 Some students learning integral calculus have fun finding I = (M/V )
ρ 2 dV for various simple
uniform-density objects of mass M and volume V , such as a ball of radius a, for which I ball =
2
5 Ma 2 ; a cylinder of radius a rotating about its axis, for which I cyl =
1
2 Ma 2 , or a stick of length l
rotated about a perpendicular axis through its center, for which I stick =
1
12 Ml 2 .
17 The increase in her kinetic energy comes from the work she did in pulling in her arms.
18 Also, the energy concept was just evolving. Gottfried Leibniz used the phrase life-force (‘visviva’) for twice the kinetic energy of the particles. Adding the potential energy came in the work
of Joseph-Louis Lagrange.
19 E. Nöther, Invariante Variationsprobleme, Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen,
Math-phys. Klasse, 235–257 (1918).
3 Mechanical Aspects of Biosystems
The displacements r i to each mass m i are calculated from the center of mass. The
symbol δ represents the ‘identity tensor’, defined so that when ‘dotted’ into a vector,
it turns that vector into itself. The factored expression in the braces depends only on
the geometric distribution of the masses, and is called the ‘moment of inertia tensor’.
For systems whose masses are distributed about the axis symmetrically, the angular
momentum L expression simplifies to I ω, wherein the moment of inertia for the
system becomes a number given by I ≡
m i ρ 2
i , where ρ i is the distance from the
axis of the mass m i . 16
An ice skater spinning on the tip of her skate has some frictional force acting
on her skate, but with a relatively small lever arm (no bigger than the radius of the
small circle the skate carves in the ice when she spins). This means the external
torque on her will be relatively small, and the value of her angular momentum,
L = I ω =
mρ 2
ω, will not change much over several rotations. If she pulls in
her arms, some mass in her arms will have a smaller distance from the axis, making
her moment of inertia smaller. From the conservation of angular momentum we
conclude that as she pulls in her arms, her angular speed must increase, keeping I ω
constant. 17
Examples of spinning objects in biology include the tail of certain bacteria,
ballerinas, and centrifuges to separate organics.
3.2.3 Conserved Quantities
Isaac Newton did not use energy conservation in solving for the motion of masses,
since, in the problems he posed, the forces were known or could be found. 18
However, as we will describe, energy conservation is deeply seated in Nature and
has wide practical purpose, even for dissipative processes if heat and microscopic
kinetic motion are recognized.
In a brilliant paper in 1918, Emmy Nöther 19 showed that for every continuous
symmetry of a system, there will be a corresponding conserved quantity, i.e. a
function of the dynamical variables which will not change in time. With Nöther’s
theorem, momentum conservation is a consequence of the deeper idea of symmetry
under translations. If the dynamics of a system does not change when the whole
16 Some students learning integral calculus have fun finding I = (M/V )
ρ 2 dV for various simple
uniform-density objects of mass M and volume V , such as a ball of radius a, for which I ball =
2
5 Ma 2 ; a cylinder of radius a rotating about its axis, for which I cyl =
1
2 Ma 2 , or a stick of length l
rotated about a perpendicular axis through its center, for which I stick =
1
12 Ml 2 .
17 The increase in her kinetic energy comes from the work she did in pulling in her arms.
18 Also, the energy concept was just evolving. Gottfried Leibniz used the phrase life-force (‘visviva’) for twice the kinetic energy of the particles. Adding the potential energy came in the work
of Joseph-Louis Lagrange.
19 E. Nöther, Invariante Variationsprobleme, Nachr. D. König. Gesellsch. D. Wiss. Zu Göttingen,
Math-phys. Klasse, 235–257 (1918).