3.2 Newton’s Laws Applied to a Biosystem
33
Newton knew these laws in 1666, and formally published them in 1687, in
his book entitled Principia. A complete description of electromagnetic fields and
forces came in 1865 with Maxwell’s paper on the subject. A new theory of gravity,
encompassing Newton’s for weak gravitational fields, was presented by Einstein in
1915. With quantum theory interpretations, no observational deviations have been
found from Maxwell’s or Einstein’s theory in over one hundred years.
3.2.2 Application of Newtonian Ideas
We can use Newtonian laws of dynamics to describe mechanical equilibrium, kinetics, and dynamics within life forms and between life forms and their environment.
We will examine several illustrative examples.
According to Newton, the external forces on every mass m within a system
determines its motion through the relation F = ma, where a is the acceleration
of the mass. By applying the left- and right-hand sides of Newton’s 2nd Law to each
element in a system of masses, and then adding, each element labeled by the number
‘i’, the right-hand side can be written d 2
m i r i /dt 2 . This expression can be recast
into the form (
m i )d 2 {(
m i r i )/(
m i )}/dt 2 , i.e., the mass of the whole system
times the acceleration of a special location within called the center-of-mass, defined
by R ≡
m i r i /(
m i ). Alternatively, the right-hand side is also the change with
time of the total linear momentum of the system, P ≡
m i v i . On the left-hand side
is the sum of external and internal forces on each element of mass in the system.
But because of Newton’s Third Law, F ij = −F ji , the internal forces cancel in
pairs, leaving the statement that only the external forces determine the motion of
the center-of-mass of a system. We have
F
ext
=
d
dt
P .
(3.1)
One evident consequence: No internal forces in a bacterium can change the
position of the center-of-mass of the bacterium! For humans, you cannot lift yourself
up by your ‘bootstraps’. (Boots once had straps! 13 )
Thus, if the external forces on the system add to zero, its momentum will be
conserved, i.e. not changed in time, and the center-of-mass, if initially at rest, will
remain at rest. However, relative motion within the system is still possible. Rotation
is a subclass of any relative motion which does not affect the distance from a
given system point to any of other system points. More generally, if the distances
between mass elements change with time, then the shape of the system changes or
is distorted. Life systems are of this kind.
13 The phrase, ‘booting your computer’ comes from this old adage. The earliest computers had
no static memory, so that an operator had to enter the code to start the machine. Figuratively, the
computer lifted itself up. (With a little help from its creators!)
33
Newton knew these laws in 1666, and formally published them in 1687, in
his book entitled Principia. A complete description of electromagnetic fields and
forces came in 1865 with Maxwell’s paper on the subject. A new theory of gravity,
encompassing Newton’s for weak gravitational fields, was presented by Einstein in
1915. With quantum theory interpretations, no observational deviations have been
found from Maxwell’s or Einstein’s theory in over one hundred years.
3.2.2 Application of Newtonian Ideas
We can use Newtonian laws of dynamics to describe mechanical equilibrium, kinetics, and dynamics within life forms and between life forms and their environment.
We will examine several illustrative examples.
According to Newton, the external forces on every mass m within a system
determines its motion through the relation F = ma, where a is the acceleration
of the mass. By applying the left- and right-hand sides of Newton’s 2nd Law to each
element in a system of masses, and then adding, each element labeled by the number
‘i’, the right-hand side can be written d 2
m i r i /dt 2 . This expression can be recast
into the form (
m i )d 2 {(
m i r i )/(
m i )}/dt 2 , i.e., the mass of the whole system
times the acceleration of a special location within called the center-of-mass, defined
by R ≡
m i r i /(
m i ). Alternatively, the right-hand side is also the change with
time of the total linear momentum of the system, P ≡
m i v i . On the left-hand side
is the sum of external and internal forces on each element of mass in the system.
But because of Newton’s Third Law, F ij = −F ji , the internal forces cancel in
pairs, leaving the statement that only the external forces determine the motion of
the center-of-mass of a system. We have
F
ext
=
d
dt
P .
(3.1)
One evident consequence: No internal forces in a bacterium can change the
position of the center-of-mass of the bacterium! For humans, you cannot lift yourself
up by your ‘bootstraps’. (Boots once had straps! 13 )
Thus, if the external forces on the system add to zero, its momentum will be
conserved, i.e. not changed in time, and the center-of-mass, if initially at rest, will
remain at rest. However, relative motion within the system is still possible. Rotation
is a subclass of any relative motion which does not affect the distance from a
given system point to any of other system points. More generally, if the distances
between mass elements change with time, then the shape of the system changes or
is distorted. Life systems are of this kind.
13 The phrase, ‘booting your computer’ comes from this old adage. The earliest computers had
no static memory, so that an operator had to enter the code to start the machine. Figuratively, the
computer lifted itself up. (With a little help from its creators!)