34
3 Mechanical Aspects of Biosystems
Just as the total external force determines the motion of the center-of-mass of the
system, the net external torque determines the pure rotation of the system. Torque
measures the effectiveness of a force in causing a rotation about an axis. It is a
vector defined in size by the magnitude of force times a ‘lever arm’, the minimum
distance from the line of the force to the line of the axis of rotation. Torque as a
vector is directed along the axis in the ‘right-hand’ sense: Curl your fingers of your
right hand in the direction that the body would rotate if the force acted alone. Then
your thumb points in the direction of the torque. This idea is conveniently expressed
by the vector cross-product of the force times the displacement from the axis to the
point where the force is acting on the system: τ ≡ r × F. The cross-product of
two vectors is another vector pointing perpendicular to the vectors in the product,
directed according to the way one’s thumb points if one turns the fingers of one’s
right hand from the first to the second vector, and has a magnitude equal to the
product of the length the two vectors times the sine of the angle between them. 14
If we construct the net torque on a system of masses, we can apply Newton’s Third Law again, collecting the internal-force terms into pairs of the form
r i − r j
× F ij . These will vanish if the forces act along the line joining the
mass elements, as do gravitational and electrical, leaving only the external torques
contributing to the net torque. 15 From Newton’s 2nd Law, this net torque is given
by
m i r i × a i = d/dt
(r i × p i ). The sum defines the net angular momentum
of the system, L, so that Newton’s 2nd Law for rotation has the form
τ
ext
net =
d
dt
L .
(3.2)
If the net external torque vanishes, the angular momentum of the system is
conserved. This observation has become very useful in analyzing the behavior of
systems with internal spin. For example, an astronaut in space who exerts a torque
on a spacecraft will cause the spacecraft to spin, and the astronaut spins, but in
the opposite direction and with a greater rotational velocity, so that the total vector
angular momentum of the two, spacecraft and astronaut, still add to zero. As another
example, the spin of an electron in hydrogen can flip by absorbing a photon, because
the photon itself carries spin. The total angular momentum of atom plus photon is
conserved.
For pure rotations about the origin, every coordinate of the mass elements
changes with time according to
14 For those sending an E-mail message to another life form on some other planet who want to
know what is ‘the right-hand rule’, tell them if they turn the first vector toward the second as
an anti-neutrino rotates, then the torque will be in the direction the anti-neutrino moves forward.
No light message, audio, or visual, can communicate handedness, because photons are their own
antiparticles. If the alien’s screens show your image reversed, we could not tell just by light signals!
15 Magnetic forces acting on moving charges do not act along the line joining the charges. There
will be added torque coming from the electromagnetic field.
3 Mechanical Aspects of Biosystems
Just as the total external force determines the motion of the center-of-mass of the
system, the net external torque determines the pure rotation of the system. Torque
measures the effectiveness of a force in causing a rotation about an axis. It is a
vector defined in size by the magnitude of force times a ‘lever arm’, the minimum
distance from the line of the force to the line of the axis of rotation. Torque as a
vector is directed along the axis in the ‘right-hand’ sense: Curl your fingers of your
right hand in the direction that the body would rotate if the force acted alone. Then
your thumb points in the direction of the torque. This idea is conveniently expressed
by the vector cross-product of the force times the displacement from the axis to the
point where the force is acting on the system: τ ≡ r × F. The cross-product of
two vectors is another vector pointing perpendicular to the vectors in the product,
directed according to the way one’s thumb points if one turns the fingers of one’s
right hand from the first to the second vector, and has a magnitude equal to the
product of the length the two vectors times the sine of the angle between them. 14
If we construct the net torque on a system of masses, we can apply Newton’s Third Law again, collecting the internal-force terms into pairs of the form
r i − r j
× F ij . These will vanish if the forces act along the line joining the
mass elements, as do gravitational and electrical, leaving only the external torques
contributing to the net torque. 15 From Newton’s 2nd Law, this net torque is given
by
m i r i × a i = d/dt
(r i × p i ). The sum defines the net angular momentum
of the system, L, so that Newton’s 2nd Law for rotation has the form
τ
ext
net =
d
dt
L .
(3.2)
If the net external torque vanishes, the angular momentum of the system is
conserved. This observation has become very useful in analyzing the behavior of
systems with internal spin. For example, an astronaut in space who exerts a torque
on a spacecraft will cause the spacecraft to spin, and the astronaut spins, but in
the opposite direction and with a greater rotational velocity, so that the total vector
angular momentum of the two, spacecraft and astronaut, still add to zero. As another
example, the spin of an electron in hydrogen can flip by absorbing a photon, because
the photon itself carries spin. The total angular momentum of atom plus photon is
conserved.
For pure rotations about the origin, every coordinate of the mass elements
changes with time according to
14 For those sending an E-mail message to another life form on some other planet who want to
know what is ‘the right-hand rule’, tell them if they turn the first vector toward the second as
an anti-neutrino rotates, then the torque will be in the direction the anti-neutrino moves forward.
No light message, audio, or visual, can communicate handedness, because photons are their own
antiparticles. If the alien’s screens show your image reversed, we could not tell just by light signals!
15 Magnetic forces acting on moving charges do not act along the line joining the charges. There
will be added torque coming from the electromagnetic field.