(Torque vector / Transformation of
torque / Axial vector)
In this section, Feynman
discusses the concept of a torque vector, the transformation of the
torque by rotation, and an axial vector.
1. Torque vector:
“…from Newton’s laws we see that we did not have to assume that the motion
was in a plane; when we differentiate xpy − ypx, we get xFy − yFx, so this theorem is still right (Feynman et al.,
1963, section 20–1 Torques in three
dimensions).”
According to
Feynman, if a rigid body is rotating in three dimensions, what we deduced for
two dimensions is still applicable. For example, it is still true that xFy − yFx is the torque in the xy-plane, or the torque is “around the z-axis.” This torque is also equal to
the rate of change of angular momentum, xpy − ypx, by using Newton’s second law of motion. Feynman’s explanation is not
sufficient because one may prefer further discussion of the difference in sign
in xFy − yFx. As a suggestion, it is good to let students derive the torque vector
by resolving a force into two components, Fx and Fy. They should be able to deduce independently that the moments due to the
two components are clockwise and anti-clockwise respectively, or vice versa.
Feynman
cautions that one may get the wrong sign for a quantity if the coordinates are
not handled in the right way. He clarifies that we may write τyz = zFy – yFz because a coordinate system may be either “right-handed” or
“left-handed.” The handedness of a coordinate system or rotation is merely a
convention in which we can assign clockwise to be the positive direction. To be
more precise, if a screw is right-handed,
it means that the screw is moved forward
when it is rotated clockwise. Clockwise and anti-clockwise are also dependent on the perspective of an
observer. An operational definition of handedness can be provided using Wu’s
experiment on parity violation that is not based on a convention (See Chapter
52).
2. Transformation
of torque:
“We wanted to
get a rule for finding torques in new axes in terms of torques in old axes, and
now we have the rule (Feynman et al., 1963, section 20–1 Torques in three dimensions).”
Feynman shows the
transformation of a torque based on the coordinates of two systems that are
related by x′ = xcos θ + ysin θ, y′ = ycos
θ – xsin θ, z′ = z. Specifically, we
can transform the torque in a new co-ordinate system rotated anti-clockwise by a
angle θ using the following equations: τx′ = τxcos θ + τysin θ, τy′ = τycos θ − τxsin θ, τz′ = τz. Some may find Feynman’s
method of direct substitution to find the torque in new
axes unnecessary or lack of insights. A shorter and simpler method is to make use of the fact that the torque
is a vector that is invariant under
rotation. Although the torque can be expressed as a sum of
vectors (e.g., τx′y′ = xFy − yFx), it is not a real vector that has all vector properties.
Feynman
elaborates that a torque is a twist on a plane and it does behave like a
vector. The torque vector is perpendicular to the plane of the twist and its
length is proportional to the strength of the twist. To a certain extent, the
three components of the torque transform like a real vector under rotation.
As an alternative, we can simply transform a torque
vector using a two dimensional rotation matrix: (i) first column: the point (1, 0) is rotated by an angle of θ anti-clockwise and moved to (cos θ, sin θ). (ii) second column: the point (0, 1) is rotated by an angle of θ anti-clockwise and moved to (−sin θ, cos θ). The rotation is a linear transformation and it
can be expressed as
R(x, y) = R[x(1, 0)+y(0, 1)] = xR(1, 0)+yR(0, 1).
3. Axial
vector:
“Vectors which
involve just one cross product in their definition are called axial vectors or pseudo vectors (Feynman
et al., 1963, section 20–1
Torques in three dimensions).”
Feynman
explains that axial vectors involve just a cross product (or vector product) in
the definition and provides examples such as torque, angular momentum, angular velocity, and magnetic
field. On the contrary, we have polar vectors such as coordinate, force, momentum, velocity, and electric
field. In Chapter 52, Feynman adds that “a ‘vector’ which, on
reflection, does not change about as the polar vector does, but is
reversed relative to the polar vectors and to the geometry of the space; such a
vector is called an axial vector (Feynman et al, 1963, section 52–5 Polar and axial vectors).” In a sense,
the term vector product is a misnomer
because it actually
produces an axial vector. Feynman says that the torque is a vector, but it
is not really a vector that it is loosely stated in books or websites.
Feynman
elaborates that the cross product is very important for representing the
features of rotation. He asks why the torque is a vector and then says that it
is a miracle of good luck that we can associate a single axis with a plane, and
thus we can associate a vector with the torque. Historically, the concept of
torque as an axial vector is based on the Gibbs vector system (Chappell et al., 2016). However, Jackson (1999) identifies a dangerous aspect of vector notation and
writes that “[t]he writing of a vector as a
does not tell us whether it is a polar or an axial vector (p. 270).” Interestingly, the
Clifford vector system includes the concept of wedge product and distinguishes
the electric and magnetic field as a vector and bivector respectively (without
the need of the axial vector).
Questions for discussion:
1. How would you define
a torque vector?
2. How would you deduce the torque vector
in another coordinate system that is rotated by an angle?
3. How would you define an axial vector?
The moral of the
lesson: a torque vector is perpendicular to the plane of
the twist, its length is proportional to the strength of the twist, and it
behaves like an axial vector.
References:
1. Chappell, J. M., Iqbal, A.,
Hartnett, J. G., & Abbott, D. (2016). The vector algebra war: a historical
perspective. IEEE Access, 4, 1997-2004.
2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and
heat. Reading, MA: Addison-Wesley.
3. Jackson, J. D. (1999). Classical
Electrodynamics (3rd ed.). John
Wiley & Sons, New York.
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