Download Sect. 5.6, Part I

Sect 5.6: Torque-Free Motion of a
Rigid Body
• Euler’s Eqtns of Motion for a Rigid Body with 1 pt. fixed:
I1(dω1/dt) - ω2ω3(I2 -I3) = N1
(1)
I2(dω2/dt) - ω3ω1(I3 -I1) = N2
(2)
I3(dω3/dt) - ω1ω2(I1 -I2) = N3
(3)
• Interesting & useful special case: Motion of a rigid
body not subject to any net forces or torques:
– Object is at rest or moving uniformly (Newton’s 1st Law!)
– Rotational motion started by some means, then force or
torque is removed.
– Can discuss in reference from where CM is at rest.
 Angular momentum comes from rotation about CM.
Euler’s eqtns, (1), (2), (3), apply with right sides = 0!
• Force & torque free conditions  Euler’s Eqtns:
I1(dω1/dt) = ω2ω3(I2 -I3)
I2(dω2/dt) = ω3ω1(I3 -I1)
I3(dω3/dt) = ω1ω2(I1 -I2)
(1)
(2)
(3)
– Note, assumed that (1), (2), (3) are written in principal axes
system.  So are most subsequent eqtns!
– Also describe motion of rigid body with one pt. fixed & no
forces, torques.
 From earlier conservation theorems can immediately know
that 2 integrals or constants of the motion are:
Total KE: T = (½)I1(ω1)2 + (½)I2(ω2)2 + (½)I3(ω3)2 = const.
Total ang. momentum: L = I1ω1 + I2ω2 + I3ω3 = const.
– Using these, can completely integrate (solve!) (1), (2), (3) in terms of
elliptic integrals. “Not very illuminating!”
• Euler’s Eqtns under force, torque free conditions:
I1(dω1/dt) = ω2ω3(I2 -I3)
(1)
I2(dω2/dt) = ω3ω1(I3 -I1)
(2)
I3(dω3/dt) = ω1ω2(I1 -I2)
(3)
T = (½)I1(ω1)2 + (½)I2(ω2)2 + (½)I3(ω 3)2  (½)Iω2 = const.
L = I1ω1 + I2ω2 + I3ω3  Iω = const.
• Alternative to a “brute force” solution to (1),(2),(3): A
geometric (& perhaps more physical?) description of motion
 Poinsot’s Construction.
• Coord system with principle axes. n  unit vector along
rotation axis. n  ω/|ω|. Consider the vector ρ  n/(I)½ from
before. I  nIn = moment of inertia. T  (½)Iω2 = const.

ρ = ρ(t) = ω/|ω(I)½| = ω/(2T)½ (time dependent!)
Define:
F(ρ)  ρIρ = (ρ1)2I1+(ρ2)2I2+(ρ3)2I3
Poinsot’s Construction
• F(ρ)  ρIρ = (ρ1)2I1+(ρ2)2I2+(ρ3)2I3 (Note: ρ = ρ(t)!)
ρ = ω/(2T)½ . Surfaces of const F(ρ)  ellipsoids in “ρ”
space. F(ρ) = 1  Inertia Ellipsoid from before.
• Body rotates.  Directions of rotation axis n & of ρ
change with time. The tip of ρ always defines a point
on the inertia ellipsoid.  The gradient of F(ρ)
evaluated at that point defines the direction of the
normal to the inertia ellipsoid at that point.
• Take the gradient of F(ρ) & use the definitions:
ρF(ρ) = 2Iρ = 2(Iω)/(2T)½
Note: L  Iω = const.

ρF(ρ) = (2/T)½ L
Defines the direction  to the inertia ellipsoid
• F(ρ)  ρIρ = (ρ1)2I1+(ρ2)2I2+(ρ3)2I3
 ρF(ρ) = (2/T)½ L (parallel to the L vector!) (A)
By the definition of , (A) defines a direction  to the inertia
ellipsoid. Also, since T & L are constant, so is ρF(ρ)!
PHYSICS: The angular
ρ = ρ(t) !
velocity vector ω always
moves such that the
corresponding  to the
inertia ellipsoid is in the
direction of the angular
momentum L. Here, we have torque free motion
 L =const. (in direction & magnitude).  The inertia ellipsoid,
fixed in the body, must move in space to preserve (A). See figure.
• Also: The distance between the ellipsoid origin
& the plane tangent to it at the point ρ is
constant in time. See figure:
Geometry  That
distance = The projection of
ρ on L or: h = (ρL)/L
But: ρ = ω(2T)-½ (= ρ(t)!)
So: h = [(ωL)(2T)½]/(L)
Note: (ωL) = ωIω = 2T

h = (2T)½/(L) = const.
(since T = const. & L = const.)
• The distance between ellipsoid origin & the plane tangent to it at ρ:
h = (2T)½/L = const.
• PHYSICS: The tangent plane is always
a fixed distance h from origin of the inertia
ellipsoid. Normal to that plane (|| L) has a
fixed direction. The tangent plane  the invariable plane
 Visualize the force & torque free motion of a rigid body in
“ρ” space as the inertia ellipsoid rolling without slipping on the
invariable plane. Ellipsoid center a const height h above plane. Rolls
without slipping because the point of contact is defined by the position
of the tip of ρ. Curve traced out on ellipsoid by the point of contact of
the ellipsoid with plane  polhode. Curve traced out on the plane by the
point of contact of the ellipsoid with the plane  herpolhode. (“The polhode
rolls without slipping on the herpolhode lying in the invariable plane!”).
• Poinsot’s construction completely describes the force
& torque free motion of a rigid body.
• The direction of the invariable plane & the height h
of origin of the inertia ellipsoid are completely
determined by the (constant) values of T & L.
– T & L are determined by the initial conditions of problem!
• Then, it’s just geometry to trace out the polhode &
the herpolhode.
– The direction of the (time dependent) angular velocity ω is
the same as the direction of ρ (= ρ(t) )
– The instantaneous orientation of the body is the same as the
orientation of the inertia ellipsoid, fixed in the body!
– Is it necessary to state that the shape of the inertia ellipsoid has nothing
whatsoever to do with the shape of the body?
Torque-Free Motion of a Symmetrical Body
(Say, symmetric about one or more axes)
• Special Case: Torque Free motion of a symmetrical body.
 Inertia Ellipsoid = Ellipsoid of Revolution
 The polhode on the ellipsoid is a circle about symmetry axis.
 The herpolhode is a circle on the invariable plane.
• An observer, fixed in the body axis system, sees the angular
velocity vector ω move on the surface of a cone ( the body
cone). Intersection of this cone with inertia ellipsoid  polhode
• An observer, fixed in the space axis system, sees ω move on the
surface of a cone ( the space cone). Intersection of this cone
with invariable plane  herpolhode
 The motion can also be described as the rolling of the body
cone on the space cone.
• Let rotation the axis be axis 3 in the principal axis
system.
– If I3 < I1, I2 the inertia ellipsoid
is prolate (~ football shaped)
as in the figure: The body
cone is outside the space cone.
– If I3 > I1, I2 the inertia ellipsoid is oblate (shaped
like football on its side). In this case, the body
cone rolls around inside the space cone.
– In both cases, the direction of the angular velocity
vector ω precesses in time about the symmetry
axis of the body.