Chapter 4: Rigid Body Kinematics
• Rigid Body  A system of mass points subject to
(holonomic) constraints that all distances between all
pairs of points remain constant throughout the
motion.
– Of course, an idealization!
– However, quite a useful concept!! 2 Chapters!
– Ch. 4: Kinematics = Description of motion without
discussing causes
• Very mathematical!
– Ch. 5: Dynamics = Causes of motion - forces, torques.
– Especially interested in rigid body rotation.
• As part of this discussion, we will discuss “fictitious”
(non-inertial) forces: Centrifugal & Coriolis
Sect. 4.1: Independent Coordinates
• How many independent coordinates does it
take to describe a rigid body?
– How many degrees of freedom are there?
• 6 indep coordinates or degrees of freedom:
– 3 external coordinates to specify position of some
reference point in body (usually CM) with respect to arbitrary
origin.
– 3 internal coordinates to specify how the body is oriented
with respect to the external coordinate axes.
– Here, we justify this.
• Rigid body, N particles

(At most) 3N degrees of freedom.
• However, constraints are that the distances
between each particle pair are fixed.
 All constraints are of the form (for pair i, j):
rij = distance between i & j = cij = const (1)
N particles, np = # pairs = (½)N(N-1) = # eqtns like (1)

Naively: s = # degrees of freedom = 3N - np
However, this is NOT valid because
– All eqtns like (1) are not independent of each other!
– ALSO: np = (½)N(N-1) > 3N
(if N  7)
np >> 3N
(N >>1)
• To fix a point in a rigid body, it is not necessary to
specify its distances from ALL other points in
body. It’s ONLY necessary to specify distances to
any three non-collinear points (figure).
See figure:
 If positions of 3
particles (figure) are
given, constraints fix
positions of all N-3
other particles. That
is, we must have # degrees of freedom s  9 (3 particles,
dimensions). However, the 3 reference points are not
all independent, but are related by eqtns like (1):
r12 = c12 = const, r23 = c23 = const, r13 = c13 = const

s=6
See figure:
Can also see s = 6 in
another way: To
establish position of
one reference point,
need 3 coords. Once
point 1 is fixed, point 2 can be specified by only 2
coords, since it is constrained to move on a sphere of
radius r12 = c12. With 2 points determined, point 3
needs only 1 coord, since r13 = c13 & r23 = c23
constrain its location.  s = 3 + 2 + 1 = 6
 A rigid body in space needs 6 independent generalized
coords to specify its configuration & to treat its dynamics, no
matter how many particles it contains.
• Also, of course, there may be additional constraints on the
body which reduce the # independent coordinates further.
• How are these 6 coordinates assigned? Configuration of rigid
body is completely specified by locating a set of Cartesian
axes FIXED IN THE RIGID BODY (primed axes  “body
axes” in figure) relative to an arbitrary set of Cartesian axes
(unprimed axes  “space” or “lab frame” or “reference” axes)
fixed in external space.
• See figure:
• See figure:
• 3 coords
(of necessary 6) :
Specify origin of
“body” (primed)
axes in “space”
(unprimed) axes
system.
• 3 coords: Specify orientation of primed axes relative
to unprimed axes (actually to axes parallel to unprimed axes
but sharing origin with primed axes). Now focus on 3
orientation coords.
• There are many ways to specify the orientation of
one Cartesian set of axes with respect to another with
a common origin. Common procedure:
• Specify the DIRECTION COSINES of the primed
axes relative to the unprimed axes.
See figure:
For example,
orientation of x´
in x, y, z system
is specified by
cosθ11, cosθ12,
cosθ13, with angles as shown in the figure.
• Notation: i, j, k  unit vectors along x, y, z.
i´, j´, k´  unit vectors along x´, y´, z´.
 Direction cosines (9 of them!): cosθ11  cos(i´i) = i´i =ii´
cosθ12  cos(i´j) = i´j =ji´
cosθ21  cos(j´i) = j´i = ij´
cosθ23  cos(j´k) = j´k = kj´
cosθ32  cos(k´j) = k´j = jk´
cosθ13  cos(i´k) = i´k = ki´
cosθ22  cos(j´j) = j´ j = jj´
cosθ31  cos(k´i) = k´i = ik´
cosθ33  cos(k´k) =k´k = kk´
Convention: 1st index is primed, 2nd is unprimed
• Relns between i, j, k 
unit vectors along x, y, z &
i´, j´, k´  unit vectors along
x´, y´, z´:
i´ = cosθ11i+ cosθ12 j+ cosθ13k
j´ = cosθ21i+ cosθ22 j+ cosθ23k
k´= cosθ31i+ cosθ32 j+ cosθ33k Inverse relns are similar.
 Can express an arbitrary point in either coord system:
r = xi + yi + zk = x´i´ + y´j´ + z´k´
• Primed coords & unprimed coords are related by:
x´ = (ri´) = cosθ11x+ cosθ12 y+ cosθ13z
y´ = (rj´) = cosθ21x+ cosθ22 y+ cosθ23z
z´ = (rk´) = cosθ31x+ cosθ32 y+ cosθ33z
Inverse relns are similar.
• Relations between components of arbitrary
vector G in the 2 systems:
• We had
x´ = (ri´) = cosθ11x+ cosθ12 y+ cosθ13z
y´ = (rj´) = cosθ21x+ cosθ22 y+ cosθ23z
z´ = (rk´) = cosθ31x+ cosθ32 y+ cosθ33z
• Procedure to get these  procedure to get
components of G:
Gx´ = (Gi´) = cosθ11Gx + cosθ12Gy+ cosθ13Gz
Gy´ = (Gj´) = cosθ21Gx+ cosθ22Gy+ cosθ23Gz
Gz´ = (Gk´) = cosθ31Gx + cosθ32Gy + cosθ33Gz
Inverse relations are similar.
• Primed axes are fixed
in body:  9 direction
cosines cosθij will be
functions of time as the
body rotates.
 Can view direction cosines as generalized
coordinates describing the orientation of the body.
However, they cannot be independent! There are 9
of them & to describe the orientation of rigid body, &
we need only 3 coordinates.
• Relns between different cosθij
Obtained using orthogonality of
unit vectors in both coord sets:
ij = jk = ki = 0,
ii = jj = kk = 1
i´j´ = j´k´ = k´i´ = 0, i´i´ = j´j´ = k´k´ = 1
 Combining i´ =cosθ11i+ cosθ12 j+ cosθ13k
j´=cosθ21i+cosθ22 j+ cosθ23k, k´ =cosθ31i+ cosθ32 j+ cosθ33k
with above dot products gives relns between cosθij:
and:
∑cosθm´ cosθm = 0 (m  m´, sum  = 1,2,3)
∑cos2θm = 1
(sum  = 1,2,3)
These  Orthogonality Relations between direction cosines
• Use the Kronecker delta
δm,m´  0
(m  m´),
δm,m´  1 (m = m´),
• Orthogonality relations become:
∑cosθm´ cosθm = δm,m´ (sum  = 1,2,3)
• 6 orthogonality relns between 9 direction cosines
 3 indep coords.  Using direction cosines as generalized
coordinates to set up Lagrangian is not possible. Instead
choose some set of 3 independent functions of the direction
cosines. There is no unique choice for this set. A common set
 The Euler Angles. Described later.
• Relations we just derived are, however, very useful. Can use
them to derive many theorems about & properties of, rigid
body motion. We do this next!