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If two bodies contact one another without slipping, and the
points in contact move along different paths, then the tangential
components of acceleration of the points will be the same; however,
the normal components will generally not be the same. For example,
consider the two meshed gears in Fig. 2-32,a. Point A is located on
gear B and a coincident point A′ is located on gear C. Due to the
rotational motion, (a A t (a A t
) =
) , however, since both points follow
′
different circular paths, (a A n (a A′ ) and therefore a A ≠ a , Fig. 2-
) ≠
A′
n
32,b.
58 Rotation About a Fixed Point
When a rigid body rotates about a fixed point, the distance r
from the point to a particle located on the body is the same for any
position of the body. Thus, the path of motion for the particle lies on
the surface of a sphere having a radius r and centered at the fixed
point. Since motion along this path occurs only from a series of
rotations made during a finite time interval, we will first develop a
familiarity with some of the properties of rotational displacements.
Euler’s Theorem. Euler’s theorem states that two “component”
rotations about different axes passing through a point are equivalent to
a single resultant rotation about an axis passing through the point. If
more than two rotations are applied, they can be combined into pairs,
and each pair can be further reduced and combined into one rotation.
Finite Rotations. If component rotations used in Euler’s
theorem are finite, it is important that the order in which they are
applied be maintained. To show this, consider the two finite rotations
θ 1 + θ 2 applied to the block in Fig. 2-33,a. Each rotation has a
magnitude of 90° and a direction defined by the right-hand rule, as
indicated by the arrow. The final position of the block is shown at the
right. When these two rotations are applied in the order θ 2 + θ 1 , as
shown in Fig. 2-33,b, the final position of the block is not the same as
it is in Fig. 2-33,a. Because finite rotations do not obey the
commutative law of addition θ 1 + θ 2 ≠ θ 2 + θ 1 , they cannot be
classified as vectors. If smaller, yet finite, rotations had been used to
illustrate this point, e.g., 10° instead of 90°, the final position of the

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