The basis of determining time derivatives of a vector using a rotating frame
is illustrated in Figure 1.3.17.

FIGURE 1.3.17 Time derivative of a  vector using a rotating reference frame.

Analysis of Velocities and Accelerations Using Rotating and Translating Frames

With the concept of general motion understood, an advantageous method
of determining velocities and accelerations is available by the method of
rotating reference frames. There are two cases in which this method
can be used. For a common origin of XYZ and xyz, with r being
a position vector to a point P,

For the origin A of xyz translating with respect XYZ:

where ? is the angular velocity of the xyz frame with respect to XYZ.
2? ? vxyz is the Coriolis acceleration.

Kinetics of Rigid Bodies in Plane Motion
Equation of Translational Motion

The fundamental equation for rigid body translation is based
on Newtons second law. In Equation 1.3.52, a is the acceleration of the
center of mass of the rigid body, no matter where the resultant force
acts on the body.  The sum of the external forces is equal to the mass
of the rigid body times the acceleration of the mass center of the rigid body,
independent of any rotation of the body.

Equation of Rotational Motion

Equation 1.3.53 states that  the sum of the external moments on the
rigid body is equal to the moment of inertia about an axis times the
angular acceleration of the body about that axis. The  angular
acceleration ? is for the rigid body rotating about an axis.
This equation is independent of rigid body translation.

where  An application is illustrated in Color Plate 2.

Applications of Equations of Motion

It is important to use the equations of motion properly. For plane motion,
three scalar equations are used to de?ne the motion in a plane.

If a rigid body undergoes only translation,

If the rigid body undergoes pure rotation about the center of mass,

Rigid body motions are categorized according to the constraints of
the motion:
1. Unconstrained Motion: Equations 1.3.54 are directly applied with all
three equations independent of one another.
2. Constrained Motion: Equations 1.3.54 are not independent of one another.
Generally, a kinematics analysis has to be made to determine how
the motion is constrained in the plane. There are two special cases:
a. Point constraint: the body has a ?xed axis.
b. Line constraint: the body moves along a ?xed line or plane.
When considering systems of rigid bodies, it is important to remember
that at most only three equations of motion are available from each
free-body diagram for plane motion to solve for three unknowns.
The motion of interconnected bodies must be analyzed using
related free-body diagrams.

Rotation about a Fixed Axis Not Through the Center of Mass

The methods presented above are essential in analyzing rigid bodies that
rotate about a ?xed axis, which is common in machines
(shafts, wheels, gears, linkages).  The mass of the rotating body may be
nonuniformly distributed as modeled in Figure 1.3.18.

FIGURE 1.3.18 Rotation of a rigid body about a ?xed axis.

Note that rC is the nearest distance between the ?xed axis O and the
mass center C. The ?gure also de?nes the normal and tangential
coordinate system used in Equations 1.3.57, which are the scalar
equations of motion using normal and tangential components.
The sum of the forces must include all reaction forces on the
rigid body at the axis of rotation.

General Plane Motion

A body that is translating and rotating is in general plane motion.
The scalar equations of motion are given by Equation 1.3.54.
If an arbitrary axis A is used to ?nd the resultant moment,

where C is the center of mass. It is a common error to forget to
include the cross-product term in the analysis.
There are two special cases in general plane motion, rolling and sliding.
Figure 1.3.19 shows pure rolling of a wheel without slipping with the
center of mass C at the geometric center of the wheel. This is called
pure rolling of a balanced wheel.

FIGURE 1.3.19 Pure rolling of a wheel.

From this ?gure the scalar equation of motion results,

For balanced wheels either sliding or not sliding, the following
schematic is helpful.

If slipping is not certain, assume there is no slipping and check whether 
? ? ?_sN. If  ? > ?_sN (not
possible; there is sliding), start the solution over using ? = ?_kN
but not using a_Cx = r?, which is not valid here.
For the problem involving unbalanced wheels
(the mass center and geometric center do not coincide),
Equations 1.3.60 result.

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