The work and energy equation for a system of particles is similar to the
equation stated for a single particle.

Momentum Methods for a System of Particles

Moments of Forces on a System of Particles. The moments of
external forces on a system of particles about a point O are given by

Linear and Angular Momenta of a System of Particles. The resultant
of the external forces on a system of particles equals the time rate of change
of linear momentum of that system.

The angular momentum equation for a system of particles about a
?xed point O is

The last equation means that the resultant of the moments of the external
forces on a system of particles equals the time rate of change of
angular momentum of that system.

Angular Momentum about the Center of Mass

The above equations work well for reference frames that are stationary,
but sometimes a special approach may be useful, noting that the
angular momentum of a system of particles about its center of mass C
is the same whether it is observed from a ?xed frame at point O or from the
centroidal frame which may be translating but not rotating. In this case

Conservation of Momentum

The Moments of Forces on a System of Particles equations for
a system of particles is analogous to that for a single particle.

Impulse and Momentum of a System of Particles

The linear impulse momentum for a system of particles is

The angular impulse momentum for a system of particles is

Kinematics of Rigid Bodies

Rigid body kinematics is used when the methods of particle kinematics
are inadequate to solve a problem. A rigid body is de?ned as one in which
the particles are rigidly connected. This assumption allows for some
similarities to particle kinematics.  There are two kinds of rigid body
motion,  translation and rotation. These motions may occur separately
or in combination.

Translation

Figure 1.3.9 models the translational motion of a rigid body.

FIGURE 1.3.9 Translational motion of a rigid body.

These equations represent an important  fact: when a rigid body is in
translation, the motion of a single point completely speci?es the motion
of the whole body.

Rotation about a Fixed Axis

Figure 1.3.10 models a point P in a rigid body rotating about a ?xed axis
with an angular velocity ?. The velocity v of point P is determined assuming
that the magnitude of r is constant,

v =  ? ? r

FIGURE 1.3.10 Rigid body rotating about a ?xed axis.

The acceleration a of point P is determined conveniently by using normal
and tangential components,

Note that the angular acceleration ? and angular velocity ? are valid for
any line perpendicular to the axis of rotation of the rigid body at
a given instant.

Kinematics Equations for Rigid Bodies Rotating in a Plane

For rotational motion with or without a ?xed axis, if displacement is
measured by an angle ?,

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