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REVIEW AND SUMMARY
In this chapter we again considered the method of work and energy
and the method of impulse and momentum. In the first part of the
chapter we studied the method of work and energy and its application to the analysis of the motion of rigid bodies and systems of rigid
bodies.
In Sec. 17.2, we first expressed the principle of work and energy for
a rigid body in the form
T1 1 U1y2 5 T2
Principle of work and energy
for a rigid body
(17.1)
where T1 and T2 represent the initial and final values of the kinetic
energy of the rigid body and U1y2 represents the work of the external
forces acting on the rigid body.
In Sec. 17.3, we recalled the expression found in Chap. 13 for the
work of a force F applied at a point A, namely
U1 y 2 5
#
Work of a force or a couple
s2
( F cos a ) ds
(17.39)
s1
where F was the magnitude of the force, a the angle it formed with
the direction of motion of A, and s the variable of integration measuring the distance traveled by A along its path. We also derived the
expression for the work of a couple of moment M applied to a rigid
body during a rotation in u of the rigid body:
U1 y 2 5
#
u2
M du
(17.5)
u1
We then derived an expression for the kinetic energy of a rigid body
in plane motion [Sec. 17.4]. We wrote
T 5 1 mv 2 1 1 I v 2
2
2
Kinetic energy in plane motion
(17.9)
where v is the velocity of the mass center G of the body, v is the
angular velocity of the body, and I is its moment of inertia about an
axis through G perpendicular to the plane of reference (Fig. 17.13)
[Sample Prob. 17.3]. We noted that the kinetic energy of a rigid body
in plane motion can be separated into two parts: (1) the kinetic
energy 1 mv 2 associated with the motion of the mass center G of the
2
body, and (2) the kinetic energy 1 I v 2 associated with the rotation of
2
the body about G.
w
G
Fig. 17.13
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Plane Motion of Rigid Bodies: Energy and
Momentum Methods
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For a rigid body rotating about a fixed axis through O with an angular
velocity V, we had
T 5 1 IOv 2
2
Kinetic energy in rotation
(17.10)
where IO was the moment of inertia of the body about the fixed axis.
We noted that the result obtained is not limited to the rotation of
plane slabs or of bodies symmetrical with respect to the reference
plane, but is valid regardless of the shape of the body or of the location of the axis of rotation.
Systems of rigid bodies
Equation (17.1) can be applied to the motion of systems of rigid
bodies [Sec. 17.5] as long as all the forces acting on the various bodies involvedinternal as well as external to the systemare included
in the computation of U1y2. However, in the case of systems consisting of pin-connected members, or blocks and pulleys connected by
inextensible cords, or meshed gears, the points of application of the
internal forces move through equal distances and the work of these
forces cancels out [Sample Probs. 17.1 and 17.2].
Conservation of energy
When a rigid body, or a system of rigid bodies, moves under the
action of conservative forces, the principle of work and energy can
be expressed in the form
T1 1 V1 5 T2 1 V2
(17.12)
which is referred to as the principle of conservation of energy [Sec.
17.6]. This principle can be used to solve problems involving conservative forces such as the force of gravity or the force exerted by a
spring [Sample Probs. 17.4 and 17.5]. However, when a reaction is
to be determined, the principle of conservation of energy must be
supplemented by the application of dAlemberts principle [Sample
Prob. 17.4].
Power
In Sec. 17.7, we extended the concept of power to a rotating body
subjected to a couple, writing
Power 5
M du
dU
5
5 Mv
dt
dt
(17.13)
where M is the magnitude of the couple and v the angular velocity
of the body.
The middle part of the chapter was devoted to the method of
impulse and momentum and its application to the solution of various
types of problems involving the plane motion of rigid slabs rigid
bodies and symmetrical with respect to the reference plane.
Principle of impulse and momentum
for a rigid body
We first recalled the principle of impulse and momentum as it was
derived in Sec. 14.9 for a system of particles and applied it to the
motion of a rigid body [Sec. 17.8]. We wrote
Syst Momenta1 1 Syst Ext Imp1y2 5 Syst Momenta2 (17.14)
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Review and Summary
Next we showed that for a rigid slab or a rigid body symmetrical with
respect to the reference plane, the system of the momenta of the
particles forming the body is equivalent to a vector m v attached at the
mass center G of the body and a couple I V (Fig. 17.14). The vector
( m)v
mv
P
=
G
Iw
Fig. 17.14
m v is associated with the translation of the body with G and represents the linear momentum of the body, while the couple I V corresponds to the rotation of the body about G and represents the
angular momentum of the body about an axis through G.
Equation (17.14) can be expressed graphically as shown in Fig.
17.15 by drawing three diagrams representing respectively the system
of the initial momenta of the body, the impulses of the external forces
acting on the body, and the system of the final momenta of the body.
y
y
y
F dt
mv2
mv1
Iw1
x
O
(a)
=
+
G
x
O
(b)
Fig. 17.15
Summing and equating respectively the x components, the y components, and the moments about any given point of the vectors shown
in that figure, we obtain three equations of motion which can be
solved for the desired unknowns [Sample Probs. 17.6 and 17.7].
In problems dealing with several connected rigid bodies [Sec.
17.9], each body can be considered separately [Sample Prob. 17.6], or,
if no more than three unknowns are involved, the principle of impulse
G
Iw2
x
O
(c)
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Plane Motion of Rigid Bodies: Energy and
Momentum Methods
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and momentum can be applied to the entire system, considering the
impulses of the external forces only [Sample Prob. 17.8].
Conservation of angular momentum
When the lines of action of all the external forces acting on a system
of rigid bodies pass through a given point O, the angular momentum
of the system about O is conserved [Sec. 17.10]. It was suggested
that problems involving conservation of angular momentum be solved
by the general method described above [Sample Prob. 17.8].
Impulsive motion
The last part of the chapter was devoted to the impulsive motion and
the eccentric impact of rigid bodies. In Sec. 17.11, we recalled that
the method of impulse and momentum is the only practicable method
for the solution of problems involving impulsive motion and that the
computation of impulses in such problems is particularly simple
[Sample Prob. 17.9].
Eccentric impact
In Sec. 17.12, we recalled that the eccentric impact of two rigid
bodies is defined as an impact in which the mass centers of the colliding bodies are not located on the line of impact. It was shown that
in such a situation a relation similar to that derived in Chap. 13 for
the central impact of two particles and involving the coefficient of
restitution e still holds, but that the velocities of points A and B where
contact occurs during the impact should be used. We have
(v9 )n 2 (v9 )n 5 e[(vA)n 2 (vB)n]
B
A
(17.19)
where (vA)n and (vB)n are the components along the line of impact
of the velocities of A and B before the impact, and (v9 )n and (v9 )n
A
B
are their components after the impact (Fig. 17.16). Equation (17.19)
n
n
A
B
A
B
vB
vA
n
(a) Before impact
v'B
n
v'A
(b) After impact
Fig. 17.16
is applicable not only when the colliding bodies move freely after the
impact but also when the bodies are partially constrained in their
motion. It should be used in conjunction with one or several other
equations obtained by applying the principle of impulse and momentum [Sample Prob. 17.10]. We also considered problems where the
method of impulse and momentum and the method of work and
energy can be combined [Sample Prob. 17.11].