Transcript Document

Chapter 17 PLANE MOTION OF RIGID BODIES:
ENERGY AND MOMENTUM METHODS
The principle of work and energy for a rigid body is expressed in
the form
T1 + U1
2=
T2
where T1 and T2 represent the initial and final values of the
kinetic energy of the rigid body and U1 2 the work of the
external forces acting on the rigid body.
The work of a force F applied at a point A is
s2
U1
2
=

(F cos a) ds
s1
where F is the magnitude of the force, a the angle it forms with
the direction of motion of A, and s the variable of integration
measuring the distance traveled by A along its path.
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The work of a couple of moment M applied to a rigid body during
a rotation in q of the rigid body is
U1
2
=

q2
M ds
q1
The kinetic energy of a rigid body in plane motion is
T=
G
w
v
1
2
2
1
2
mv + Iw2
where v is the velocity of the mass center G of
the body, w the angular velocity of the body,
and I its moment of inertia about an axis
through G perpendicular to the plane of
reference.
T=
1
2
mv 2 + 12 Iw2
G
The kinetic energy of a rigid body in plane
motion may be separated into two parts:
w
v
(1) the kinetic energy 12 mv 2 associated
with the motion of the mass center G of the
1
body, and (2) the kinetic energy 2 Iw2 associated with the rotation
of the body about G.
For a rigid body rotating about a fixed axis through O with an
angular velocity w,
O
1
T = 2 IOw2
w
where IO is the moment of inertia of the body
about the fixed axis.
When a rigid body, or a system of rigid bodies, moves under
the action of conservative forces, the principle of work and
energy may be expressed in the form
T1 + V1 = T2 + V2
which is referred to as the principle of conservation of energy.
This principle may be used to solve problems involving
conservative forces such as the force of gravity or the force
exerted by a spring.
The concept of power is extended to a rotating body subjected
to a couple
dU M dq
Power =
=
= Mw
dt
dt
where M is the magnitude of the couple and w is the angular
velocity of the body.
The principle of impulse and momentum derived for a system of
particles can be applied to the motion of a rigid body.
Syst Momenta1 + Syst Ext Imp1
2=
Syst Momenta2
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 mv attached to the
mass center G of the body and a couple Iw. The vector mv is
associated with translation of the body with G and represents the
linear momentum of the body, while the couple Iw corresponds
to the rotation of
(Dm)v
the body about G
mv and represents the
P
angular momentum
Iw
of the body about
an axis through G.
The principle of impulse and momentum can be expressed
graphically by drawing three diagrams representing
respectively the system of initial momenta of the body, the
impulses of the external forces acting on it, and the system of
the final momenta of the body. Summing and equating
respectively the x components, the y components, and the
moments about any given point of the vectors shown in the
figure, we obtain three equations of motion which may be
solved for the desired unknowns.
 Fdt
y
mv1
y
y
G
G
Iw2
Iw1
O
mv2
x
O
x
O
x
 Fdt
y
mv1
y
y
G
G
Iw2
Iw1
O
mv2
x
O
x
O
x
In problems dealing with several connected rigid bodies each
body may be considered separately or, if no more than three
unknowns are involved, the principles of impulse and
momentum may be applied to the entire system, considering
the impulses of the external forces only.
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.
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. In such a
situation a relation for the impact
involving the coefficient of
restitution e holds, and the
velocities of points A and B
where contact occurs during the
impact should be used.
n
B
vB
A
n
vA
(a) Before impact
n
B
A
(v’B)n - (v’A)n = e[(vA)n - (vB)n]
n
v’B
v’A
(b) After impact
n
n
B
A
n
B
vB
vA
(a) Before impact
A
n
v’B
v’A
(b) After impact
(v’B)n - (v’A)n = e[(vA)n - (vB)n]
where (vA)n and (vB)n are the components along the line of impact
of the velocities of A and B before impact, and (v’A)n and (v’B)n
their components after impact. This equation is applicable not
only when the colliding bodies move freely after impact but also
when the bodies are partially constrained in their motion.