Transcript Chapter 3 Impulse

KINETICS OF PARTICLES: ENERGY AND
MOMENTUM METHODS
s2
A
dr
a
F
A1
s1
A2
ds
Consider a force F acting on
a particle A. The work of F
corresponding to the small
displacement dr is defined as
dU = F dr
s
Recalling the definition of scalar product of
two vectors,
dU = F ds cos a
where a is the angle between F and dr.
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s2
A2
ds
A
a
The work of F during a finite
F
displacement from A1 to A2 ,
denoted by U1 2 , is obtained
by integrating along the path described by
the particle.
A2
dr
A1
s1
dU = F dr = F ds cos a
s
U1
2
=
ٍ F dr
A1
For a force defined by its rectangular components, we write
A2
U1
2
=
ٍ (F dx + F dy + F dz)
x
A1
y
z
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A2
The work of the weight
W of a body as its
center of gravity moves
from an elevation y1 to
y2 is obtained by setting
Fx = Fz = 0 and
Fy = - W .
W
dy
y2
A
A1
y1
y
y2
U1
2=
-
ٍ
Wdy = Wy1 - Wy2
y1
The work is negative when the elevation increases, and
positive when the elevation decreases.
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The work of the force F exerted by a
spring on a body A during a finite
displacement of the body from A1 (x = x1)
to A2 (x = x2) is obtained by writing
spring undeformed
B
AO
dU = -Fdx = -kx dx
x2
B
U1
A1
x1
2=
ٍ
-
x1
F
B
=
A
x
x2
k x dx
A2
1
2
2
kx1
-
2
1
kx
2 2
The work is positive when
the spring is returning to.
its undeformed position.
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A2
A’
dr
r2
A
r
F
dq
-F
q r1
M
O
m
The work of the gravitational force
F exerted by a particle of mass M
located at O on a particle of mass
m as the latter moves from A1 to
A2 is obtained from
r2
U1
A1
2
=
ٍ
r1
GMm dr
r2
GMm
GMm
=
r2
r1
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The kinetic energy of a particle of mass m moving with a
velocity v is defined as the scalar quantity
T=
1
2
mv2
From Newton’s second law the principle of work and energy
is derived. This principle states that the kinetic energy of a
particle at A2 can be obtained by adding to its kinetic energy at
A1 the work done during the displacement from A1 to A2 by the
force F exerted on the particle:
T1 + U1
2=
T2
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The power developed by a machine is defined as the time rate
at which work is done:
dU
Power =
=F v
dt
where F is the force exerted on the particle and v is the velocity
of the particle. The mechanical efficiency, denoted by h, is
expressed as
power output
h = power input
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When the work of a force F is independent of the path followed,
the force F is said to be a conservative force, and its work is
equal to minus the change in the potential energy V associated
with F :
U1
2=
V1 - V2
The potential energy associated with each force considered
earlier is
Force of gravity (weight):
Vg = Wy
Gravitational force:
GMm
Vg = r
Elastic force exerted by a spring:
Ve =
1
2
kx2
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U1
2=
V1 - V2
This relationship between work and potential energy, when
combined with the relationship between work and kinetic
energy (T1 + U1 2 = T2) results in
T1 + V 1 = T2 + V 1
This is the principle of conservation of energy, which states that
when a particle moves under the action of conservative forces,
the sum of its kinetic and potential energies remains constant.
The application of this principle facilitates the solution of
problems involving only conservative forces.
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v
f
P
v0
r
f0
r0
O
P0
(HO)0 = HO :
T0 + V0 = T + V :
When a particle moves under a
central force F, its angular momentum
about the center of force O remains
constant. If the central force F is also
conservative, the principles of
conservation of angular momentum
and conservation of energy can be
used jointly to analyze the motion of
the particle. For the case of oblique
launching, we have
r0mv0 sin f0 = rmv sin f
1
2
mv02
GMm
=
r0
1
2
mv2
GMm
r
where m is the mass of the vehicle and M is the mass of the earth.
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The linear momentum of a particle is defined as the product mv
of the mass m of the particle and its velocity v. From Newton’s
second law, F = ma, we derive the relation
t2
mv1 +
ٍ F dt = mv
t1
2
where mv1 and mv2 represent the momentum of the particle at a
time t1 and a time t2 , respectively, and where the integral defines
the linear impulse of the force F during the corresponding time
interval. Therefore,
mv1 + Imp1
2=
mv2
which expresses the principle of impulse and momentum for a
particle.
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When the particle considered is subjected to several forces, the
sum of the impulses of these forces should be used;
mv1 + SImp1
2=
mv2
Since vector quantities are involved, it is necessary to consider
their x and y components separately.
The method of impulse and momentum is effective in the study
of impulsive motion of a particle, when very large forces, called
impulsive forces, are applied for a very short interval of time Dt,
since this method involves impulses FDt of the forces, rather
than the forces themselves. Neglecting the impulse of any
nonimpulsive force, we write
mv1 + SFDt = mv2
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In the case of the impulsive motion of several particles, we write
Smv1 + SFDt = Smv2
where the second term involves only impulsive, external forces.
In the particular case when the sum of the impulses of the
external forces is zero, the equation above reduces to
Smv1 = Smv2
that is, the total momentum of the particles is conserved.
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Line of
Impact
vB
B
A
Before Impact
vA
v’B
In the case of direct central impact,
two colliding bodies A and B move
along the line of impact with
velocities vA and vB , respectively.
Two equations can be used to
determine their velocities v’A and v’B
after the impact. The first represents
the conservation of the total
momentum of the two bodies,
mAvA + mBvB = mAv’A + mBv’B
B
A
v’A
After Impact
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Line of
Impact
mAvA + mBvB = mAv’A + mBv’B
vB
B
A
Before Impact
vA
v’B
B
The second equation relates the
relative velocities of the two bodies
before and after impact,
v’B - v’A = e (vA - vB )
The constant e is known as the
coefficient of restitution; its value lies
between 0 and 1 and depends on the
material involved. When e = 0, the
impact is termed perfectly plastic; when
e = 1 , the impact is termed perfectly
elastic.
A
v’A
After Impact
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Line of
Impact
n
t
vB
B
A
In the case of oblique central impact,
the velocities of the two colliding
bodies before and after impact are
resolved into n components along the
line of impact and t components
along the common tangent to the
surfaces in contact. In the t direction,
Before Impact
vA
v’B
n
t
v’A
B
vB
A
vA After Impact
(vA)t = (v’A)t
(vB)t = (v’B)t
while in the n direction
mA (vA)n + mB (vB)n =
mA (v’A)n + mB (v’B)n
(v’B)n - (v’A)n = e [(vA)n - (vB)n]
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Line of
Impact
n
t
(vA)t = (v’A)t
mA (vA)n + mB (vB)n =
mA (v’A)n + mB (v’B)n
vB
B
A
Before Impact
vA
v’B
B
(v’B)n - (v’A)n = e [(vA)n - (vB)n]
n
t
v’A
(vB)t = (v’B)t
vB
A
vA After Impact
Although this method was
developed for bodies moving freely
before and after impact, it could be
extended to the case when one or
both of the colliding bodies is
constrained in its motion.
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