Download kinetics of a particle: impulse and momentum

KINETICS OF A PARTICLE:
IMPULSE AND MOMENTUM
Introduction:
In the previous chapter we integrated the equations of motion with respect to displacement. We found
that the velocity changes could be expressed directly in terms of the work done or in terms of the changes in
the mechanical energy. In this chapter, we will integrate the equations of motion with respect to time and
thereby obtain the principle of impulse and momentum. The resulting equation will be useful for solving
problems involving force, velocity, and time.
I. Principle of linear impulse and momentum:
P
Consider a particle of mass m subjected to a system of forces represented by the resultant F~R = F~ .
Newton’s second law applied in an inertial frame can be written as
~
P
d
dV
F~R = F~ = (m~v ) = m
= m~a
dt
dt
~ =V
~1 at t1 and V
~ =V
~2 at t2 gives
rearranging the terms and integrating between the limits V
XZ
t2
t1
F~ dt = m
Z
~2
V
~1
V
~ =mV
~2 − m V
~1
dV
• This is referred to as the principle of linear impulse and momentum. It provides a direct means
~2 after a specified time period when the particle’s initial
of obtaining the particle’s final velocity V
velocity is known and the forces acting on the particle are either constant or can be expressed as a
function of time.
~
• If
a two step process would be necessary; i.e., apply
PV~2 was determined using the equations of motion,
~ /dt to obtain V
~2 .
F = m ~a to obtain ~a, then integrate ~a = dV
~ is referred to as the particle’s linear momentum. It has the same direction as the
• The vector m V
particle’s velocity and has a unit of mass times unit of speed, (SI: kg · m/s), (US: slug · f t/s )
Rt
• The vector I~ = t12 F~ dt is referred to as linear impulse. It measures the effect of a force F~ during
the time the force acts. It acts in the same direction as the force, its magnitude has the unit of force
times time (SI: N · s = kg · m/s), (US: lb · s = slug · f t/s ). Particularly the magnitude of the impulse
Z t2
~
I=
F~ dt can be represented experimentally by the shaded area under the curve of force versus
time.
t1
1
• The principle of linear impulse and momentum can be illustrated graphically, the result is known as
as the impulse and momentum diagram.
• Principle of linear impulse in scalar form: If each of the vectors is resolved into its x, y, z
components, we can write symbolically the following three scalar equations
m(Vx )1 +
P
t2
Z
Fx dt = m(Vx )2
t1
m(Vy )1 +
P
t2
Z
Fy dt = m(Vy )2
t1
m(Vz )1 +
P
Z
t2
Fz dt = m(Vz )2
t1
I-1. Procedure for analysis:
For applications of the principle of linear impulse and momentum it is suggested that the following
procedure be used.
• Establish the x, y, z inertial frame of reference and draw the particle’s free body diagram in order to
account for all the forces that produce impulses on the particle.
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• The direction and sense of the particle’s initial and final velocities should be established
• If the vector is known, assume that the sense of its components is in the direction of the positive
inertial coordinates
• As an alternative procedure, draw the impulse and momentum diagrams for the particle.
• In accordance with the established coordinate system apply the principle of impulse and momentum.
• Forces that are function of time must be integrated to obtain the impulse. Graphically, the impulse
is equal to the area under the force-time curve.
• If the problems involves the dependent motion of several particles, use the method described in
chapter 3 to relate their velocities. Make sure the positive coordinate directions used for writing these
kinematic equations are the same as those used for writing the equations of impulse and momentum
II. Principle of linear impulse and momentum for a system of particles:
Starting with equation of motion for a s system of particles moving relative relative ti an inertial frame
of reference.
~i
P~
P dV
Fi = mi
dt
The term on the left side represents only the sum of external forces acting on the system of particles.
~i = (vecVi )1 and t = t2 , V
~i =
Multiplying both sides by dt and integrating between the limits t = t1 , V
(vecVi )2 yields
P
~i )1 + P
mi (V
t2
Z
F~i =
X
~i )2
mi (V
t1
Which states that the initial momentum of the system plus the impulses of all the external forces acting
on the system from t1 to t2 are equal to the system final linear momentum. Alternatively, in terms of the
mass center G of the system, we have.
P
~G = P mi V
~i
( mi ) V
X
mi
~G )1 +
(V
XZ
t2
t1
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F~i =
X
mi
~G )2
(V
III. Conservation of linear momentum for a system of particles:
• When the sum of the external impulses is zero the principle of linear impulse and momentum becomes
P
mi (Vi )1 =
P
mi (Vi )2
which expresses the conservation of linear momentum; that is the linear momentum of a system of
particles remains constant during the time period t1 to t2 .
• In terms of the mass center G of the system, we have
(VG )1 = (VG )2
• For application, a careful study of the free-body diagram for the entire system of particles should be
made to identify the forces which create external impulses and thereby determine in which direction
linear momentum is conserved.
– If the time period over which the motion is studied is very short, some of the external impulses
may also be neglected or considered approximately zero. The forces causing these negligible
impulses are called non-impulsive forcese.g., weight of a body or any force which is very small
compared to the other larger (impulsive) forces (which change the system’s momentum drastically). When making the distinction between impulsive and non-impulsive forces, it is important
to realize that this applies only during the (very short) time interval t1 to t2 .
III-1. Procedure for analysis:
Generally, the principle of linear impulse and momentum or the conservation of linear momentum is
applied to a system of particles in order to determine the final velocities of the particles just after the time
period considered. By applying these equations to the entire system, the internal impulses acting within
the system, which may be unknown are eliminated from the analysis. For application it is suggested that
the following procedure be used.
• Establish the x, y, z inertial frame of reference and draw the free body diagram for each particle of
the system in order to identify the internal and external forces
• The conservation of linear momentum applies to the system in a given direction when no external
forces or if non-impulsive forces on the system in that direction Establish the direction and sense of
the particle’s initial and final velocities. If the sense is unknown, assume it is along a positive inertial
coordinate axis
• As an alternative procedure, draw the impulse and momentum diagrams for each particle of the
system
• Apply the principle of linear impulse and momentum or the conservation of linear momentum in the
appropriate directions
• If it is necessary to determine the internal impulse acting on only one particle of a system, then the
particle must be isolated (free body diagram) and the principle of linear impulse and momentum
must be applied to the particle
• After the impulse is calculated, and provided the time ∆t for which the impulse acts is known, the
average impulsive force F~av can be determined from
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F~av =
R
F~ dt
∆t
IV. Impact:
Impact occurs when two bodies collide with each other during a very short period of time, causing
relatively large (impulsive) forces to be exerted between the bodies. In general there are two types of
impact:
• Central impact: When the direction of motion of the mass centers of the two colliding particles is
along a line (line of impact) passing through the mass centers of the particles
• Oblique impact: When the motion of one or both particles is at an angle from the line of impact
IV-1. Central impact:
Consider the central impact of two particles A and B of masses mA and mB respectively
• The particles have the initial momenta shown in the figure below part (a). Provided (vA > vB ) the
collision will eventually occur.
• During the collision the particles must be thought of as deformable or nonrigid. The particles will
undergo
a period of deformation such that they exert an equal but opposite deformation impulse
R
P~ dt on each other (same figure part (b)).
~,
• Only at the instant of maximum deformation will both particles move with a common velocity V
since their relative motion is zero (same figure part (c))
• Afterward a period of restitution occur, in which case the particles will either return to Rtheir original
~ pushes
shape or remain permanently deformed. The equal but opposite restitution impulse Rdt
the particles apart from one another (same figure part (d))
• Just after separation the particles will have the final momenta shown in part (e)
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In most problems the initial velocities of the particles will be known, and it will be necessary to
determine their final velocities (vA )2 , and (vB )2 . in this regard, the momentum of system of particles is
conserved because during the collision the internal impulses of collision and restitution cancel and
(vA )1 + vB )1 = (vA )2 + vB )2
In order to obtain the second equation necessary to solve for (vA )2 , and (vB )2 . We must apply the
principle of impulse and momentum to each particle:
R
mA (vA )1 − P dt = mA v
mA v −
R
R dt = mA (vA )2
The ratio of the restitution impulse to the deformation impulse is called the coefficient of restitution.
e. From the above equations this value for particle A is
R
R dt
v − (vA )2
e= R
=
(vA )1 − v
P dt
In a similar manner we can establish e by considering particle B and the result is
R
R dt
v − (vB )2
R
e=
=
v − (vB )1
P dt
If the unknown v is eliminated in the previous equations, the coefficient of restitution e can be expressed
as
e=
(vB )2 − (vA )2
(vA )1 − (vB )1
Provided a value of e is specified the original problem can be solved for (vA )2 , and (vB )2 . In general e
has a value between 0 and 1, and one should be aware of the physical meaning of these two limits
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R
R
• Elastic Impact (e = 1) Deformation impulse ( P dt) is equal to the restitution impulse ( R dt)
R
• Plastic Impact (e = 0) No restitution impulse ( R dt = 0) so that after collision, both particles
stick together and move with a common velocity
From the above derivation it should be evident that the principle of work and energy cannot be used
for the analysis of impact problems since it is not possible to know how the internal forces of deformation
and restitution vary or displace during the the collision. By knowing the particle’s velocities before and
after collision, however, the energy loss during thePcollision
Pcan be calculated on the basis of the difference
in the particle’s kinetic energy. This energy loss ( T2 − T1 ), occurs because some of the kinetic energy
of the particle is transformed into thermal energy as well as creating sound and local deformation of the
material when collision occurs. In particular if the collision is perfectly elastic, no energy is lost in the
collision; whereas if the collision is plastic, the energy lost during the collision is a maximum.
IV-2. Oblique impact:
When oblique impact occurs between two smooth particles, the particles move away from each other
with velocities having unknown directions as well as unknown magnitudes. Provided the initial velocities
are known, four unknown are present in the problem. As shown in the figure below, these unknowns may
be represented with their magnitudes and directions or with their components.
IV-2-1. Procedure for analysis:
• If the y-axis is established within the plane of contact and the x-axis along the line of impact, the
impulsive forces of deformation and restitution act only in the x direction. Resolving the velocity or
momentum into components along the x and y axes, it is possible to write four independent scalar
equations in order to determine (vAx )2 , (vAy )2 , (vBx )2 , (vBy )2 .
• Momentum is conserved along the line of impact, x-axis.
• The coefficient of restitution e = [(vBx )2 − (vAx )2 ]/[(vAx )1 − (vBx )1 ], relates the relative-velocity
components of the particles along the line of impact (x-axis)
• Moment of particle A is conserved along the y-axis, perpendicular to the line of impact, since no
impulse acts on particle A in this direction
• Moment of particle B is conserved along the y-axis, perpendicular to the line of impact, since no
impulse acts on particle B in this direction
V. Angular Momentum:
~ o of a particle about point O is defined as the ”moment” of the particle’s
The angular momentum H
linear momentum about O. It is sometimes referred to as the moment of the momentum.
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~ as
• Scalar Formulation: Consider a particle of mass m moving in the x-y plane with a velocity V
shown in the figure below. The magnitude of the angular momentum of this particle about point O
~ | where d is the moment arm or the perpendicular distance from
has a magnitude equal to d m | V
~ . The direction can be found using the right hand rule Curling your
O to the line of action of mV
~ from the reference O. The extended thump points in the direction
right hand in the direction of mV
~ o . Common units for H
~ o are kg · m2 /s or slug · f t2 /s.
of H
• Vector Formulation: If the particle is moving along a space curve and ~r is a position vector drawn
from point O to the particle P then
~ o = ~r × mV
~
H
~ o is perpendicular to the plane made of ~r and mV
~
As shown in the figure below H
In the cartesian coordinate system where all vectors are expressed in terms of the unit vector along
the x, y and z axes the angular momentum is determined by evaluating the determinant:
î
rx
mvx
ĵ
ry
mvy
k̂
rz
mvz
VI. Relation between moment of a force and angular momentum:
• Consider a particle of constant mass m moving in an inertial frame as shown in the figure below.
Newton’s second applied to the particle states that
P~
d(m~v )
F =
= m ~a
dt
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The moment of the forces about point O
X
~ o = ~r ×
M
X
P ~
Mo can be expressed using Newton’s second law as
~˙ = d (~r × mV
~ ) − ~r˙ × mV
~ = d (~r × mV
~)=H
~˙ o
F~ = ~r × mV
dt
dt
This equation states that the resultant moment about point O of all the forces acting on the particle
is equal to the time rate of change of the particle’s angular momentum about point O.
• System of particles: Consider a system of particles of masses {mi } moving in an inertial system
of coordinates as shown in the figure below
Using the result of the previous paragraph on particle i, the resultant moments of the external forces
F~i and internal forces f~i on particle i about O is equal to the time time rate of change of the particle’s
angular momentum about point O.
~˙ i )o
(~ri × F~i ) + (~ri × f~i ) = (H
Summing over all the particles in the system gives
P
P
P ~˙
(~ri × F~i ) + (~ri × f~i ) = (H
i )o
The second term is zero since the internal forces occur in equal but opposite collinear pairs and hence
the moment of each pair about point O is zero. The above equation reduces to
P
P ~˙
(~ri × F~i ) = (H
i )o
Which states that the sum of the moments about point O of all the external forces acting on a system
of particles is equal to the time rate of change of the total angular momentum of the system about point
O. Although O has been chosen as the origin of coordinates, it actually can represent any fixed point in
the inertial frame of reference.
VII. Angular Impulse and Momentum Principles:
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• Principle of angular impulse
and momentum: If the last equation in the previous paragraph
P ~
~ o and integrated, we have assuming that at tine t = t1 ,
is rewritten in the form
Mo dt = dH
~ o = (H
~ o ) and at time t = t2 , H
~ o = (H
~ o)
H
1
2
XZ
t2
t1
~ o dt = (H
~ o ) − (H
~ o)
M
2
1
This equation is referred to as the principle of angular impulse and momentum. The initial and final
~ o = (H
~ o ) and H
~ o = (H
~ o ) are defined as the moment of teh linear momentum
angular momenta H
1
2
X Z t2
~ o dt
M
of the particle at the instant t1 and t2 respectively. The second term on the left side
t1
is called the angular impulse. It is determined by integrating with respect to time, the moments of
all the forces acting on the particle over the time period t1 to t2
t2
Z
~ o dt =
M
angular impulse =
Z
t2
(~r × F~ ) dt
t1
t1
• Principle of angular impulse and momentum for a system of particles: In a similar manner,
the principle of angular impulse and momentum for a system of particles may be written as
X
~ o) +
(H
1
XZ
t2
~ o dt =
M
t1
X
~ o)
(H
2
P~
Here the first and third terms represent the angular momenta of the system of particles [
Ho =
P
(~ri × m~vi ) ] at the instants t1 and t2 . The second term is the sum of the angular impulses given
to all the particles from t1 to t2 . Recall that these impulses are created only by the moments of the
~ o = ~ri × F~i .
external forces acting on the system where, for the ith particle, M
• Impulse and Momentum principles:
– Impulse and momentum principles lead to the following two vector equations describing the
particle’s motion
X Z t2
m ~v1 +
F~ dt = m ~v2
t1
~ o) +
(H
1
XZ
t2
t1
~ o dt = (H
~ o)
M
2
– If the particle is confined to move in the x-y plane, these two vector equations can be written
as three scalar equations:
X Z t2
m (vx )1 +
Fx dt = m (vx )2
t1
m (vy )1 +
(Ho )1 +
t2
XZ
t1
XZ
Fy dt = m (vy )2
t2
t1
Mo dt = (Ho )2
The first two of these equations represent the principle of linear impulse and momentum in the x
and y directions. The third equation represents the principle of angular impulse and momentum
about the z-axis
• Conservation of Angular Momentum:
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– When the angular impulses acting on a particle are all zero during the time t1 to t2 we have
~ o ) = (H
~ o)
(H
1
2
which is known as conservation of angular momentum
– If no external impulse is applied to the particle, both linear a nd angular momentum will be
conserved.
– Conservation of angular momentum for a system of particles is given by
X
X
~ o) =
~ o)
(H
(H
1
2
where the summation must include the angular momenta of all particles in the system
• Procedure for solving problems: When applying the principles of angular impulse and momentum, or the conservation of angular momentum, the following procedure should be used
– Free-body diagram:
∗ Draw the particle’s free-body diagram in order to identify any axis about which angular
momentum may be conserved. For this to occur, the moments of the forces (or impulses)
must be parallel or pass through the axis so as to create zero moment throughout the time
period t1 to t2 .
∗ The direction and sense of the particles’ initial and final velocities should also be established
∗ As an alternative procedure, draw the impulse and momentum diagrams for the particle
– Momentum equation:
∗ Apply the principle of angular impulse and momentum, or if appropriate, the conservation
of angular momentum.
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