Download Chapter 15: Kinetics of a Particle: Impulse and

Chapter 15: Kinetics of a Particle:
Impulse and MomentumTextbook: Engineering MechanicsSTATICS and DYNAMICS- 11th Ed., R. C. Hibbeler and A. Gupta
Course Instructor: Miss Saman Shahid
Impulse and momentum principles are
required to predict the motion of this
golf ball.
“the product of the average force acting upon a
body and the time during which it acts, equivalent to the
change in the momentum of the body produced by such a
force.
Principle of Linear Impulse and Momentum
 Linear momentum of the particle:
L=mv
 The integral of I is referred to as
linear impulse.
 This term is a vector quantity
which measures the effect of a force
during the time the force acts.
For problem solving:
 Initial momentum+ sum of all
impulses = final momentum
Principle of Linear Impulse and
Momentum for a System of Particles
Conservation of Linear Momentum for a System of
Particles
 When the sum of the external impulses acting
on a system of particles is zero, eq., 15.6
reduces to:
The conservation of linear momentum is often applied when
particles collide or interact.
Internal impulses for the system will always cancel out, since they
occur in equal but opposite collinear pairs.
If the time period over which the motion is studies is very short,
some of the external impulses may also neglected or considered
approximately equal to zero.
Impulsive and Non-Impulsive Forces
 The forces causing negligible impulses are called nonimpulsive
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forces.
Forces which are very large and act for a very short period of time
produce a significant change in momentum and are called
impulsive forces.
Impulsive forces normally occur due to an explosion or the
striking of one body, the force imparted by a slightly deformed
spring having a relatively small stiffness, or for that matter is very
small compared to other larger (impulsive) forces.
Consider the tennis ball with a racket. During the very short time
of interaction, the force of the racket on the ball is impulsive since
it changes the ball’s momentum drastically.
Ball’s weight will have negligible effect on the change in
momentum, and therefore it is nonimpulsive. Consequently, it can
be neglected from an impulse-momentum analysis during this
time.
If impulse- momentum analysis is considered, then the impulse of
the ball’s weight is important since it, along with air resistance,
causes the change in the momentum of the ball.
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.
Examples:
1) String of a hammer on a nail,
2) Golf club on a ball
Two types of impact:
Central Impact: It occurs when the direction of
motion of the mass centers of the particles. This line
is called the line of impact.
Oblique Impact: When the motion of one or both
of the particles is at an angle with the line of impact.
1- Central Impact
•Two particles have the initial
momenta and vA1 > vB1, collision
will eventually occur.
•During collision the particles will
be deformed or non-rigid.
•At the instant of maximum
deformation, both particles move
with a common velocity.
•The particles will return to their
original shape and will have equal
and opposite restitution impulse.
•Just after separation the particles
will have the final momenta and
vB2>vA2
Momentum for the System of Particles is Conserved, since during collision the
internal impulses of deformation and restitution cancel
•In order to obtain second equation necessary to
solve for vA2 and vB2, we must apply the
principle of impulse and momentum to each
particle. During the deformation phase for
particle A.
•The ratio of the restitution impulse to the
deformation impulse is called the coefficient of
restitution e.
•If the unknown v is eliminated from two
equations of e then then coefficient of restitution
can be expressed in terms of particles’ initial and
final velocities.
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Coefficient of Restitution:
 Experimentally it has been found that e varies appreciably
with impact velocity as well as with the size and shape of
the colliding bodies.
 In general e has a value between zero and one.
 Elastic Impact (e=1): If the collision between the two
particles is perfectly elastic, the formation impulse (Pdt) is equal
and opposite to the restitution impulse (Rdt). Although in reality
this can never be achieved e=1 for an elastic collision. If the
impact is perfectly elastic, no energy is lost in the collision.
 Plastic Impact (e=0): The impact is said to be inelastic or
plastic when e=0. In this case there is no restitution impulse given
to the particles (Rdt=0), so that after collision both particles
couple or stick together and move with a common velocity. If the
collision is plastic, the energy lost during collision is a maximum.
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 unknowns may be
represented either as vA2, vB2,
θ2 , Ф2.
Angular Momentum
 The angular momentum of a particle about point O is
defined as the “moment” of the particle’s linear momentum
about O. Since this concept is analogues to finding the
moment of a force about a point, the angular momentum
Ho, is sometimes referred to as the “moment of
momentum”.
 Scalar Formulation: if a particle moves along a curve lying in
the x-y plane, the angular momentum at any instant can be
determined about point O (actually the z-axis) by using a scalar
formulation.
 Here d is the moment arm or perpendicular distance from O to the
line of action of mv.
 Right Hand Rule:The curl of the fingers of the right hand
indicates the sense of rotation of mv about O, so that in this case the
thumb or (Ho) is directed perpendicular to the x-y plane along
the+z-axis.
Angular Momentum- vector formulation
 If the particle moves along a space curve, the
vector cross product can be used to determine the
angular momentum about O. Here r denotes a
position vector drawn from point O to the
particle. As shown in the figure, Ho is
perpendicular to the shaded plane containing r
and mv.
Relation between
Moment of a Force and Angular Momentum
Another way of expressing Newton’s Law
 The moments about point O of all forces acting on the
particle may be related to the particle’s angular momentum
by using the equation of motion.
 The moments of the forces about point O can be obtained by
performing a cross-product multiplication of each side of
this equation by the position vector r, which is measured in
the x,y,z inertial frame of reference.
 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.
 L=mv, (F=ma=mdv/dt=d(mv)/dt) , so the
resultant force acting on the particle is equal to the time
rate of change of the particle’s linear momentum.
Angular Impulse and Momentum Principles
 Equation can be rewritten in the form below and after integrating it
we get the
principle of angular impulse and momentum.
The quanitiy (Modt) is called Angular Impulse.
Vector Formulation: Using impulse and
momentum principles it is therefore
possible to write two equations which
define the particle’s motion:
Scalar Formulation: In general, the
above equations may be expressed in
x,y,z component form, yielding a total of
six independent scalar equations.
First two equations represent “principle
of linear impulse” and third equation
represents the “principle of angular
momentum about the z-axis.
Conservation of Angular Momentum:
When the angular impulses acting on a
particle are all zero during time t1 to t2,
we can write:
It states that from t1 to t2, the particle’s
angular momentum remains constant.
Obviously, if no external impulse is
applied to the particle, both linear and
angular momentum will be conserved.
In some cases, however, the particle’s
angular momentum will be conserved and
linear momentum may not.