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.
Chapter 15: Kinetics of a Particle: Impulse and Momentum Textbook:
Engineering Mechanics- STATICS and DYNAMICS- 11 th Ed., R. C.
Hibbeler and A. Gupta Course Instructor: Miss Saman Shahid
Slide 2
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.
Slide 3
Slide 4
For problem solving: Initial momentum+ sum of all impulses =
final momentum
Slide 5
Slide 6
Principle of Linear Impulse and Momentum for a System of
Particles
Slide 7
Conservation of Linear Momentum for a System of Particles 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. When the sum of the external impulses
acting on a system of particles is zero, eq., 15.6 reduces to:
Slide 8
Impulsive and Non-Impulsive Forces The forces causing
negligible impulses are called nonimpulsive 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 balls momentum drastically. Balls 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 balls weight is
important since it, along with air resistance, causes the change in
the momentum of the ball.
Slide 9
Slide 10
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.
Slide 11
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
Slide 12
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.
Slide 13
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.
Slide 14
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, 2, 2.
Slide 15
Angular Momentum The angular momentum of a particle about point
O is defined as the moment of the particles 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.
Slide 16
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.
Slide 17
Relation between Moment of a Force and Angular Momentum Another
way of expressing Newtons Law The moments about point O of all
forces acting on the particle may be related to the particles
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
particles 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 particles
linear momentum.
Slide 18
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.
Slide 19
Vector Formulation: Using impulse and momentum principles it is
therefore possible to write two equations which define the
particles 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.
Slide 20
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 particles 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 particles angular momentum will be
conserved and linear momentum may not.