Conservation of Energy
WORK AND KINETIC ENERGY
Definition of Work
dU = F. dr
dU = work done by force ( ) on
particle during displacement ( dr )
dU =F ds cos
dU may be interpreted as
displacement times force component
Ft = F cos in the dir
KINETICS OF SYSTEMS OF PARTICLES
1. Rigid body is a solid system of particles where the distance between particles
2. Nonrigid body is a defined mass of liquid or gaseous particles flowing at
Generalized Newtons Second L
ANGULAR IMPULSE AND ANGULAR MOMENTUM
In addition to the properties of linear impulse and linear momentum, there is a
parallel set of equations for angular impulse and angular momentum.
Particle P of mass m moving along a curve in space, located by its pos
IMPULSE AND MOMENTUM
A particles momentum G ( m V ) is a vector quantity describing its
tendency to continue in its current path.
The Impulse applied to an object over a time period is equal to the change
As long as
STEADY MASS FLOW
Steady Flow of non-rigidly connected particles fluid flow.
Conservation of Mass
Conservation of Momentum
Analysis of Flow Through a Rigid Container
Consider a rigid container into which mass flows in a steady stream at the rate
Particle A of mass m is observed from a set of axes x-y-z which translate with
respect to a fixed reference frame X-Y-Z.
The acceleration of origin B of x-y-z axes is
The acceleration of A as observed relative to x-y-z is
a = a/ B= r /
CENTRAL FORCE MOTION
When a particle moves under the influence of a force directed toward a fixed
center of attraction, the motion is called central-force motion.
Motion of a Single Body
Consider a particle of mass m moving under the action of the central
The linear momentum for a representative particle of mass m i of the system
Gi=mi v i where v i =r i
The linear momentum of the system is defined as the vector sum of the linear
momenta of all of its particles
G= mi v i
Sometimes convenient to use a frame or reference that is moving.
Choice of Coordinate System
The motion of the moving coordinate system is specified with respect to a fixed
Fixed coordinate system is a system whose absol
KINETICS OF PARTICLES
Kinetics is the study of the relations between unbalanced forces and the resulting
changes in motion.
There are three general approaches to the solution of kinetics problems:
Direct application of Newtons second law (called the force
Plane Curvilinear Motion
motion along a curved path lieing in a single plane.
If in x-y plane, z and are zero and r is equivalent to R
calculation will require time derivatives of vector quantities.
At time t the particle is at position A
At time t+ t
AMME1550 Dynamics 1
Can describe motion in x and y components independently. Final vector can be
reassembled by combination of the scalar components
are unit vectors in the x and y-directions respectively
= x y
e r and
e are unit vectors
position defined by angle ( ) and radius ( r )
=r e r
During time dt the coordinate directions rotate through the angle d .
The unit vectors also rotate through the same angle.
d e r= e d
d e = e r d
KINEMATICS OF PARTICLES
-Motion of bodies without reference to the forces.
-Geometry of motion.
-Design of Cams. Gears, Linkages, Machine Elements to produced desired motion
needed as a prerequisite to Kinetic
NORMAL AND TANGENTIAL COORDINATES
Using path variables
(n & t) orthogonal coordinates n normal, t tangential
positive n always toward center of curvature,
positive t always in direction of motion.
e n uni
SPACE CURVILINEAR MOTION Three Dimensional Motion
Rectangular Coordinates (x-y-z) (Cartesian)
= x y
V = = x y z = u v w
i j k
= V = = x y z = u v w = ax a y z
i j k
We can write Newtons second law equation for all the forces acting on a particle
which moves along plane curvilinear path in one of the three ways - depending
on the choice of the coordinate system which is most appropriate to specify t
CONSTRAINED MOTION OF CONNECTED PARTICLES
Constrains or limits may need to be applied to the equations of motion of a
One Degree of Freedom
To analyze the motion of A and B, we establish position coordinates x and y
measured from a convenient fi
When the mass within the boundary of a system under consideration is not
constant, the previous relationships are no longer valid.
Equation of Motion
Linear motion of a system whose mass varies with time.
Body which gains mass by overtaking