KINETICS OF PARTICLES - DEUkisi.deu.edu.tr/binnur.goren/Dynamics2016G/11_Work_Energy... · The...
Transcript of KINETICS OF PARTICLES - DEUkisi.deu.edu.tr/binnur.goren/Dynamics2016G/11_Work_Energy... · The...
WORK : Figure shows a force acting on a particle at A which
moves along the path. The position vector measured from
some convenient origin O locates the particle as it passes
point A, and is the differential displacement from A to A.
The work done by the force during the displacement is
defined as
rdFdU
F
r
rd
F
The magnitude of this dot product is dU=Fdscosa, where ds
is the magnitude of . This expression may be interpreted
as the displacement multiplied by the force component
Ft=Fcos a in the direction of the displacement.
rd
With this definition of work, it should be noted that the
component Fn=Fsina normal to the displacement does no
work.
The SI units of work are those of
force (N) times displacement (m)
of N.m. This unit is given Joule (J).
Calculation of Work
During a finite movement of the point of application of a force,
the force does an amount of work equal to
In order to carry out this integration, it is necessary to know the
relations between the force components and their respective
coordinates or the relation between Ft and s.
dzFdyFdxFrdFU zyx
or
dsFU t
We consider the common linear spring of stiffness k where the force required to stretch or
compress the spring is proportional to the deformation x. We wish to determine the work
done on the body by the spring force as the body undergoes an arbitrary displacement
from an initial position x1 to a final position x2.
x1 x x2
F=kx P
F=kx
F
Undeformedposition
Stretched
x1 x x2
F=kx
F
Undeformedposition
Compressed
F=kxPx x
(a)(b)
21
2221
21
2
12
1
xxkdx.kxU
FdxU
x
x
Work done on the block by the spring is negative since spring force is in the
opposite direction of the displacement.
Work done on the block by the spring
We assume that g is constant, that is the altitude variation is sufficiently small so that the
acceleration of gravity may be considered constant.
1221
2
1
yymgmgdyU
y
y
If the body rises (perhaps due to other forces not shown), then ( y2 y1) > 0
and this work is negative. If the body falls, ( y2 y1) < 0 and the work is positive.
Work done on the body by the weight mg as the
body is displaced from an altitude y1 to a final
altitude y2 is
We now consider the work done on a particle of mass m moving along acurved path under the action of the force F, which is the resultant of allforces acting on the particle.
The work done by F during a finite movement of the particle from point 1to point 2 is
2
1
2
1
21 dsFrdFU t
When we substitute Newton’s second law , the expression becomes
2
1
2
1
21 rdamrdFU
where at is the tangential component of the acceleration. In terms of the
velocity atds=vdv. Thus the expression for work becomes
dsarda t
2
1
21
22
2
1
212
1v
v
vvmmvdvrdFU
The kinetic energy T of the particle is defined as2
2
1mvT
and is the total work which must be done on the particle to bring it from a
state of rest to a velocity v. Kinetic energy T is a scalar quantity with the units
of “N.m” or “Joule (J)” in SI units. Kinetic energy is always positive, regardless
of the direction of the velocity.
Expression for the work may be written as
TTTU 1221
which is the work-energy equation of particle. The equation states that the
total work done by all forces acting on a particle as it moves from point 1 to
point 2 equals the corresponding change in kinetic energy of the particle.
Although T is always positive, the change T may be negative, positive or zero.
We consider first the motion of a particle of mass m in close proximity to the
surface of the earth, where the gravitational attraction (weight) mg is essentially
constant. The gravitational potential energy Vg of the particle is defined as
mghVg
hmghhmgVg 12
The change in potential energy is
The second example of potential energy occurs in the deformation of an
elastic body, such as a spring. Elastic potential energy is
2
02
1kxkxdxV
x
e
21
22
2
1xxkVe
The change in elastic potential energy is
With the elastic member included in the system, we may write the work-energyequation as
where U1-2 term is for the work of all forces other than gravitational forces andspring forces.
22221111 egeg VVTUVVT or
eg VVTU 21
We may rewrite the alternative work-energy equation as
eg VVTU 21
Where E=T+Vg+Ve is the total mechanical energy of the particle. For problemswhere U1-2 term is zero, and the energy equation becomes
0E or E=constant
This equation expresses the law of conservation of dynamical energy.
The capacity of a machine is measured by the time rate at
which it can do work or deliver energy. The total work or
energy output is not a measure of this capacity since a motor,
no matter how small, can deliver a large amount of energy if
given sufficient time. Thus, the capacity of a machine is rated
by its power, which is defined as the time rate of doing
work.
Power (GÜÇ)