Transcript Work and Energy

WORK AND ENERGY
PRINCIPLE
Work and Kinetic Energy
In the
article we applied Newton’s second
 previous

law F  ma to various problems of particle motion to
establish the instantaneous relationship between
the net force acting on a particle and the resulting
acceleration of the particle. When we needed to
determine the change in velocity or the
corresponding displacement of the particle, we
integrated the computed acceleration by using the
appropriate kinematic equations.
There are two general classes of problems in which
the cumulative effects of unbalanced forces acting
on a particle are of interest. These cases involve
(1) integration of forces with respect to the
displacement of the particle and (2) integration of
forces with respect to the time they are applied.
We may incorporate the results of these
integrations directly into the governing equations of
motion so that it becomes unnecessary to solve
directly for the acceleration.
Integration with respect to displacement leads to
the equations of work and energy.
Definition of Work
Let’s now develop the quantitative
meaning of “work”.

The figure shows a force F acting on a particle at A

which moves along the path shown. The position vector r
measured from some convenient origin O locates the

particle as it passes A and dr is the differential
displacement associated with an infinitesimal
movement from A to A'.
The work
 done by the
force F during the

displacement dr is defined
as
 
dU  F  dr
The magnitude of this dot product
is dU=F ds cosa,


where a is the angle between F and dr and where ds

is the magnitude of dr . This expression may be
interpreted as the displacement multiplied by the
force component Ft=Fcosa in the direction of the
displacement.
Alternatively, the work is dU
may be interpreted as the force
multiplied by the displacement
component ds cosa in the
direction of the force.
dU  Ft  ds
or dU  F ds cos a 
So dU may be computed as
 
dU  F dr cos a  F ds cos a
With this definition of work, it should be noted that
the component Fn=Fsina normal to the displacement
does no work.Thus, the work dU may be written as
dU  Ft  ds
Work is positive if the working component Ft is in the
direction of the displacement and negative if it is in
the opposite direction. Forces which do work are
termed as active forces. Constraint forces which do
no work are termed as reactive forces.
Unit of Work
The SI units of work are those of force (N) times
displacement (m) or N·m. This unit is given the special
name joule (J), which is defined as the work done by a
force of 1 N acting through a distance of 1 m in the
direction of the force.
Calculation of Work
During a finite movement of the
point of application of a force,
the force does an amount of
work equal to
 
dU  F  dr ,
U   Fx d x  Fy d y  Fz d z 
2
1
s2
s2
s1
s1
U   F cos a   Ft ds
or
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. If the functional relationship is not known as a
mathematical expression which can be integrated but is
specified in the form of approximate or experimental
data, then we can compute the work by carrying out a
numerical or graphical integration as represented by
the area under the curve of Ft versus s.
Examples of Work
When work must be calculated, we
always begin
 must

with the definition of work U   F  dr , insert 
appropriate vector expressions for theforce F and
the differential displacement vector dr , and carry
out the required integration.
Among the most frequently met forces will be
constant forces, spring forces and weights.
Work Associated with a Constant External Force




Consider the constant force P  Pxi  Py j  Pz k
applied to the body as it moves from position 1 to
position 2. The work done by this force is
x2
y2
z2
x1
y1
z1
U  Px  dx  Py  dy  Pz  dz
 Px x2  x1   Py  y2  y1   Pz z2  z1 
As can be seen, the work done
depends only on the coordinates of
positions 1 and 2, there is no
reference to the path travelled
between these positions. The same
work will be done whether the
particle has followed path a or path b.
Work Associated with a Spring Force
We consider here 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.
The force exerted
bythe spring

on the body is F   kxi .From the
definition of work
U1 2  
2
1
 


2
F  dr    kxi  dxi
1
 
x2
x1

1
2
2
kxdx  k x1  x2
2

If the initial position is the position of zero spring
deformation so that x1=0, then the work is negative for
any final position x2≠0.This is verified by recognizing that
if the body begins at undeformed spring position and then
moves to the right, the spring force is to the left; if the
body begins at x1=0 and moves to the left, the spring
force is to the right. On the other hand, if we move from
an arbitrary initial position x1≠0 to the undeformed final
position x2=0, we see that the work is positive. In any
movement toward the undeformed spring position, the
spring force and the displacement are in the same
direction.
In the general case, neither x1
nor x2 is zero. The magnitude of
the work will be equal to the
shaded trapezoidal area under
the line.
In calculating the work done on
a body by a spring force, care
must be taken to ensure that
the units of k and x are
consistent.
motion
motion
original
position
stretched
position
original
position
compressed
position
In the case of a block pulled by a force P, since the extension
or compression of the spring by an amount x is in opposite
direction to the displacement of the body, the work done by the
spring will be negative.
x2
1
2
2
U 1 2    Fdx    kxdx   k x 2  x1
x1
2
On the other hand, when a precompressed or preextended
spring returns to its unstrethced length, since the spring force
and the displacement will be in the same direction work will be
positive.


Work Associated with Weight
We assume that g is constant,
that is the altitude variation is
sufficiently small so that the
acceleration of gravity may be
considered constant.
The work done by the weight
W is
U   Fx dx  Fy dy  Fz dz 
0  0  W  dy  0  0  Wy2   Wy1 
U1 2  W  y1  y2   W  y2  y1   Wy

y2
y1
If y < 0 work will be positive, if y > 0 work will be
negative.
Work and Curvilinear Motion
We now consider the work
done on a particle of mass m,
moving along a curved path
under
the action of the force

for the
F , which stands

resultant F of all forces
acting on the particle. The
position of m is specified by

the position vector r , and its
displacement along its path
during the time dt is

represented by the change dr
in its position vector.

The work done by F during a
finite movement of the
particle from point 1 to point
2 is
U1 2  
2
1
 
s2
F  dr   Ft ds
s1
where the limits specify the
initial and final end points of
the motion.
From Newton’s second law


F  ma
and
U1 2  
2
1
 
s2
 
F  dr   madr
s1
 
Since a  dr  at ds , in the relationship ads = vdv, a=at
(at = acosq).
U1 2  
2
1

v2
v1
 
s2
s2
 
F  dr   madr   mat ds
s1
s1

1
2
2
mvdv  m v2  v1
2

where the integration is carried out between points 1
and 2 along the curve, at which points the velocities
have the magnitudes v1 and v2, respectively.
Principle of Work and Kinetic Energy
The kinetic energy T of the particle is defined as
1 2
T  mv
2
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 joules (J) in SI units. Kinetic energy is always
positive, regardless of the direction of the velocity.
Work – energy equation can be stated as
U12  T2  T1  T
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 positive, negative or zero. When
written in this concise form, the equation tells us that
the work always results in a change of kinetic energy.
Alternatively, the work – energy relation may be
expressed as the initial kinetic energy T1 plus the work
done U1-2 equals the final kinetic energy T1, or
T1  U12  T2
When written in this form, the terms correspond to the
natural sequence of events.
Advantages of the Work – Energy Method
A major advantage of the method of work and energy is
that it avoids the necessity of computing the
acceleration and leads directly to the velocity changes
as functions of the forces which do work. Further, the
work energy equation involves only those forces which
do work and thus give rise to changes in the magnitude
of the velocities.
If we consider a system of two particles joined
together by a connection which is frictionless and
incapable of any deformation, the forces in the
connection are equal and opposite, and their points of
application have identical displacement components in
the direction of forces. Therefore, the net work done
by these internal forces is zero during any movement of
the of the system. Thus, the equation of work – energy
is applicable to the entire system, where U1-2 is the
total or net work done on the system by forces
external to it and T is the change, T2 – T1, in the total
kinetic energy of the system.
The total kinetic energy is the sum of the kinetic
energies of both elements of the system. So, another
advantage of the work – energy method is that it
enables to analyze a system of particles joined without
dismembering the system.
Application of the work – energy method requires
isolation of the particle or system under consideration.
Therefore, correct drawing of the free body diagram is
of vital importance in the solution of any problem.
Power
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 input 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, thus the capacity of a machine is rated
by its power, which is defined as the time rate of
doing work.

When a force F does work an amount U, the power
P will be
 
dU F  dr
P

dt
dt
 
P  F v

dr 
v
dt
Power is a scalar quantity. In SI system it has the
units of N∙m/s = Joule /s = Watt (W).
Potential Energy
The work done on a particle must be calculated
separately for each force. The work done on a
particle by forces named as “conservative forces”
can be determined by a potential energy function
which depends only on the position of the particle.
Spring forces and forces of gravitational
attraction are named as conservative forces and
when the work by these are calculated using the
potential energy concept, they provide
simplification in the analysis of many problems.
A) Gravitational Potential Energy
Let’s consider the motion of a
particle 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 the work mgh done
against the gravitational field to elevate a particle a
distance h above some arbitrary reference plane (called
a datum), where Vg is taken to be zero. Thus, the
potential energy is written as
Vg=mgh
This work is called potential energy because it may
be converted into energy if the particle is allowed
to do work on a supporting body while it returns to
its lower original datum plane. In going from one
level at h=h1 to a higher level at h=h2, the change
in potential energy becomes
Vg = mg (h2 – h1)=mgh
The corresponding work done by the gravitational
force on the particle is –mgh. Thus, the work
done by the gravitational force is the negative of
the change in potential energy.
B) Elastic Potential Energy
The second example of potential energy occurs in
the deformation of an elastic body, such as a
spring. The work which is done on the spring to
deform it is stored in the spring and is called the
elastic potential energy Ve. This energy is
recoverable in the form of work done by the
spring on the body attached to its movable end
during the release of the deformation of the
spring.
The force supported by a linear spring at any
deformation x, tensile or compressive, from its
undeformed position is F=kx.
Thus the elastic potential energy of the spring as
the work done on it to deform it an amount x is
defined as
Ve  
x
0
1 2
kxdx  kx
2
If the deformation, either tensile or compressive,
of a spring increases from x1 to x2, during the
motion, then the change in potential energy of the
spring is

1
2
2
Ve  k x2  x1
2
which is positive.

Conversely, if the deformation of a spring decreases
during the motion interval, then the change in
potential energy of the spring becomes negative.
Because the force exerted on the spring by the
moving body is equal and opposite to the force F
exerted by the spring on the body, it follows that the
work done on the spring is the negative of the work
done on the body. Therefore, we may replace the
work U done by the spring on the body by –Ve, the
negative of the potential energy change for the
spring, provided that the spring is included in the
system.
Work – Energy Equation
With the elastic member included in the system, we
can modify the work – energy equation to account for
the potential energy terms. If Uʹ1-2 stands for the
work of al external forces other than gravitational
forces and spring forces, than we can write
U12  T  V
or
U12  T  Vg  Ve
where V is the change in total potential energy.
Alternatively, we may write
T1  Vg1  Ve1  U1 2  T2  Vg 2  Ve 2
or
U12   T  Vg  Ve 
For problems where the only forces are gravitational,
elastic and nonworking constraint forces, the Uʹ term
is zero and the equation becomes
T1  V1  T2  V2
or
E1  E2
Where E=T + V is the total mechanical energy of the
particle and its attached spring. When E is constant
E  0
The transfers of energy between kinetic and
potential may take place as long as the total
mechanical energy T + V does not change. This
equation expresses “the law of conservation of
dynamical energy”.