The Work-Energy Theorem
Now
that we have a definition of work, we can apply the concept to kinematics.
Just as force was related to acceleration through F = ma ,
so is work related to velocity through the Work-Energy Theorem.
Derivation of the Work-Energy Theorem
It
would be easy to simply state the theorem mathematically. However, an
examination of how the theorem was generated gives us a greater understanding
of the concepts underlying the equation. Because a complete derivation requires
calculus, we shall derive the theorem in the one-dimensional case with a
constant force.
Consider
a particle acted upon by a force as it moves from x o to x f .
Its velocity also increases from v o to v f .
The net work on the particle is given by:
W net = F net(x f - x o)
Using Newton's Second Law we can substitute
for F:
W net = ma(x f - x o)
Given
uniform acceleration, v f 2 - v I 2 =
2a(x f - x o) .
Substituting for a(x f - x o)into
our work equation, we find that:
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W net =
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This equation is one form of the work-energy
equation, and gives us a direct relation between the net work done on a
particle and that particle's velocity. Given an initial velocity and the amount
of work done on a particle, we can calculate the final velocity. This is
important for calculations within kinematics, but is even more important for
the study of energy, which we shall see below.
Kinetic Energy and the Work-Energy Theorem
As
is evident by the title of the theorem we are deriving, our ultimate goal is to
relate work and energy. This makes sense as both have the same units, and the
application of a force over a distance can be seen as the use of energy to
produce work. To complete the theorem we define kinetic energy as the energy of
motion of a particle. Taking into consideration the equation derived just
previously, we define the kinetic energy numerically as:
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K =
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Thus we can substitute K in
our work energy theorem:
W net =
mv f 2 -
mv I 2 = K f - K o
Implying
that
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W net = ΔK
|
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This is our complete Work-Energy theorem. It
is powerfully simple, and gives us a direct relation between net work and
kinetic energy. Stated verbally, the equations says that net work done by
forces on a particle causes a change in the kinetic energy of the particle.
Though
the full applicability of the Work-Energy theorem cannot be seen until we study
the conservation
of energy, we can use the theorem now to calculate the velocity of
a particle given a known force at any position. This capability is useful,
since it relates our derived concept of work back to simple kinematics. A
further study of the concept of energy, however, will yield far greater uses
for this important equation.
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