On the left we have the work W, so:
W = m a r
∆
On the right we have two quantities used to characterize the motion of a particle so we have
certainly met our goal of relating work to motion, but we can untangle things on the right a bit if
we recognize that, since we have a constant force, we must have a constant acceleration. This
means the constant acceleration equations apply, in particular, the one that (in terms of r rather
than x) reads:
v 2 = v 2
o
+ 2a∆r
Solving this for a∆r gives
1
1
2
2
a ∆r =
v − v
o
2
2
Substituting this into our expression for W above (the one that reads W = m a r
∆ ) we obtain
1
1
2
2
W = m v −
vo
2
2
which can be written as
1
1
2
2
W =
v
m
−
v
m o
2
2
1
Of course we recognize the
2
v
m o as the kinetic energy of the particle before the work is done
2
1
on the particle and the
2
v
m
as the kinetic energy of the particle after the work is done on it.
2
To be consistent with the notation we used in our early discussion of the conservation of
mechanical energy we change to the notation in which the prime symbol ( ′ ) signifies “after” and
no super- or subscript at all (rather than the subscript “o”) represents “before.” Using this
notation and the definition of kinetic energy, our expression for W becomes:
W = K ′ − K
Since the “after” kinetic energy minus the “before” kinetic energy is just the change in kinetic
energy ∆K, we can write the expression for W as:
W = K
∆
(24-2)
This is indeed a simple relation between work and motion. The cause, work on a particle, on the
left, is exactly equal to the effect, a change in the kinetic energy of the particle. This result is so
important that we give it a name, it is the Work-Energy Relation. It also goes by the name: The
Work-Energy Principle. It works for extended rigid bodies as well. In the case of a rigid body
that rotates, it is the displacement of the point of application of the force, along the path of said
point of application, that is used (as the ∆r ) in calculating the work done on the object.
1
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