Math 21a Supplement 1 on Work and Energy
Newton’s law asserts that the position vector
r
(t) of a particle of mass m under the
influence of a force
F
obeys the equation
m
r
´´ =
F
.
(1)
a)
Work
Suppose that the components of the force vector
F
do not depend on time or the position of
the particle.
Thus,
F
has components (a, b, c) which are numbers, not functions.
(For example,
F
= (1, 2, 3).)
Then, the
work
done in moving the particle from position
r
0
to position
r
1
is (by
definition)
W
≡
F
•(
r
1

r
0
).
(2)
Note that this notion of work can be generalized to apply to any force vector
F
, constant or not; but
we are not ready at this point in the course for the generalization.
b) Energy
The
energy
of a particle at position
r
with velocity vector
r
´ is the function
e
≡
2
−
1
m 
r
´
2

F
•
r
.
(3)
Note that Newton’s law (Equation (1)) implies that the energy function does not change as time
evolves.
Indeed, we can differentiate (3) to find that
e
´ = m
r
´´•
r
´ 
F
•
r
´ =
F
•
r
´ 
F
•
r
´ = 0 ,
(4)
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 Fall '11
 GregoryJ.Galloway
 Math, Multivariable Calculus, Force, Mass, 1 m, force vector

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