2Curves and Line
IA Vector Calculus
2.4 Work and potential energy
Definition
(Work and potential energy)
.
If
F
(
r
) is a force, then
R
C
F ·
d
r
is
the work done by the force along the curve
C
. It is the limit of a sum of terms
F(r) ·δr, i.e. the force along the direction of δr.
Consider a point particle moving under
F
(
r
) according to Newton’s second
law: F(r) = m
¨
r.
Since the kinetic energy is defined as
T (t) =
1
2
m
˙
r
2
,
the rate of change of energy is
d
dt
T (t) = m
˙
r ·
¨
r = F ·
˙
r.
Suppose the path of particle is a curve C from a = r(α) to b = r(β), Then
T (β) −T (α) =
Z
β
α
dT
dt
dt =
Z
β
α
F ·
˙
r dt =
Z
C
F · dr.
So the work done on the particle is the change in kinetic energy.
Definition
(Potential energy)
.
Given a conservative force
F
=
−∇V
,
V
(
x
) is
the potential energy. Then
Z
C
F · dr = V (a) − V (b).
Therefore, for a conservative force, we have
F
=
∇V
, where
V
(
r
) is the
potential energy.
So the work done (gain in kinetic energy) is the loss in potential energy. So
the total energy T + V is conserved, i.e. constant during motion.
We see that energy is conserved for conservative forces. In fact, the converse
is true — the energy is conserved only for conservative forces.