Substituting Eq. 8-5 into Eq. 8-1, we find that the change in potential energy
due to the change in configuration is, in general notation,
(8-6)
Gravitational Potential Energy
We first consider a particle with mass m moving vertically along a y axis (the
positive direction is upward). As the particle moves from point y i to point y f ,
the gravitational force does work on it. To find the corresponding change in
the gravitational potential energy of the particle – Earth system, we use Eq. 8-6
with two changes: (1) We integrate along the y axis instead of the x axis, because
the gravitational force acts vertically. (2) We substitute Ϫmg for the force symbol F,
because has the magnitude mg and is directed down the y axis. We then have
which yields
⌬U ϭ mg(y f Ϫ y i ) ϭ mg ⌬y.
(8-7)
Only changes ⌬U in gravitational potential energy (or any other type of
potential energy) are physically meaningful. However, to simplify a calculation or
a discussion, we sometimes would like to say that a certain gravitational potential
value U is associated with a certain particle – Earth system when the particle is at
a certain height y.To do so, we rewrite Eq. 8-7 as
U Ϫ U i ϭ mg(y Ϫ y i ).
(8-8)
Then we take U i to be the gravitational potential energy of the system when it is
in a reference configuration in which the particle is at a reference point y i .
Usually we take U i ϭ 0 and y i ϭ 0. Doing this changes Eq. 8-8 to
U( y) ϭ mgy (gravitational potential energy).
(8-9)
This equation tells us:
⌬U ϭ Ϫ͵
y f
y i
(Ϫmg) dy ϭ mg ͵
y f
y i
dy ϭ mg ΄ y ΅
y f
y i
,
F
:
g
F
:
g
⌬U ϭ Ϫ͵
x f
x i
F(x) dx.
182
CHAPTE R 8 POTE NTIAL E N E RGY AN D CONSE RVATION OF ENERGY
Elastic Potential Energy
We next consider the block – spring system shown in Fig. 8-3, with the block
moving on the end of a spring of spring constant k. As the block moves from
point x i to point x f , the spring force F x ϭ Ϫkx does work on the block. To find the
corresponding change in the elastic potential energy of the block – spring system,
we substitute Ϫkx for F(x) in Eq. 8-6. We then have
or
(8-10)
To associate a potential energy value U with the block at position x, we
choose the reference configuration to be when the spring is at its relaxed length
and the block is at x i ϭ 0. Then the elastic potential energy U i is 0, and Eq. 8-10
⌬U ϭ
1
2 kx f
2 Ϫ
1
2 kx i
2
.
⌬U ϭ Ϫ͵
x f
x i
(Ϫkx) dx ϭ k ͵
x f
x i
x dx ϭ
1
2 k ΄ x
2
΅
x f
x i
,
The gravitational potential energy associated with a particle – Earth system
depends only on the vertical position y (or height) of the particle relative to the
reference position y ϭ 0, not on the horizontal position.