9781118230725.pdf

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 mmoving vertically along a yaxis (the

positive direction is upward). As the particle moves from point yito point yf,

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 yaxis instead of the xaxis, because

the gravitational force acts vertically. (2) We substitute mgfor the force symbol F,

because has the magnitude mgand is directed down the yaxis. We then have

which yields

Umg(yfyi)mgy. (8-7)

OnlychangesUin 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

valueUis 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

UUimg(yyi). (8-8)

Then we take Uito be the gravitational potential energy of the system when it is

in a reference configurationin which the particle is at a reference pointyi.

Usually we take Ui0 and yi0. Doing this changes Eq. 8-8 to

U(y)mgy (gravitational potential energy). (8-9)

This equation tells us:

U

yf

yi

(mg)dymg

yf

yi

dymg y

yf

yi

,

F

:

g

F

:

g

U

xf

xi

F(x)dx.

182 CHAPTER 8 POTENTIAL ENERGY AND CONSERVATION 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

pointxito point xf, the spring force Fxkxdoes work on the block. To find the

corresponding change in the elastic potential energy of the block – spring system,

we substitute kxforF(x) in Eq. 8-6. We then have

or (8-10)

To associate a potential energy value Uwith 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 xi0. Then the elastic potential energy Uiis 0, and Eq. 8-10

U^12 kxf^2 ^12 kxi^2.

U

xf

xi

(kx)dxk

xf

xi

xdx^12 k x^2

xf

xi

,

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 y0, not on the horizontal position.