-
8/17/2019 5.4 - Work
1/13
Work
-
8/17/2019 5.4 - Work
2/13
Known FactsForce = (mass)(acceleration) = ma
Work = (force)(distance) = Fd
Units: F in Newtons, d in meters, W Joules
OR: F in pounds, d in feet, W in foot pound
!ssumptions: mo"ement is alon# a strai#$t line,acceleration is
constant, force is constant%
-
8/17/2019 5.4 - Work
3/13
&aria'le Force
Want: to find t$e work done w$en t$eforce is "aria'le, 'ut t$e
mo"ement is stillon a strai#$t line%
ro'lem to consider: !ssume a force f(x) mo"es an o' ect alon#
t$e x-axis from x= ato x = b, w$ere f is continuous on * a, b +
-
8/17/2019 5.4 - Work
4/13
olution
artition * a, b + into n su'inter"als of e-uallen#t$ and select
a point in eac$ su'inter"al%
.f n is lar#e, t$en t$e len#t$ of eac$ su'inter"alis "er/ small,
so f is almost constant in eac$su'inter"al, so
],[,,... 1*
110 iiiiin x x x x x xb x x xa −− ∈−=∆=
-
8/17/2019 5.4 - Work
5/13
0efinition of Work
1ence, '/ takin# t$e limit we #et
∫ ∑ =∆==∞→
b
a
n
iin
dx x f x x f W )()(lim1
*
f continuous on * a, b +%
-
8/17/2019 5.4 - Work
6/13
23ample ! leak/ 4 l' 'ucket is lifted from t$e #round
into t$e air '/ pullin# in 56 ft of rope at aconstant speed% 7$e
rope wei#$s 6%68 l'9ft%7$e 'ucket starts wit$ #allons of water
(;<l') and leaks at a constant rate% .t finis$esdrainin# ust as
it reac$es t$e top% 1ow muc$work was spent liftin# t$e water,
'ucket, and
rope
-
8/17/2019 5.4 - Work
7/13
Water !lone7$e force re-uired in t$is case is t$e
water>swei#$t w$ic$ "aries steadil/ from ;< to 6 l'o"er t$e
56 ft lift%
.f t$e 'ucket is x units from t$e #round, t$en
−
= 2020
16)( x
x f
Initial weight Proportion left at elevation x
5
416
x
−=
-
8/17/2019 5.4 - Work
8/13
?ucket !lone
7$e force needed to lift a 4 l' wei#$t an/distance x is f(3) = 4
l'%
Rope !lone7$e force needed to lift t$e rope x feet fromt$e
#round is f(x) = (6%68)(56 @ x )
-
8/17/2019 5.4 - Work
9/13
Water, ?ucket, and Rope
7$e total wei#$t at x units from t$e #round is
)20)(08.0(520
2016)( x x x f −++ −=
water bucket rope
.nte#ratin# t$is force will #i"e us t$e work%
-
8/17/2019 5.4 - Work
10/13
Work 0one
1ence, t$e work needed to lift all is 5A< ft l'
276)20)(08.0(5
20
2016
20
0
=−++
−= ∫ dx x xW
-
8/17/2019 5.4 - Work
11/13
1ooke>s Baw
7$e force re-uired to maintain a sprin#
stretc$ed x units 'e/ond its naturallen#t$ is proportional to x
: f(x) = kx k is t$e sprin# constant and it ispositi"e%
-
8/17/2019 5.4 - Work
12/13
23ample! sprin# $as natural len#t$ ; m% ! force of 5C Nstretc$es
t$e sprin# to a len#t$ of ;%8 m% 1owmuc$ work will it take to
stretc$ t$e sprin# 5 m'e/ond its natural len#t$
First, we need to find t$e force constant%
N/m308.0
24)8.0(24 ==⇒= k k
1ence, f(x) = 30x
-
8/17/2019 5.4 - Work
13/13
20,30)( ≤≤= x x x f
60302
0 ==∫ dx xW
7$e work done to stretc$ t$e sprin# 5 m'e/ond its natural len#t$
is
