Section 6.4
Work
Work
In physics the word “work” is used to describe the work a
force has done on an object to move it some distance.
Work done = Force · Distance or W = F · D
Units
International
British
Force
Newton (N)
Pound (lb)
 Lifting object vertically:
Distance
Meter (m)
Foot (ft)
Work
Joule (j)
Foot-pound (ft-lb)
F  Weight  V  (density )
 Example:
What is the work done in lifting 120 pound object 4 feet off the
ground?
Variable Force
 If a force F(x) varies along a to b, then the work done by
the force as the object is moved from a to b is:
W   F  x  dx
b
a
 Hooke’s Law: The force required to maintain a spring
stretched x units beyond its natural length is proportional
to x:
F=kx
where k is called the spring constant
http://www.intmath.com/applications-integration/7work-variable-force.php
Example
A spring has a natural length of 1 m.
A force of 24 N stretches the spring to 1.8 m.
F  kx
24  k .8
a Find k:
30  k
F  30x
b How much work would be needed to stretch the spring
3m beyond its natural length?
W   F  x  dx
b
a
3
W   30 x dx
0
W  15x
2 3
0
W  135 newton-meters
Example
20
A 5 lb bucket is raised 20 feet. How much
work is done if the rope weighs 0.08 lb/ft?
Work:
Bucket: 5 lb  20 ft  100 ft-lb
Rope: F  x    20  x  0.08
0
W 
20
0
1.6  .08x  dx
20 F  1.6 lb
Check: At x 2 0,
W  1.6 x  .04 x  16 ft-lb
0
At x  20,
F 0
Total:
100  16  116 ft-lb
5 ft
Example
4 ft
I want to pump the water out of this
tank. How much work is done?
(Water weighs 62.5 lb per cubic foot)
10 ft
Consider the work to move one “slab”
of water:
4 ft
0
weight of slab  density  volume
 62.5    52 dy
 1562.5 dy
10 ft
distance  y  4
dy
10
5 ft
5 ft
4 ft
work   y  4 1562.5 dy
10 ft
distance
W 
10
0
4 ft
0
A 1 horsepower pump,
rated at 550 ft-lb/sec,
could empty the tank
10 ft
in just under 14 dx
minutes!
10
5 ft
force
 y  41562.5
dy
10
1 2

W  1562.5  y  4 y 
2
0
W  1562.5 50  40
W  441, 786 ft-lb
Example
10 ft
2 ft
10 ft
A conical tank is filled to within 2 ft
of the top with salad oil weighing 57
lb/ft3. How much work is required to
pump the oil to the rim?
Consider one slice (slab) first:
W y   F  d
10  y
5,10 W  density  volume  distance
 y
x
y  2x
y
x
1
y
2
W y 
  1 2 
 57   y   dy10  y 
  2  
10 ft
1 2
W   10  y  57  y dy
0
4
57 8
2
3
W
10
y

y
dy

4 0
8
2 ft
10 ft
57
W
4
5,10
10  y
x
y  2x
y
x
1
y
2
8
10 3 y 
 y  
4 0
3
W  30,561 ft-lb
4
Examples
1) A spring has a natural length of 20 cm. If a 25 N
force is required to keep it stretched to 30 cm, how
much work is required to stretch it from 20 cm to
25 cm?
2) A tank full of water has a triangular cross section
that is 5m high, 3m wide at the top and 8m long.
Given that the density of water is 9800 N/m3, how
much work is required in order to empty the tank?