Section 6.5: Work, Hydrostatic Force,
Moments and Center of Mass
Work = Force * Distance (if force is constant)
Example: 2 lb book is moved up 5 ft.
10 ft-lb of work
English Units: ft-lb
Metric Units: Newton-meter, or Joule
dyne-cm, or erg
Suppose the force on a particle is F ( x )  x N
2
How much work is done to move it from x  0 to x  5?
Δx
Split [0,5] into small intervals.
Pretend F ( x ) is constant on each interval.

F ( xk* ) x
5
125
0 x dx  3
2
Hooke's Law:
The force on a spring is proportional to the displacement.
F(x) = k x,
where x = displacement
and k depends on the particular spring
1
A spring is normally 5ft. long. A 4 lb Force stretches it ft.
2
Find k.
Force = k * Displacement
1
4=k  
2
k 8
How much work is done to stretch it from 6ft. to 8ft.?
3
1 8 x dx
4x
2 3
1
 32 ft-lb
An aquarium measures 1ft. x 2ft. x 3ft.
Find the work done in emptying the tank.
1
Given: Water weighs 62.4 lb/ft 3 .
3
0
Let x = distance to top
= distance we have to move the water
x
3
3

x
62.4 (1) (2) dx
0
Distance
Density
2
Volume
 561.6 ft-lb
Force
Warning: It’s possible to choose your coordinates
many different ways. The resulting integrals may look
different, but all give the same answer.
0
x
3
Let x = distance to bottom of tank
3
 3 xx
62.4 (1) (2) dx
0
Distance
Density
Volume
 561.6 ft-lb
Force
Cylindrical tank 10 ft high with radius 6 ft.
Full of water. Find to work required to empty it.
0
x
10
Let x = distance to top
= distance we have to move the water
10

x
62.4
 6 dx
2
0
Distance
Density
Volume
Force
Cylindrical tank 10 ft high with radius 6 ft.
Full of water. Find to work required to empty it.
Let x = distance to bottom of the tank
10
x
0
10

10  x 62.4
 6 dx
2
0
Distance
Density
Volume
Force
Cylindrical tank 10 ft high with radius 6 ft.
What if the tank is half full and there is a 7 ft pipe out the top?
-7
Let x = distance to top of the tank
0
5
x
10
10

x  7 62.4
 6 dx
2
5
Distance
Density
Volume
Force
Cylindrical tank 10 ft high with radius 6 ft.
What if the tank is half full and there is a 7 ft pipe out the top?
0
7
12
x
17
Let x = distance to top of the pipe
17

x
62.4
 6 dx
2
12
Distance
Density
Volume
Force
Cylindrical tank 10 ft high with radius 6 ft.
What if the tank is half full and there is a 7 ft pipe out the top?
17
Let x = distance to bottom of the tank
10
5
x
0
5

17  x 62.4
 6 dx
2
0
Distance
Density
Volume
Force
Cylindrical tank 10 ft high with radius 6 ft.
What if the tank is half full and there is a 7 ft pipe out the top?
-12
Let x = distance to the top of the water
-5
0
x
5
5

x  12 62.4
 6 dx
2
0
Distance
Density
Volume
Force
Conical tank 10 ft high with diameter 6 ft.
Full of water. Find to work required to empty it.
0
Let x = distance to the top of the cone
x
10
10

x
62.4
 r 2 dx
0
Distance
Density
Volume
Force
3
Similar Triangles
3
r

tan  
10 10  x
r
10
3
 3
(10
 10
(10xx) )
10

2
0
r
10-x
x
10
10

x
62.4
 r2
dx
0
Distance
Density
Volume
Force