Section 6.5: Work, Hydrostatic Force,
Moments and Center of Mass
Work = Force x 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
2
Suppose the force on a particle is F(x) = x N
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(x*k ) 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x)x)
10

2
0
r
10-x
x
10
10

x
62.4
 r2
dx
0
Distance
Density
Volume
Force
Hydrostatic Force
Force = Density x Depth x Area
A gate on a dam is a 4 x 8 ft rectangle.
Find the force on the gate if the top is 2 ft under the water.
0
2
Let x = depth
x
6
6

62.4
x
(8) dx
2
Density
Depth
Area
A gate on a dam is a 4 x 8 ft rectangle.
Find the force on the gate if the top is 2 ft under the water.
-2
0
Let x = distance to top of gate
x
4
4

62.4 x  2 (8) dx
0
Density
Depth
Area
A gate on a dam is a 4 x 8 ft rectangle.
Find the force on the gate if the top is 2 ft under the water.
6
4
Let x = distance to bottom of gate
x
0
4

62.4 6  x (8) dx
0
Density
Depth
Area
A dam is a triangle 6 ft wide, 5 ft high.
Find the force on the gate if the top is 2 ft under the water.
-2
0
w
x
5
Let x = distance to top of dam
5

62.4 x  2
w
dx
0
Density
Depth
Area
A dam is a triangle 6 ft wide, 5 ft high.
3
-2
0
½w
w
5
x
5-x
5
1
w
2 3
5 x 5
6
w  (5
(5xx) )
55
5

62.4 x  2
w
dx
0
Density
Depth
Area
A dam is a triangle 6 ft wide, 5 ft high.
Find the force on the gate if the top is 2 ft under the water.
0
2
w
x
7
Let x = depth
7

62.4
x
w
dx
2
Density
Depth
Area
A dam is a triangle 6 ft wide, 5 ft high.
3
0
2
½w
w
5
x
7-x
7
1
w
2 3
7x 5
6
w  (7
(7xx) )
55
7

62.4
x
w
dx
2
Density
Depth
Area
A semicircle with radius 5 ft is submerged level with top of water.
0
x 2  y 2  52 is a circle
with radius 5 centered at origin
½w
w
y
5
x  25  y 2
1
w  x  25  y 2
2
2
w  22 25
25 yy 2
5

62.4
y
w
0
Density
Depth
Area
dy
Moments and Center of Mass
Moments measure the tendency of a body to rotate about an axis.
M y  Moment about y-axis
M x  Moment about x-axis
Moment of a point about an axis = Distance to axis * Mass
_
_
Center of mass = ( x , y )
_
_
My
Mx
x
y
M
M
We will assume our bodies have no thickness
and density  1 so mass = area.
Region bounded by y = 2x, y = x
4
2
Moment of a point about an axis
= Distance to axis * mass
x
My 
2

x
(2 x  x 2 ) dx
Mx 
0
Distance to
y-axis
2

0
Mass
2 x  x2
2
(2 x  x ) dx
2
Distance to
x-axis
2
M   (2 x  x 2 ) dx
0
2
Mass
Region bounded by y = x+6, y = x
2
Moment of a point about an axis
= Distance to axis * mass
3
3

x (( x  6)  x ) dx
2
2
Distance to
y-axis
x
3
Mx 
My 

-2
2
Mass
( x  6)  x (( x  6)  x 2 ) dx
2
2
Distance to
x-axis
2
M   ( x  6)  x 2 dx
0
Mass