Work and Fluid Pressure
Lesson 7.7
Work
• Definition
The product of

The force exerted on an object
 The distance the object is moved by the force
• When a force of 50 lbs is exerted to move
an object 12 ft.

600 ft. lbs. of work is done
50
2
12 ft
Hooke's Law
• Consider the work done to
stretch a spring
• Force required is proportional to distance

When k is constant of proportionality
 Force to move dist x = k • x = F(x)
• Force required to move through i th
interval, x
x

W = F(xi) x
a
b
3
Hooke's Law
• We sum those values using the definite
integral
• The work done by a continuous force F(x)

Directed along the x-axis
 From x = a to x = b
b
W   F ( x ) dx
a
4
Hooke's Law
• A spring is stretched 15 cm by a force of
4.5 N

How much work is needed to stretch the
spring 50 cm?
F  kx
• What is F(x) the force function?
0.5
• Work done? W   30 x dx
4.5  k  0.15
30  k
F ( x)  30 x
0
5
Winding Cable
• Consider a cable being wound up by a
winch
20
W   2  50  y  dy
Cable is 50 ft long
0
 2 lb/ft
 How much work to wind in 20 ft?

• Think about winding in y amt
y units from the top  50 – y ft hanging
 dist = y
 force required (weight) =2(50 – y)

6
Pumping Liquids
• Consider the work needed to pump a
liquid into or out of a tank
• Basic concept:
Work = weight x dist moved
• For each V of liquid

Determine weight
 Determine dist moved
 Take summation (integral)
7
Pumping Liquids – Guidelines
• Draw a picture with the
b
coordinate system
a
• Determine mass of thin
horizontal slab of liquid
• Find expression for work needed to lift this
slab to its destination
• Integrate expression from bottom of liquid
a
to the top
2
r
W       r (b  y ) dy
0
8
Pumping Liquids
• Suppose tank has
8
4

r=4
 height = 8
 filled with petroleum (54.8 lb/ft3)
• What is work done to pump oil over top
Weight  54.8   16  y
Disk weight?
 Distance moved? (8 – y)
8
 Integral?
Work  54.8   16 (8  y )y


0
9
Fluid Pressure
• Consider the pressure of fluid
against the side surface of the container
• Pressure at a point

Density x g x depth
• Pressure for a horizontal slice

Density x g x depth x Area
• Total force
d
F    h( y )  L( y ) dy
c
10
Fluid Pressure
• The tank has cross section
of a trapazoid
(-4,2.5)
(4,2.5)
2.5 - y

Filled to 2.5 ft with water
 Water is 62.4 lbs/ft3
• Function of edge
• Length of strip
• Depth of strip
2 (0.8y + 2)
(2,0)
(-2,0)
y = 1.25x – 2.5
x = 0.8y + 2
2.5 - y
2.5
• Integral
F  62.4  (2.5  y )(1.6  4) dy
0
11
Assignment
• Lesson 7.7a
• Page 307
• Exercises 1 – 13 odd, 21
• Lesson 7.7b
• Page 307
• Exercises 23 – 35 odd
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