Work, Pumping liquid and Fluid Pressure

Lesson 7.7

Work

• Work is the product of force and distance.

2

How much work is done in lifting a mass of 1.2 kg a distance of 12 meters?The weight of the mass is (1.2)(9.8) Newton, so the work is (1.2)(9.8)(12) = 141.12 joules.

3

Work

• DefinitionThe 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

12 ft

4

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❧ ❒W = F(xi) ❒x

a b

❒x

5

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

a

W F x dx= ∫

Find the work required to compress a spring from its natural length of 1 ft to a length of 0.75 ft if the force constant is K=16lb/ft.

8

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?

• What is F(x) the force function?

• Work done?

4.5 0.15

30

( ) 30

F k x

k

k

F x x

= ×= ×

==

0.5

0

30W xdx= ∫

9

11

Winding Cable

• Consider a cable being wound up by a winch❧ Cable is 50 ft long❧ 2 lb/ft❧ How much work to wind in 20 ft?

• Think about winding in ❒y❧ y units from the top → 50 – y ft hanging ❧ dist = ❒y ❧ force required (weight) =2(50 – y)

( )20

0

2 50W y dy= −∫

12

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)

13

• Draw a picture with thecoordinate system

• Determine mass of thinhorizontal slab of liquid

• Find expression for work needed to lift this slab to its destination

• Integrate expression from bottom of liquid to the top

-a

r-y

-b

14

Pumping Liquids – Guidelines

• Draw a picture with thecoordinate system

• Determine mass of thinhorizontal slab of liquid

• Find expression for work needed to lift this slab to its destination

• Integrate expression from bottom of liquid to the top

ab

r

2

0

( )a

W r b y dyρ π= × × −∫

y

15

Pumping Liquids

• Suppose tank has❧ r = 4❧ height = 8❧ filled with petroleum (54.8 lb/ft3)

• What is work done to pump oil over top❧ Disk weight?❧ Distance moved?❧ Integral?

84

54.8 16Weight yπ= × × ×∆(8 – y)

8

0

54.8 16 (8 )Work y yπ= × × × − ∆∫

16

Pumping Liquids

17

Pumping Liquids A hemispherical bowl with radius 10 ft is filled with water to a level of 6 ft. Find the work done to the nearest ft.lb to pump all the water to the top of the bowl.Density of water = 62.4 lb/ft3.

18

Pumping Liquids

19

• A spherical tank of radius 8 feet is half of oil that weight 50 pounds per cubic foot. Find the work required to pump oil out through a hole in the top of the tank.

20

21

Fluid Pressure

• Consider the pressure of fluidagainst 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

c

F h y L y dyρ= ×∫

22

( ) ( )d

c

F h y L y dyρ= ×∫

23

24

25

Fluid Pressure

• The tank has cross sectionof a trapazoid❧ Filled to 2.5 ft with water❧ Water is 62.4 lbs/ft3

• Function of edge• Length of strip• Depth of strip

(-2,0) (2,0)

(-4,2.5) (4,2.5)

2.5 - y

26

Fluid Pressure

• The tank has cross sectionof a trapazoid❧ Filled to 2.5 ft with water❧ Water is 62.4 lbs/ft3

• Function of edge• Length of strip• Depth of strip

• Integral

(-2,0) (2,0)

(-4,2.5) (4,2.5)

2.5 - y

y = 1.25x – 2.5x = 0.8y + 22 (0.8y + 2)

2.5 - y