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6.5 WorkConstant Force

W = F D.Work = Force Distance.

AlgebraicApplication!

.If force (lbs.) and distance (feet),

then work (ft lbs).A force can be thought of as a push or a pull; a force changes the

state of rest or state of motion of a body.

In the U.S. measurement system, work is typically expressed in foot­pounds, inch­pounds, or foot­tons. In the centimeter­gram­second (C­G­S) system, the basic unit of force is the dyne ­ the force required to produce an acceleration of 1 centimeter per second per second on a mass of 1 gram. In this system, work is typically expressed in dyne­centimeters (ergs) or newton­meters (joules), where 1 joule = 107 ergs.

F.Y.I.

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Illustrate:

It takes a force of about 1 newton (1 N) to lift an apple from a table. If you lift it 1 meter, you have done about 1 newton­meter (N m) of work on the apple!

.

Work is measured in:

foot­poundsnewton­meters (joules)

force­distance unit

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Variable Forceie. stretching or compressing a spring!

takes Calculus to measure the work done by a variable force!

natural state of spring

compressing:x

x

a = 0b

x

stretching:

a = 0 bx

take,

F(x) = function representing a continually varying force,and

x = distance of any partition in [a, b]

W = dx F(x)a

bWork done by a variable force.

Def.

W = .i x iF(c )iW = iW

i = 1

n. x iF(c )i

i = 1

n

W = . x iF(c )ii = 1

n

lim|| || 0

W = dx F(x)a

b

Work = Area under Force function!Geometrically

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says that the amount of force, F(x) it takes to stretch or compress a spring x length units from its natural length is proportional to x.

Hooke's Law for Springs

F(x) = k x. x = spring displacement from natural lengthk = spring constant ("restoring force")and

,

W = dx F(x)a

b

= a

b

k x dx .

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Examples:

1.)

Spring ProblemA force of 8 lbs. is required to stretch a spring from its natural length of 10 inches to a length of 15 inches. How much work is done in stretching the spring from a length of 12 inches to a length of 20 inches?

How far beyond its natural length will a 16 lb. force stretch the spring?

Find the spring constant!

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2.)

Pumping Liquids from Containers ProblemA tank in the shape of an inverted cone is full of water. The tank has a diameter of 20 feet at the top and is 15 feet deep. If it is emptied by pumping the water over the rim, how much work is done?

x

y

x

y

10 ft.

15 ft.(water weighs

~62.5 lbs/ft3)

"imagine lifting out one thin horizontal slab of liquid at a time, applying W = F D to that slab, and then summing the work needed to lift all of the slabs!"

.

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Assignmentp456 #1-4 (constant force), #5-8 (concepts), #9, 12-15 (springs), #21-27 (pumping liquids)

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3.)

Moving a Space Module into Orbit ProblemA space module weighs 15 tons on the surface of earth. How much work is done in propelling the module to a height of 800 miles above earth, as shown in Figure 6.50? (Use 4000 miles as the radius of earth. Do not consider the effect of air resistance or the weight of the propellant.)

Newton's Law of Universal Gravitation

The weight of a body varies inversely as the square of its distance from the center of earth the Force exerted by gravity is:

F(x) = Cx2 , C is the constant of proportionality.

Not drawn to scale!

.

4000miles

800miles

p453

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4.)

Lifting a ChainA 20­foot chain weighing 5 pounds per foot is lying coiled on the ground. How much work is required to raise one end of the chain to a height of 20 feet so that it is fully extende, as shown in Figure 6.52?

p455

y

ground(x­axis)

y

| ||

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Students ‐ on your own!See p455 Example 6

Work Done by an Expanding Gas p455

Boyle's Law of an Expanding Gas

The pressure of gas is inversely proportional to the volume.

p = kV

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Assignmentp456 #17-20 (propulsion), #31-38 (lifting a chain), #39,40 (expanding gas)