6.5 Work
Constant Force
Algebraic
Application!
.
Work = Force Distance
.
W = F D
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.
F.Y.I.
In the U.S. measurement system, work is typically expressed in foot­pounds, inch­
pounds, 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.
1
Work is measured in:
foot­pounds
newton­meters (joules) force­distance unit
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!
2
3
Variable Force
ie.
takes Calculus to measure the work done by a variable force!
stretching or compressing a spring!
natural state of spring
x
stretching:
x
a = 0
b
x
compressing:
x
b
a = 0
take,
F(x) = function representing a continually varying force,
and
x = distance of any partition in [a, b]
W = F(c ) . x W = F(c ) .
W i
i
i
n
n
i
i = 1
i = 1
i
x i
W = lim
|| || 0
n
i = 1
F(c i ) . x i
b
W = F(x) dx a
Def.
Work done by a variable force.
b
W = F(x) dx a
Geometrically
Work = Area under Force function!
4
Hooke's Law for Springs
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.
F(x) = k. x
,
x = spring displacement from
natural length
and k = spring constant ("restoring force")
b
b
W = F(x) dx = k x dx .
a
a
5
Examples:
1.)
Spring Problem
A 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!
6
Pumping Liquids from Containers Problem
2.)
A 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
{
10 ft.
x
15 ft.
y
(water weighs 3
~62.5 lbs/ft )
"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!"
7
Assignment
p456 #1-4 (constant force),
#5-8 (concepts),
#9, 12-15 (springs),
#21-27 (pumping liquids)
8
Moving a Space Module into Orbit Problem
3.)
p453
A 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
}
}
.
800
miles
4000
miles
Not drawn to scale!
The weight of a body varies inversely as the square of its distance from the center of earth the Force exerted by gravity is:
C is the F(x) = C2 , constant of x
proportionality.
9
Lifting a Chain
4.)
p455
A 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?
||
|
y
y
ground
(x­axis)
10
Work Done by an Expanding Gas
p455
Boyle's Law of
an Expanding Gas
The pressure of gas is inversely proportional to the volume.
Students ‐ on your own!
See p455
Example 6
p = k
V
11
Assignment
p456 #17-20 (propulsion),
#31-38 (lifting a chain),
#39,40 (expanding gas)
12