Transcript Limits and Derivatives
Chapter 6 – Applications of Integration
6.4 Work 1 6.4 Work Erickson
Work
In everyday language, work means the total amount of effort required to perform a task.
In physics, work has a technical meaning that depends on the idea of force.
If an object moves along a straight line with position function
s
(
t
), then the force,
F
, on the object is given by
F
m
2
d s dt
2 2 6.4 Work Erickson
Work
Units Mass is measured in kilograms (kg) Displacement is measured in meters (m) Time is measured in seconds (s) Force is measured in Newtons (𝑁 = 𝑘𝑔∙𝑚 𝑠 2 ) 3 6.4 Work Erickson
Work – Equation 2
Work measured in Newton-meters called Joules (J) if F is measured in meters.
Measured in foot-pounds (ft-lb) if F is measured in pounds
W
Fd
NOTE: In this case, F is a constant.
4 6.4 Work Erickson
Example – pg. 449
1.
A 300-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in
(a)
10 seconds
(b)
5 seconds 5 6.4 Work Erickson
Work
What happens if force is a variable?
We define the work done in moving the object from
a
to
b
as
W
n
lim
i n
1
b a
x
6 6.4 Work Erickson
Example – pg. 449
3.
A variable force of 5
x
-2 pounds moves an object along a straight line when it is
x
feet from the origin. Calculate the work done in moving the object from
x
= 1 to
x
= 10 ft.
7 6.4 Work Erickson
Hooke’s Law
Hooke’s Law states that the force required to maintain a spring stretched
x
units beyond its natural length is proportional to
x
:
f
(
x
) =
kx
where
k
is a positive constant called the spring constant. Hooke’s law holds provided that
x
is not too large.
8 6.4 Work Erickson
Examples – pg. 450
7.
A force of 10 lb is required to hold a string stretched 4 in beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?
9.
Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to 42 cm.
(a)
How much work is needed to stretch the spring from 35 cm to 40 cm?
(b)
How far beyond its natural length will a force of 30 N keep the spring stretched?
9 6.4 Work Erickson
Example – pg. 450
22.
A tank is full of water. Find the work required to pump the water out of the spout. Note that you might need the fact that water weighs 62.5 lb/ft 3 .
10 6.4 Work Erickson