L39– Work
Let f (x) be the force acting on an object at each
point x in [a, b]. Then the work, w, needed to move
the object from x = a to x = b is:
w = lim
n
X
n→∞
w=
i=1
Rb
a
f (xi)∆x
f (x)dx
To calculate work of a non-constant force,
you integrate over the force function.
Unit: f (x) in newtons, d in meters,→ W in joule(J)).
f (x) in pounds, d in feet, → W in foot-pound.
We will calculate the work required · · ·
1. in Spring Problems(Hook’s Law)
2. in cable/rope problem
3. required to Drain a Tank
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Calculating the Work required in Spring
Problems in conjunction with Hooke’s Law
Hooke’s Law states that the force required to hold
a spring stretched 0x0 units beyond its natural length
is proportional to the amount/distance x stretched:
f (x) = kx
(k = spring constant)
ex. A force of 10 lb is required to hold a spring
stretched 4in beyond its natural length. How much
work is done in stretching it from its natural length
to 6in beyond its natural length? (Use foot-pounds!)
Want to calculate:
Given:
2
Calculating the Work Required in a
Cable/Rope problem
1. A heavy rope, 50 feet long, weighs 2 lb/ft and
hangs over the edge of a building 120 feet high.
(a) How much work is done in pulling the rope to
the top of the building?
3
(b) How much work is done in pulling half the
rope to the top of the building?
4
2. A paint bucket is being lifted from the ground to
the third floor of an building where it is removed
from the rope and the rest of the rope is hauled
to the top of the building.
The distance from the ground to the third floor is
30 feet and the building is 90 feet tall.
The bucket alone weighs 2 pounds and the 5 gallons of paint in it weights 45 pounds.
The rope weighs approximately 1 pound, for each
3 feet.
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(a) find the work in lifting the bucket from the
ground to the third floor.
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(b) Then find the work in lifting the rope from the
third floor to the top of the building.
7
Calculating the Work Required to Drain a
Tank
R
W = F ·d
1. Find the Volume of a Generic Cross Sectional Slice
2. Force: Multiply Volume by Density:
ft : 62.5lb/ft3,
Meters : 9.8 × 1000,
(assuming filled with water)
3. Generic Displacement
4. Limits of Integration
5. Integrate
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ex. A tank is full of water. Find the work required
to pump the water out of the spout.
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NYTI:
1. Suppose 2J of work is needed to stretch a spring
from its natural length of 30cm to a length of
42cm. How much work is required to stretch it
from 35cm to 40cm?
Calculate:
Given:
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Continue with the rope problem on page 7:
(c) Unfortunately, the bucket had a hole at the
bottom and some of the paint was lost during the
haul up to the third floor.
If there was only 3 gallons of paint left over once
the bucket reached the third floor and assuming
that the paint leaked out at a constant rate, find
the work in lifting the bucket from the ground to
the third floor.
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