Math 152 – Spring 2016
Section 7.4
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Section 7.4– Work
F orce = mass · acceleration
W ork = F orce · distance
Units
• mass: kilograms
• displacement: meters
• Force: Newtons (N = kg ∗ m/s2 )
• Work: Joule, sometimes called newton-meter (J = N ∗ M) or foot-pound (ft-lb)
Example 1. How much work is done lifting a 75 kg weight 2 m off the ground?
Example 2. How much work is done lifting a 150 lb weight 3 ft off the ground?
Question. These questions are simple because the force (weight) remains constant.
What if the force changes?
Suppose an object is moved along the x-axis in the positive direction from x = a to
x = b and a force f (x) acts on the object where f is continuous. Find a formula for
the work done.
W = lim
||P ||→0
n
X
i=1
f (x∗i )4xi =
Z
b
f (x) dx
a
Math 152 – Spring 2016
Section 7.4
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Example 3. When a particle is at a distance x meters from the origin, a force of
cos(πx/3) acts on it. How much work is done in moving the particle from x = 12 to
x = 1?
Springs
Theorem (Hooke’s Law). The force f (x) required to hold a spring stretched x units
beyond its natural length is proportional to x, i.e.,
f (x) = kx
where k is a positive constant (the spring constant).
Remark. Notice that x is the distance beyond the springs natural length, NOT the
length of the spring.
Theorem. The work done stretching a spring with spring constant k from a distance
a to b past its natural length (with b ≥ a ≥ 0) is equal to
Z
W =
b
kx dx
a
Example 4. Suppose a spring has a natural length of 5 cm, and a force of 15 N is
required to hold the spring stretched to 8 cm. How much work is done stretching the
spring from 11 cm to 13 cm.
Example 5. Suppose the work required to stretch a spring from its natural length to
2 ft beyond its natural length is 12 ft-lb. How much work is required to stretch it 18
in. beyond its natural length?
Math 152 – Spring 2016
Section 7.4
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Ropes
Example 6. Suppose a rope that weighs 10 lb and is 50 ft long is hanging off a roof.
(a) How much work is required to pull the rope to the top?
(b) How much work is required to pull 10 ft of the rope to the top?
Example 7. A bucket that weighs 5 lb and carries 25 lbs of water is used to draw
water from a well 100 ft deep. If the rope weighs .25 lb per ft, find the amount of work
required to lift one bucket of water to the surface.
Math 152 – Spring 2016
Section 7.4
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Tanks of Water
Example 8. A tank is full of water. Find the work required to pump the water out of
the outlet. Water weighs 62 ft/lb2 or 1000 kg/m2 .
(a) (Sideways cylindrical tank with radius r, length a, spout b, all in meters)
Solution Method
• Find the volume for a typical cross-section.
• Multiply by 1000 kg/m3 ∗9.8 m/s2 or 62.5 lb/f t3 to get force for cross-section.
• Work for cross-section equals force times the distance the cross-section needs
to travel.
• Convert to an integral and solve.
(b) (Upside down pyramid with base 5 ft by 5 ft square, and height 10 ft.)