Math 152 Class Notes
September 17, 2015
7.4 Work
In physics the term work has a technical meaning that depends on the idea of a force.
Intuitively, we can think of a force as describing a push or pull on an object. If an
object is moved by a constant force F , then the work done is dened to be the product
of the force F and the distance that the object moves:
W = Fd
Work = Force · Distance
There are two unit systems. In the SI metric system, the force is measured in newtons
(N) and distance in meters (m). By Newton's Second Law of Motion, the force (N)
is the product of the mass (kg) and the acceleration (m/s2 ). The unit of work is a
newton-meter (N-m), which is called a Joule (J).
In the US Customary system, the force is measured in pound (lb) and distance in feet
(ft), then the unit for work is a foot-pound (ft-lb), which is about 1.36 J.
Example 1. How much work is done by lifting a 1.2 kg book o the oor to put it on a
desk that is 0.7 m high? Use the fact that the acceleration due to gravity is 9.8 m/s2 .
Example 2. How much work is done in lifting a 20 lb weight 6 ft o the ground?
When the force is not constant, we will use integration to nd the work done. Suppose
that the object moves along the x-axis in the positive direction, from x = a to x = b,
and at each point between and a force f (x) acts on the object. Then the work done
by moving the object from a to b is
ˆ
W =
b
f (x)dx
a
Example 3. When a particle is located a distance x feet from the origin, a force of
x2 + x pounds acts on it. How much work is done in moving it from x = 1 to x = 3?
: The force required to maintain a spring stretched x units beyond its
natural length is proportional to x:
Hooke's Law
f (x) = kx
where k is a positive constant called the spring
constant
.
Example 4. A force of 40 N is required to hold a spring that has been stretched from
its natural length of 10 cm to a length of 15 cm. How much work is done in stretching
the spring from 15 cm to 18 cm?
Example 5. Suppose that 2 J of work are needed to stretch a spring from its natural
length of 30 cm to a length of 42 cm. How much work is needed to stretch it from 35
cm to 40 cm?
Example 6. A rectangular tank 10 m long, 3 m wide and 2 m deep is lled with water.
Find the work required to pump all the water to the top of the tank. (The density of
water is 1000 kg/m3 .)
Example 7. A spherical tank of radius 4 ft is half full of water. Find the work done in
pumping the water to the top of the tank. (The weight of water is 62.5 lb/ft3 )
Example 8. A tank has the shape of an inverted circular cone with height 10 m and
base radius 4 m. It is lled with a liquid to a height of 8 m, which has density of 900
kg per cubic meter. Find the work required to empty the tank by pumping all of the
water to the top of the tank.
Example 9. A 200 lb cable is 100 ft long and hangs vertically from the top of a tall
building. How much work is required to lift the cable to the top of the building?
Example 10. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft
500 ft deep. Find the work done.