6.4. PHYSICS APPLICATIONS: WORK, FORCE, AND PRESSURE
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the total work.
Consider a representative cylindrical slice that sits on the surface of the water at a
depth of x feet below the top of the crock. It follows that the approximate volume of that
slice is given by
V slice = π f (x)
2 △x = π(1.5 − 0.1875x)
2 △x.
Since water weighs 62.4 lb/ft 3 , it follows that the approximate weight of a representative
slice, which is also the approximate force the pump must exert to move the slice, is
F slice = 62.4 · V slice = 62.4π(1.5 − 0.1875x)
2 △x.
Because the slice is located at a depth of x feet below the top of the crock, the slice being
moved by the pump must move x feet to get to the level of the basement floor, and then,
as stated in the problem description, be moved another 9 feet to reach the drain at ground
level outside a basement window. Hence, the total distance a representative slice travels is
d slice = x + 9.
Finally, we note that the work to move a representative slice is given by
W slice = F slice · d slice = 62.4π(1.5 − 0.1875x)
2 △x · (x + 9),
since the force to move a particular slice is constant.
We sum the work required to move slices throughout the tank (from x = 0 to x = 4),
let △x → 0, and hence
W =
4
0
62.4π(1.5 − 0.1875x)
2 (x + 9) dx,
which, when evaluated using appropriate technology, shows that the total work is W =
10970.5π foot-pounds.
The preceding example demonstrates the standard approach to finding the work
required to empty a tank filled with liquid. The main task in each such problem is to
determine the volume of a representative slice, followed by the force exerted on the slice,
as well as the distance such a slice moves. In the case where the units are metric, there
is one key difference: in the metric setting, rather than weight, we normally first find the
mass of a slice. For instance, if distance is measured in meters, the mass density of water
is 1000 kg/m 3 . In that setting, we can find the mass of a typical slice (in kg). To determine
the force required to move it, we use F = ma, where m is the object’s mass and a is the
gravitational constant 9.81 N/kg 3 . That is, in metric units, the weight density of water is
9810 N/m 3 .
Activity 6.11.
In each of the following problems, determine the total work required to accomplish the
371
the total work.
Consider a representative cylindrical slice that sits on the surface of the water at a
depth of x feet below the top of the crock. It follows that the approximate volume of that
slice is given by
V slice = π f (x)
2 △x = π(1.5 − 0.1875x)
2 △x.
Since water weighs 62.4 lb/ft 3 , it follows that the approximate weight of a representative
slice, which is also the approximate force the pump must exert to move the slice, is
F slice = 62.4 · V slice = 62.4π(1.5 − 0.1875x)
2 △x.
Because the slice is located at a depth of x feet below the top of the crock, the slice being
moved by the pump must move x feet to get to the level of the basement floor, and then,
as stated in the problem description, be moved another 9 feet to reach the drain at ground
level outside a basement window. Hence, the total distance a representative slice travels is
d slice = x + 9.
Finally, we note that the work to move a representative slice is given by
W slice = F slice · d slice = 62.4π(1.5 − 0.1875x)
2 △x · (x + 9),
since the force to move a particular slice is constant.
We sum the work required to move slices throughout the tank (from x = 0 to x = 4),
let △x → 0, and hence
W =
4
0
62.4π(1.5 − 0.1875x)
2 (x + 9) dx,
which, when evaluated using appropriate technology, shows that the total work is W =
10970.5π foot-pounds.
The preceding example demonstrates the standard approach to finding the work
required to empty a tank filled with liquid. The main task in each such problem is to
determine the volume of a representative slice, followed by the force exerted on the slice,
as well as the distance such a slice moves. In the case where the units are metric, there
is one key difference: in the metric setting, rather than weight, we normally first find the
mass of a slice. For instance, if distance is measured in meters, the mass density of water
is 1000 kg/m 3 . In that setting, we can find the mass of a typical slice (in kg). To determine
the force required to move it, we use F = ma, where m is the object’s mass and a is the
gravitational constant 9.81 N/kg 3 . That is, in metric units, the weight density of water is
9810 N/m 3 .
Activity 6.11.
In each of the following problems, determine the total work required to accomplish the