374
6.4. PHYSICS APPLICATIONS: WORK, FORCE, AND PRESSURE
x − 5
△x
x
(25, 30)
45
y = f (x)
x +
y +
Figure 6.18: A trapezoidal dam that is 25 feet tall, 60 feet wide at its base, 90 feet wide at
its top, with the water line 5 feet down from the top of its face.
is given by P = 62.4d, we have that at any point on the representative slice
P slice = 62.4(x − 5).
Knowing both the pressure and area, we can find the force the water exerts on the slice.
Using F = P A, it follows that
F slice = P slice · A slice = 62.4(x − 5) · 2(45 −
3
5
x)△x.
Finally, we use a definite integral to sum the forces over the appropriate range of x-values.
Since the water rises to within 5 feet of the top of the dam, we start at x = 5 and slice all
the way to the bottom of the dam, where x = 30. Hence,
F =
x=30
x=5
62.4(x − 5) · 2(45 −
3
5
x) dx.
Using technology to evaluate the integral, we find F ≈ 1.248 × 10 6 pounds.
Activity 6.12.
In each of the following problems, determine the total force exerted by water against
the surface that is described.
(a) Consider a rectangular dam that is 100 feet wide and 50 feet tall, and suppose
that water presses against the dam all the way to the top.
(b) Consider a semicircular dam with a radius of 30 feet. Suppose that the water
6.4. PHYSICS APPLICATIONS: WORK, FORCE, AND PRESSURE
x − 5
△x
x
(25, 30)
45
y = f (x)
x +
y +
Figure 6.18: A trapezoidal dam that is 25 feet tall, 60 feet wide at its base, 90 feet wide at
its top, with the water line 5 feet down from the top of its face.
is given by P = 62.4d, we have that at any point on the representative slice
P slice = 62.4(x − 5).
Knowing both the pressure and area, we can find the force the water exerts on the slice.
Using F = P A, it follows that
F slice = P slice · A slice = 62.4(x − 5) · 2(45 −
3
5
x)△x.
Finally, we use a definite integral to sum the forces over the appropriate range of x-values.
Since the water rises to within 5 feet of the top of the dam, we start at x = 5 and slice all
the way to the bottom of the dam, where x = 30. Hence,
F =
x=30
x=5
62.4(x − 5) · 2(45 −
3
5
x) dx.
Using technology to evaluate the integral, we find F ≈ 1.248 × 10 6 pounds.
Activity 6.12.
In each of the following problems, determine the total force exerted by water against
the surface that is described.
(a) Consider a rectangular dam that is 100 feet wide and 50 feet tall, and suppose
that water presses against the dam all the way to the top.
(b) Consider a semicircular dam with a radius of 30 feet. Suppose that the water