Solução_Calculo_Stewart_6e

F.
TX.10 SECTION 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING ¤ 363
8.3 Applications to Physics and Engineering
1. The weight density of water is δ =62.5lb/ft 3 .
(a) P = δd ≈ (62.5lb/ft 3 )(3 ft) = 187.5lb/ft 2
(b) F = PA ≈ (187.5lb/ft 2 )(5 ft)(2 ft) = 1875 lb.
(A is the area of the bottom of the tank.)
(c) As in Example 1, the area of the ith strip is 2(∆x) and the pressure is δd = δx i.Thus,
F = 3
0 δx · 2 dx ≈ (62.5)(2) 3
0 xdx=125 1
2 x2 3
0 = 125 9
2
=562.5lb.
In Exercises 3–9, n is the number of subintervals of length ∆x and x ∗ i is a sample point in the ith subinterval [x i−1 ,x i ].
3. Set up a vertical x-axisasshown,withx =0at the water’s surface and x increasing in the
downward direction. Then the area of the ith rectangular strip is 6 ∆x and the pressure on
the strip is δx ∗ i (where δ ≈ 62.5lb/ft 3 ). Thus, the hydrostatic force on the strip is
δx ∗
i · 6 ∆x and the total hydrostatic force ≈ n δx ∗ i · 6 ∆x. The total force
F = lim
n
n→∞ i=1
i=1
δx ∗ i · 6 ∆x = 6
δx · 6 dx =6δ 6
xdx=6δ 1
x2 6
=6δ(18 − 2) = 96δ ≈ 6000 lb
2 2 2 2
5. Set up a vertical x-axis as shown. The base of the triangle shown in the figure
has length 3 2 − (x ∗ i )2 ,sow i =2 9 − (x ∗ i )2 ,andtheareaoftheith
rectangular strip is 2 9 − (x ∗ i )2 ∆x. Theith rectangular strip is (x ∗ i − 1) m
below the surface level of the water, so the pressure on the strip is ρg(x ∗ i − 1).
The hydrostatic force on the strip is ρg(x ∗ i − 1) · 2 9 − (x ∗ i )2 ∆x and the total
force on the plate ≈ n ρg(x ∗ i − 1) · 2 9 − (x ∗ i )2 ∆x. The total force
F =lim n
i=1
i=1
ρg(x ∗ i − 1) · 2 9 − (x ∗ i )2 ∆x =2ρg 3
1 (x − 1) √ 9 − x 2 dx
=2ρg 3
x √ 9 − x
1 2 dx − 2ρg 3
√
1 9 − x2 dx 30 3 x
=2ρg − 1 (9 − √ x
3
3 x2 ) 3/2 − 2ρg 9 − x2 + 9
1 2
2 sin−1 3 1
=2ρg √
0+ 1 3 8 8 − 2ρg 0+
9 · π
2 2 − 1
√
2 8+
9
sin−1
1
2 3
√
= 32
3 2 ρg −
9π
ρg +2√ 2
2 ρg +9 sin −1
1
3 ρg = 38
3
≈ 6.835 · 1000 · 9.8 ≈ 6.7 × 10 4 N
√
2 −
9π
2 +9sin−1 1
3
ρg
Note: If you set up a typical coordinate system with the water level at y = −1,thenF = −1
−3 ρg(−1 − y)2 9 − y 2 dy.