Hydrostatic pressure and force calculus

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Current Score : – / 57 Due : Sunday, August 5 2018 11:00 PM PDT

1. –/2 pointsSCalcET6 8.3.028.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. (Do this on paper. Your instructor may ask you to turn in this work.) Then find the exact coordinates of the centroid. (Enter the argument of the logarithmic function in parentheses.)

y = 0, x = 3, x = 12 (

,

)

2. –/4 pointsSCalcET6 8.3.035.

Calculate the moments Mx and My and the center of mass of a lamina with density ρ = 7 and the given shape.

Mx =

My =

(x, y) = ( , )

(3, 1)

8.3 (Homework)

Zhikun Deng BYU IS Math 113­2014, Fall 2014 Instructor: Steven McKay

WebAssign

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3. –/4 pointsSCalcET6 8.3.023.MI.

The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the

system.

m1 = 6, m2 = 5, m3 = 9;

P1 = (1, 5), P2 = , P3 =

Mx =

My =

(x, y) = ( , )

4. –/1 pointsSCalcET6 8.3.013.

A trough is filled with a liquid of density 845 kg/m3. The ends of the trough are equilateral triangles with sides 10 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough. (Use 9.8

m/s2 for the acceleration due to gravity.) N

5. –/2 pointsSCalcET6 8.3.031.

Find exact coordinates of the centroid of the region bounded by the given curves.

y = 2sin(3x) y = 2cos(3x) x = 0, x = π/12

(

,

)

(3, −3) (−2, −2)

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6. –/4 pointsSCalcET6 8.3.024.

The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the

system.

m1 = 6, m2 = 5, m3 = 1, m4 = 4;

P1(1, ­2), P2(3, 4), P3(­3, ­7), P4(6, ­1)

Mx =

My =

(x, y) = ( , )

7. –/2 pointsSCalcET6 8.3.032.

Find the centroid of the region bounded by the given curves. Give your answers correct to two decimal places.

y = x3

x + y = 30 y = 0

( , )

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8. –/4 pointsSCalcET6 8.3.017.

A swimming pool is 30 ft wide and 60 ft long and its bottom is an inclined plane, the shallow end having a depth of 4 ft and the deep end, 7 ft. Assume the pool is full of water. (Round your answers to the

nearest whole number. Recall that the weight density of water is 62.5 lb/ft3.)

(a) Estimate the hydrostatic force on the shallow end. lb

(b) Estimate the hydrostatic force on the deep end. lb

(c) Estimate the hydrostatic force on one of the sides. lb

(d) Estimate the hydrostatic force on the bottom of the pool. lb

9. –/1 pointsSCalcET6 8.3.003.

A 3 ft high and 5 ft width rectangular plate is submerged vertically in water so that the top is 1 ft below the surface. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Give your answer correct to the nearest whole number.)

lb

10.–/2 pointsSCalcET6 8.3.033.

Find the exact coordinates of the centroid of the region bounded by the given curves.

x = 8 ­ y2

x = 0 ( , )

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11.–/1 pointsSCalcET6 8.3.Tut.01.

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12.–/1 pointsSCalcET6 8.3.011.

A trapezoidal plate with height h and bases a and 2a is submerged vertically in water as shown. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Use delta for δ, the weight density constant.)

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13.–/6 pointsSCalcET6 8.3.AE.01.

50 m

30 m

20 m

Video Example Tutorial

Online Textbook

EXAMPLE 1 A dam has the shape of a trapezoid shown in the figure. The height is 30 m, and the width is 50 m at the top and 20 m at the bottom. Find the force on the dam due to hydrostatic pressure if the water level is 4 m from the top of the dam.

SOLUTION We place the origin at the surface of the water with the x­axis pointing downward. The width at the surface of the water (x = 0) is 46 m and the width at x = 26 is

m. So a function for the width at x m is

f(x) =

The depth of the water is 26 m, so we divide the interval [0, 26] into horizontal strips that are approximated by a rectangle with height Δx and width 46 ­ xi. If Ai is the area of

the ith strip, then

Ai

Δx

If Δx is small, then the pressure Pi on the ith strip is almost

constant and we can write

Pi 1000gxi

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The hydrostatic force Fi acting on the ith strip is the product

of the pressure and the area:

Fi = PiAi 1000gxi(

)Δx

Adding these forces and taking the limit as n ∞, we obtain the total hydrostatic force on the dam:

= 9800

N

14.–/1 pointsSCalcET6 8.3.016.

A dam is inclined at an angle of 30° from the vertical and has the shape of an isosceles trapezoid 140 ft wide at the top and 70 ft wide at the bottom and with a slant height of 100 ft. Find the hydrostatic force on the dam when it is full of water. (Round your answer to the nearest whole number. Recall that the

weight density of water is 62.5 lb/ft3.) lb

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15.–/1 pointsSCalcET6 8.3.Tut.07.

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16.–/1 pointsSCalcET6 8.3.Tut.03.

17.–/2 pointsSCalcET6 8.3.026.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. (Do this on paper. Your instructor may ask you to turn in this work.) Then find the exact coordinates of the centroid.

5x + 6y = 30 x = 0, y = 0

( , )

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18.–/2 pointsSCalcET6 8.3.030.

Find exact coordinates of the centroid of the region bounded by the given curves.

y = x + 2

y = x2

( , )

19.–/1 pointsSCalcET6 8.3.019.

A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Use Simpson's Rule to estimate the force of the water against the plate. Give your answer correct to the nearest whole number.

Depth (m) 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Plate width (m) 0 0.7 1.8 2.2 2.6 3.3 3.9

N

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20.–/7 pointsSCalcET6 8.3.AE.03.

Video Example Tutorial

Online Textbook

EXAMPLE 3 Find the moments and center of mass of the system of objects that have masses 4, 4, and 8 at the point (­1, 1), (2, ­1), and (3, 2), respectively.

SOLUTION We use the following equations to compute the moments:

My = Σ mixi = 4(­1) + 4(2) + 8(3) =

Mx = Σ miyi = 4(1) + 4(­1) + 8(2) =

Since m = 4 + 4 + 8 = , we can obtain

x = My

m

=

y = Mx

m

=

Thus the center of mass is ( , ).

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21.–/3 pointsSCalcET6 8.3.002.

A tank is 6 m long, 4 m wide, 5 m high, and contains kerosene with density 820 kg/m3 to a depth of 4.5

m. (Use 9.8 m/s2 for the acceleration due to gravity.)

(a) Find the hydrostatic pressure on the bottom of the tank. Pa

(b) Find the hydrostatic force on the bottom of the tank. N

(c) Find the hydrostatic force on one end of the tank. N

22.–/2 pointsSCalcET6 8.3.022.

Point­masses mi are located on the x­axis as shown. m1 = 25, m2 = 50, and m3 = 10. Find the moment

M of the system about the origin and the center of mass x.

M =

x =

23.–/1 pointsSCalcET6 8.3.012.

lb

A large tank is designed with ends in the shape of the region between the curves y = x2 and y = 28,

measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 20 ft with gasoline. (Assume the gasoline's density is 42.0 lb/ft3. Round your answer to one decimal place.)

1 5

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24.–/1 pointsSCalcET6 8.3.007.

An equilateral triangle with sides 3 m is submerged vertically in water so that the base is even with the surface. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Give your answer correct to the nearest whole number.)

N

25.–/1 pointsSCalcET6 8.3.008.

A trapezoid with bases 3 m and 7 m is partially submerged vertically in water so that the top is 2 m above the surface and the bottom is 2 m below the surface. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Give your answer correct to the nearest whole number.)

N

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