If the density of a fluid varies with depth, then the pressure at depth y is p(y) (which

How is pressure defined?

Fluid pressure is proportional to depth. What is the factor of proportionality?

When fluid force acts on the side of a submerged object, in which direction does it act?

Why is fluid pressure on a surface calculated using thin horizontal strips rather than thin vertical strips?

If a thin plate is submerged horizontally, then the fluid force on one side of the plate is equal to pressure times area. Is this true if the plate is submerged vertically?

The plate R in Figure 8, bounded by the parabola y = x2 and y = 1, is submerged vertically in water (distance in meters). (a) Show that the width of R at height y is f (y) = 2 y and the fluid force on a side of a horizontal strip of thickness y at height y is approximately (g)2y1/2(1 y)y. ((b) Write a Riemann sum that approximates the fluid force F on a side of R and use it to explain why F = g  1 0 2y1/2(1 y) dy (c) Calculate F. Water surface 1 y 1 ( y, y) 1 1 y = x2 f(y) R y x y

Let F be the fluid force on a side of a semicircular plate of radius r meters, submerged vertically in water so that its diameter is level with the waters surface (Figure 9). (a) Show that the width of the plate at depth y is 2 r2 y2. (b) Calculate F as a function of r using Eq. (2).

Calculate the force on one side of a circular plate with radius 2 m, submerged vertically in a tank of water so that the top of the circle is tangent to the water surface.

A semicircular plate of radius r meters, oriented as in Figure 9, is submerged in water so that its diameter is located at a depth of m meters. Calculate the fluid force on one side of the plate in terms of m and r.

A semicircular plate of radius r meters, oriented as in Figure 9, is submerged in water so that its diameter is located at a depth of m meters. Calculate the fluid force on one side of the plate in terms of m and r.

Figure 10 shows the wall of a dam on a water reservoir. Use the Trapezoidal Rule and the width and depth measurements in the figure to estimate the fluid force on the wall. Depth (ft) 20 0 600 900 1100 1400 1650 1800 (ft) 40 60 80 10

Calculate the fluid force on a side of the plate in Figure 11(A), submerged in water.

Calculate the fluid force on a side of the plate in Figure 11(B), submerged in a fluid of mass density = 800 kg/m3. 3 m 4 m 7 m 2 m (A) (B) 2 m 2 m 4 m

Find the fluid force on the side of the plate in Figure 12, submerged in a fluid of density = 1200 kg/m3. The top of the plate is level with the fluid surface. The edges of the plate are the curves y = x1/3 and y = x1/3. 8 2 Water level 8 y = x 1/3 y = x 1/3

Let R be the plate in the shape of the region under y = sin x for 0 x 2 in Figure 13(A). If R is rotated counterclockwise by 90 and then submerged in a fluid of density 1100 kg/m3 with its top edge level with the surface of the fluid as in (B), find the fluid force on a side of R. 2 1 (A) (B) Fluid level y = sin x R Fluid level

In the notation of Exercise 15, calculate the fluid force on a side of the plate R if it is oriented as in Figure 13(A). You may need to use Integration by Parts and trigonometric substitution.

Calculate the fluid force on one side of a plate in the shape of region A shown Figure 14. The water surface is at y = 1, and the fluid has density = 900 kg/m3. y = ln

Calculate the fluid force on one side of the infinite plate B in Figure 14, assuming the fluid has density = 900 kg/m3.

Figure 15(A) shows a ramp inclined at 30 leading into a swimming pool. Calculate the fluid force on the ramp.

Calculate the fluid force on one side of the plate (an isosceles triangle) shown in Figure 15(B). 4 3 10 y f(y) Vertical change y 6 Water surface (A) (B) Water surface

The massive Three Gorges Dam on Chinas Yangtze River has height 185 m (Figure 16). Calculate the force on the dam, assuming that the dam is a trapezoid of base 2000 m and upper edge 3000 m, inclined at an angle of 55 to the horizontal (Figure 17).

A square plate of side 3 m is submerged in water at an incline of 30 with the horizontal.

The trough in Figure 18 is filled with corn syrup, whose weight density is 90 lb/ft3. Calculate the force on the front side of the trough

Calculate the fluid pressure on one of the slanted sides of the trough in Figure 18 when it is filled with corn syrup as in Exercise 23.

The end of the trough in Figure 19 is an equilateral triangle of side 3. Assume that the trough is filled with water to height H. Calculate the fluid force on each side of the trough as a function of H and the length l of the trough.

A rectangular plate of side  is submerged vertically in a fluid of density w, with its top edge at depth h. Show that if the depth is increased by an amount h, then the force on a side of the plate increases by wAh, where A is the area of the plate.

Prove that the force on the side of a rectangular plate of area A submerged vertically in a fluid is equal to p0A, where p0 is the fluid pressure at the center point of the rectangle.

If the density of a fluid varies with depth, then the pressure at depth y is p(y) (which need not equal wy as in the case of constant density). Use Riemann sums to argue that the total force F on the flat side of a submerged object submerged vertically is F =  b a f (y)p(y) dy, where f (y) is the width of the side at depth y.