a rectangular plate 12ft long and 8 ft wide is submerged vertically with the longer edge in the surface of the water. Find the force on one side of the plate.

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To find the force on one side of the submerged rectangular plate, we need to calculate the hydrostatic force acting on it.

The hydrostatic force is equal to the pressure multiplied by the area over which the force is acting.

1. The pressure exerted by the fluid at a certain depth is given by the formula:
Pressure = density of fluid * gravitational acceleration * depth

2. The depth of the submerged plate is equal to the width of the plate, which is 8 ft.

3. The area over which the force acts is equal to the width of the plate multiplied by the height of the plate. Since the longer edge is in the surface of the water, the height of the plate is equal to the length of the plate, which is 12 ft.

Now, let's plug in the values into the formula and calculate the force:

Pressure = density of fluid * gravitational acceleration * depth
= density of water * 32.2 ft/s^2 * 8 ft

The density of water is approximately 62.4 lb/ft^3.

Pressure = 62.4 lb/ft^3 * 32.2 ft/s^2 * 8 ft

Now, we need to calculate the area over which the force acts:

Area = width * height
= 8 ft * 12 ft

Finally, we can calculate the force using the formula:

Force = Pressure * Area

Substituting in the values:

Force = (62.4 lb/ft^3 * 32.2 ft/s^2 * 8 ft) * (8 ft * 12 ft)

To find the force on one side of the plate, we need to calculate the pressure exerted by the water on the surface of the plate.

The pressure exerted by a fluid at a certain depth is given by the equation:

Pressure = Density × gravitational acceleration × depth

In this case, the plate is submerged vertically with the longer edge in the water surface. This means that the depth is 12 ft, which is the height of the plate.

First, let's find the pressure exerted by the water on the plate. We will need to know the density of water and the acceleration due to gravity.

The density of water is approximately 62.4 lbs/ft³, and the acceleration due to gravity is 32.2 ft/s².

Now, we can calculate the pressure:

Pressure = Density × gravitational acceleration × depth
= 62.4 lbs/ft³ × 32.2 ft/s² × 12 ft

The unit of pressure will be pounds per square foot.

To find the force on one side of the plate, we need to multiply the pressure by the area of that side. The area of one side of the rectangular plate is 8 ft × 12 ft.

Force = Pressure × Area
= (62.4 lbs/ft³ × 32.2 ft/s² × 12 ft) × (8 ft × 12 ft)

Now, we can calculate the force on one side of the plate using the given values.

Hydrostatic pressure at depth h

=ρgh
where ρ=density of water = 1000 kg/m³
g=acceleration due to gravity=9.8 m/s²
Total Force = pressure * area
=∫ ∫ ρgh dh dx
x from 0 to 12
h from 0 to 8