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Understanding hydrostatic pressure is essential to tackling problems involving submerged objects. It represents the pressure exerted by a fluid at any given point below its surface due to the gravitational force on the fluid. This pressure increases with depth because more fluid is stacked above it, exerting gravitational force downwards. To express hydrostatic pressure mathematically, use the formula: \[ P = P_0 + \rho g h \] where:- \( P_0 \) is the atmospheric pressure at the surface,- \( \rho \) is the fluid's density (for water, typically \(1000\, \text{kg/m}^3 \)),- \( g \) is the acceleration due to gravity (approximately \(9.81\, \text{m/s}^2 \)),- \( h \) is the depth below the fluid surface.
For example, in the context of our underwater lamp problem, the pressure on the lamp at a depth of 3 meters is calculated by substituting the known values into the equation, resulting in a pressure of \( 130730\, \text{Pa} \). This includes both atmospheric pressure and the pressure from the weight of the water above.

Once the hydrostatic pressure is understood, the next step is calculating the force exerted by this pressure. The force on a submerged surface depends on the pressure acting on it and the area of the surface covered by the fluid. The total force \( F \) is given by:\[ F = \Delta P \times A \]where:- \( \Delta P \) is the difference between external and internal pressures,- \( A \) is the area of the surface receiving the pressure.In our problem, the pressure difference \( \Delta P \) is \( 29430\, \text{Pa} \) and the hemispherical surface area \( A \) is computed to be \(0.1413\, \text{m}^2\). By multiplying these values, the total force exerted by the water on the lamp amounts to \( 4156.5\, \text{N}\). This calculation is vital in various applications, such as designing aquatic equipment or safety protocols for underwater activities.

When dealing with objects like a hemispherical lamp cover submerged in a fluid, calculating the surface area correctly is crucial for determining the forces involved. A hemisphere has a distinctive curved surface, which is exactly half of a full sphere minus the base.The surface area of a hemisphere, excluding its base, is given by:\[ A = 2 \pi r^2 \]where \( r \) is the radius of the sphere.
In the given exercise, the diameter of the hemisphere is \(30\, \text{cm}\), which converts to a radius \( r = 0.15\, \text{m} \). Substituting this radius into the formula, the hemispherical surface area is found to be approximately \(0.1413\, \text{m}^2\).This value plays a crucial role when determining the force due to hydrostatic pressure, as these forces act over the entire submerged surface area.