Total force on a curved surface is given by
Calculating Total Force Magnitude on a Curved Surface
When dealing with forces acting on a curved surface, such as the pressure exerted by a fluid, the total force is a resultant vector quantity. This force can be resolved into components acting in different directions, typically horizontal and vertical.
For a curved surface subjected to fluid pressure, the total hydrostatic force is often calculated by finding its horizontal and vertical components separately.
- Horizontal Component (\(F_x\)): This is usually equal to the hydrostatic force on the vertical projection of the curved surface.
- Vertical Component (\(F_y\)): This is usually equal to the weight of the fluid volume vertically above the curved surface up to the free surface level.
The total force (\(F\)) on the curved surface is the vector sum of these orthogonal components (\(F_x\) and \(F_y\)). To find the magnitude of this resultant force, we use the Pythagorean theorem, which is applicable for finding the magnitude of a vector given its perpendicular components.
The formula for the magnitude of the total force \(F\) based on its horizontal component \(F_x\) and vertical component \(F_y\) is given by:
\[ \rm F = \sqrt{F_x^2 + F_y^2} \]
This formula represents the Euclidean norm or magnitude of the resultant force vector obtained by combining the horizontal and vertical force vectors.
Why Other Options Are Incorrect
- \(\rm F=\sqrt{F_x^2 - F_y^2}\): Subtracting the squares of the components is not the method for finding the magnitude of a resultant vector.
- \(\rm F=\sqrt{F_x^2 * F_y^2}\): Multiplying the squares of the components is not a standard vector operation for finding magnitude.
- \(\rm F=\sqrt{F_x^2 / F_y^2}\): Dividing the squares of the components is not a standard vector operation for finding magnitude.
Therefore, the correct way to combine the horizontal and vertical components to find the magnitude of the total force is using the square root of the sum of their squares, as derived from vector addition principles and the Pythagorean theorem.
Important Questions from Hydrostatic Force
The centre of pressure of a plane submerged surface
In the context of hydrostatics, the resultant hydrostatic force acting on a submerged plane surface passes through which of the following points?
The depth of the center of pressure on a vertical rectangular gate (4 m wide and 3 m high) with water up to top surface is
If a planar surface is immersed in a liquid, the resultant liquid pressure acts at a point called ___________.
The resultant of all normal pressure acts
