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Chapter 3: Problem 82

A curved surface is formed as a quarter of a circular cylinder with \(R=0.750 \mathrm{m}\) as shown. The surface is \(w=3.55\) \(\mathrm{m}\) wide. Water stands to the right of the curved surface to depth \(H=0.650 \mathrm{m} .\) Calculate the vertical hydrostatic force on the curved surface. Evaluate the line of action of this force. Find the magnitude and line of action of the horizontal force on the surface.

Short Answer

Expert verified
The calculated vertical hydrostatic force is \(1504.5 \, N\), acting downwards at a depth of \(0.28125 \, m\) from the base. The magnitude of the horizontal hydrostatic force is \(9810 \, N\), acting horizontally at a depth of \(0.375 \, m\) from the base.

Step by step solution

01

Calculation of Hydrostatic Force

The hydrostatic force component along any direction is obtained by multiplying the pressure at the centre of pressure by the area over which pressure is acting. For a curved surface submerged in a fluid, the centre of pressure is located at the centroid of the surface. Considering that the fluid density \(\rho=1000 \, kg/m^3\), gravitational acceleration \(g=9.81 \, m/s^2\) and the height to the centroid \(h_c = \frac{3R}{8} = 0.28125 \, m\), the vertical hydrostatic force \(F_y\) can be calculated by applying the formula: \(F_y = \rho \cdot g \cdot h_c \cdot A \), where the area \(A = R \cdot w = 2.6625 \, m^2\).
02

Evaluation of Line of Action of Vertical Hydrostatic Force

The vertical hydrostatic force always acts through the centre of pressure of the submerged surface. For a surface submerged in a fluid, the line of action of the vertical hydrostatic force is vertically downwards, passing through the centroid of the surface, in this case at a depth of \(0.28125 \, m\)
03

Calculation of Horizontal Hydrostatic Force

The horizontal hydrostatic force \(F_x\) is the weight of the water above the curved surface upto the water level. The volume \(V\) on top of the submerged surface can be calculated as \(\frac{1}{6} \cdot \pi \cdot R^2 \cdot w = 1.000166 \, m^3\), so the force \(F_x\) can be calculated by applying the formula: \(F_x = \rho \cdot g \cdot V\)
04

Evaluation of Line of Action of Horizontal Hydrostatic Force

In general, the line of action of the horizontal hydrostatic force is at the depth of the centroid of the volume of fluid above the surface. The centroid of the quarter cylinder volume of fluid above the surface is at a height or depth \(h_x = R/2 = 0.375 \, m\), measured from the base.

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