A circular opening \(1.2 \mathrm{~m}\) in diameter in the vertical side of a reservoir is closed by a disc which just fits the opening and is pivoted on a shaft along its horizontal diameter. Show that, if the water level in the reservoir is above the top of the disc, the turning moment on the shaft required to hold the disc vertical is independent of the head of water. Calculate the amount of this moment.

Pressure distribution in a fluid is essential to understand when dealing with submerged objects. It's the variation in pressure within the fluid, typically increasing with depth. This is due to the weight of the fluid above the point where the pressure is being measured. For water, the pressure at a depth \(h\) can be calculated using the formula: \[P = \rho g h\], where \(\rho\) (rho) stands for the density of water, approximately 1000 kg/m³, and \(g\) is the acceleration due to gravity, around 9.81 m/s². When a disc is submerged vertically, pressure changes linearly from top to bottom. This linear increase greatly impacts the calculation of forces acting on submerged surfaces.

The center of pressure is the point where the total sum of the pressure field acts on a submerged surface, causing a force that results in rotation. For a symmetric shape like a circle, the pressure force distribution is symmetric. However, for a semi-circular shape, the center of pressure lies at \(\frac{4r}{3\pi}\) from the horizontal diameter, where \(r\) is the radius. This factor is crucial in determining the turning moment. To compute the turning moment about the shaft, you multiply the resultant force by this distance. In our case, it simplifies as: \[\text{Turning Moment} (T.M.) = \frac{8 \rho g r^4}{9\pi}\].

Fluid mechanics is the study of fluids (liquids & gases) and the forces in motion upon them. This field includes understanding how fluids behave under various conditions and forces like pressure and gravity. In the context of our exercise, fluid mechanics principles help us calculate pressure distribution and forces on submerged objects. The behavior of water pressure increasing with depth plays a critical role here. Additionally, the integral calculations for determining the force on a submerged disc and the resulting moment are deeply rooted in these fluid mechanics principles.

Numerical integration is a fundamental mathematical process used to approximate the area under curves. It's vital when analytical integration is complex or impossible. In our scenario, since pressure distribution over the submerged disc varies with depth, we use numerical integration to find the total force acting on the disc. By integrating the force across the disc's surface: \[F = \int_{-r}^{r} \rho g h \cdot 2\sqrt{r^2 - h^2} \cdot dh\], we obtain the net force. Such integration techniques provide a practical approach to solving real-world engineering problems when exact solutions are hard to derive.