A plate in the shape of an isosceles triangle with base 1 m and height 2 m is submerged vertical

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A plate in the shape of an isosceles triangle with base 1 m...

A plate in the shape of an isosceles triangle with base 1 $\mathrm{m}$ and height 2 $\mathrm{m}$ is submerged vertically in a tank of water so that its vertex touches the surface of the water (Figure 7$)$ $$ \begin{array}{l}{\text { (a) Show that the width of the triangle at depth } y \text { is } f(y)=\frac{1}{2} y \text { . }} \\ {\text { (b) Consider a thin strip of thickness } \Delta y \text { at depth } y . \text { Explain why the }} \\ {\text { fluid force on a side of this strip is approximately equal to } \rho g \frac{1}{2} y^{2} \Delta y}\\{\text { (c) Write an approximation for the total fluid force } F \text { on a side of the }} \\ {\text { plate as a Riemann sum and indicate the integral to which it converges. }} \\ {\text { (d) Calculate } F .}\end{array} $$

A plate in the shape of an isosceles triangle with base 1 $\mathrm{m}$ and height 2 $\mathrm{m}$ is submerged vertically in a tank of water so that its vertex touches the surface of the water (Figure 7$)$
$$
\begin{array}{l}{\text { (a) Show that the width of the triangle at depth } y \text { is } f(y)=\frac{1}{2} y \text { . }} \\ {\text { (b) Consider a thin strip of thickness } \Delta y \text { at depth } y . \text { Explain why the }} \\ {\text { fluid force on a side of this strip is approximately equal to } \rho g \frac{1}{2} y^{2} \Delta y}\\{\text { (c) Write an approximation for the total fluid force } F \text { on a side of the }} \\ {\text { plate as a Riemann sum and indicate the integral to which it converges. }} \\ {\text { (d) Calculate } F .}\end{array}
$$

Key Concepts

Riemann Sum and Integration

The Riemann sum is a method for approximating the total accumulation of a quantity, such as force, by summing the contributions of small segments or slices. As the thickness of these slices approaches zero, the Riemann sum converges to a definite integral, which gives the exact total force. This method is fundamental in calculus for evaluating variable quantities over continuous intervals, especially when the quantity varies in a non-uniform manner.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity, and it increases linearly with depth. This concept is crucial for understanding how the pressure acting on a submerged surface varies with depth, and it is typically expressed as p = ?g y, where ? represents the density of the fluid, g is the acceleration due to gravity, and y is the depth below the surface.

Pressure Force on Submerged Surfaces

The pressure force on a submerged surface is calculated by multiplying the pressure at a given depth by the area over which the pressure is acting. In scenarios where pressure varies with depth, the force is determined by integrating the pressure distribution over the entire surface area. This concept is essential when dealing with objects of irregular shape, as the pressure and consequently the force vary continuously over the surface.

Geometric Similarity and Proportionality

Geometric similarity involves using similar triangles to relate different parts of an object’s dimensions. In problems where an object's dimensions change with depth (such as the width of a triangular plate that changes linearly with y), similar triangles help establish the proportional relationship between these dimensions and the depth. This technique is crucial when deriving functions that describe how specific lengths or widths vary with depth.

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