Problem 1026 - JEE Physics

Question

A triangular lamina of area \(\mathrm{A}\) and, height \(\mathrm{h}\) is immersed in a liquid of density \(\mathrm{S}\) in a vertical plane with its base on the surface of the liquid. The thrust on lamina is (A) \((1 / 2)\) Apgh (B) \((1 / 3)\) Apgh (C) \((1 / 6)\) Apgh (D) \((1 / 3)\) Apgh

Step-by-Step Solution

Answer

The thrust formula on the triangular lamina is given by \[ F = (1/3)ASgh \], which corresponds to option (B).

Step 1: 1. Calculate the pressure at the center of pressure

To find the pressure on the center of pressure, we need to find the distance of the center of pressure from the surface of the liquid. For a triangle, the center of pressure (COP) is located at the distance of (h/3) from the base and 'h' is the distance of the lowest point of triangle from the base. The pressure at the COP can be calculated using the formula: \[ P = Sg(h/3) \] where, P = pressure at the center of pressure S = density of the liquid g = acceleration due to gravity h = distance of the lowest point of the triangle from the surface of the liquid

Step 2: 2. Calculate the thrust on the lamina

Now we can find the thrust (F) on the lamina by multiplying the pressure at the center of pressure with the area of the lamina. \[ F = PA \] \[ F = (Sg(h/3))A \]

Step 3: 3. Simplify the expression for thrust

We can simplify the expression for thrust: \[ F = (h/3)ASg \] Now we see that the expression for thrust is matching with option (B) of the given problem. Therefore, the correct thrust formula on lamina is: Thrust F = \((1 / 3) \cdot A Sgh \) So, the correct answer is (B).

Key Concepts

Center of Pressure

The center of pressure (COP) is a critical point in fluid mechanics, especially when dealing with submerged surfaces. It represents the point at which the resultant hydrostatic force acts. In simple terms, it's the point where you can imagine all the water pressure acting when a surface is submerged. This point is crucial for ensuring stability in underwater structures.
For shapes like a triangle, the center of pressure is not the same as the geometric center (center of gravity), because pressure distribution under a liquid is not uniform. For a triangular surface with its base at the waterline and submerged vertically, its COP is located a third ( \( \frac{h}{3}\) ) of the way from the base to the tip. This is because pressure increases with depth.
Understanding the location of the COP allows engineers to predict how a structure will react to water pressure and is vital for constructing stable bridges, ships, and dams.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at equilibrium, due to the force of gravity. It is an essential part of fluid mechanics and applies to fluids in containers, dams, and even the human body.
In a scenario where a lamina (or flat surface) is submerged in a liquid, the pressure exerted on any point of the lamina increments with depth due to the increasing weight of the liquid above that point. The formula to compute the hydrostatic pressure at a given depth is: \[ P = ho g h \] where:

• \( \rho \) is the density of the liquid,
• \( g \) is the acceleration due to gravity,
• \( h \) is the depth of the point below the liquid's surface.

Recognizing how hydrostatic pressure increases with depth is vital, especially when determining the forces on submerged surfaces, as it affects how materials and structures will withstand underwater conditions.

Thrust on Submerged Surfaces

Thrust on a submerged surface is the total force exerted by a fluid on that surface due to hydrostatic pressure. Understanding how to calculate thrust is crucial for designing objects that interact with fluids, like boats or dams.
When a triangular lamina is submerged in a fluid, the thrust can be calculated by multiplying the pressure at the center of pressure with the area of the surface. For our triangular lamina:

1. The pressure at the COP is determined based on its depth ( \( \frac{h}{3} \) from the surface).
2. The thrust formula becomes \( F = PA = \left( Sg \frac{h}{3} \right)A \).

Simplifying this, the thrust is calculated as:\[ F = \frac{1}{3} ASgh \]This is the same as the pressure distribution integrated over the area of the lamina.
Understanding thrust helps in optimizing design and ensuring safety in structural engineering. By calculating thrust accurately, engineers can predict how forces act on submerged surfaces, ensuring stability and structural integrity.