JNTU B.Tech II MECHANICS OF FLUIDS Previous Question Paper (October - 2020 - R13)

Consider a plane surface of area A inclined at an angle θ with the free surface of a liquid. Let G be the centroid of the area and P be the centre of pressure. Let h_bar be the depth of the centroid and h_cp be the depth of the centre of pressure from the free surface.

The total pressure force F acting on the surface is given by:

The moment of this force about the free surface must equal the sum of the moments of the elemental forces. Taking moments about the intersection of the plane and the free surface (axis O-O):

Here, ∫ y² dA represents the moment of inertia of the area about the axis O-O, denoted as I_OO. Using the parallel axis theorem, I_OO = I_G + A y_bar², where I_G is the moment of inertia about the centroidal axis parallel to O-O and y_bar is the distance of the centroid from O along the incline.

Substituting F and I_OO into the moment equation:

Simplifying for y_cp:

Since the depth h = y sinθ, the depth of the centre of pressure h_cp is:

This expression shows that the centre of pressure always lies below the centroid of the submerged inclined plane surface.

A manometer is a device that measures pressure by balancing the fluid column against gravity, typically using a liquid like mercury or water. In contrast, a mechanical gauge measures pressure by using the elastic deformation of a sensing element, such as a diaphragm or tube, to actuate a mechanical linkage and pointer.

Key differences include:

• Manometers are generally more accurate and serve as primary standards for calibration, but they are bulky, fragile, and limited to low or moderate pressure ranges.
• Mechanical gauges are compact, portable, and capable of measuring very high pressures, but they require periodic calibration against a standard because the elastic element can suffer from hysteresis or fatigue over time.

The primary types of mechanical pressure gauges are:

1. Bourdon tube pressure gauge: Uses a curved, elliptical cross-section metal tube that tends to straighten when pressurized, moving a pointer via a gear mechanism.
2. Diaphragm pressure gauge: Employs a thin, flexible metal disk that deforms under pressure, with the displacement transmitted to a dial indicator.
3. Bellows pressure gauge: Utilizes a corrugated metal cylinder that expands or contracts axially in response to pressure changes.
4. Dead-weight pressure gauge: A highly accurate device that balances a known weight against the fluid pressure acting on a piston of known area, primarily used for calibrating other gauges.