Explanation:

• Meta Centre is defined as the point about which a body starts oscillating when body is tilted by a small angle.
• The meta-centre may also be defined as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement.
• The distance MG i.e. the distance between the meta-centre of a floating body and the centre of gravity of the body is called meta-centric height. 
• It is measured along the line BG.

CONCEPT:

• Pressure: The force per unit area is called pressure. It is denoted by P.
• Piezometer: A device that is used to measure liquid pressure in a system by measuring the height to which a column of the liquid rises against gravity or a device that measures the pressure (more precisely, the piezometric head) of groundwater at a specific point is called piezometer.

• One end of the piezometer is connected to the point where the pressure is to be measured and the other end is open to the atmosphere.

EXPLANATION:

• Piezometer is used to measure Pressure in pipe channels etc. at a point. So option 1 is correct.
• As one end is open to the atmosphere so it can’t be used for measuring the gas pressure (or atmospheric pressure). So option 2 is not correct.
• Gas pressure cannot be measured by means of piezometers because a gas forms no free atmosphere surface and it can’t be used when large pressures in the lighter liquid are to be measured.
• It can be used to measure medium or low pressure only. So option 3 is not correct.
• It can't measure the pressure difference between two points.

The pressure at any point in a fluid at rest is obtained by the Hydrostatic Law which states that the rate of increase of pressure in a vertically downward direction must be equal to the specific weight of the fluid at that point.

\(\frac{{dP}}{{dh}} = \rho g\)

\(\smallint dP = \rho g\smallint dh\)

P = ρgh

Explanation:

For a plane surface of arbitrary shape immersed in a liquid in such a way that the plane of the surface makes an angle θ with the free surface of the liquid:

A = Total area of the inclined surface

h̅ = Depth of centre of gravity of inclined area from the free surface

h* = Distance of centre of pressure from the free surface of the liquid

\({h^*} = \frac{{{I_G}{{\sin }^2}\theta }}{{A\bar h}} + \bar h\)

For vertical plane surface: θ = 90°

\({h^*} = \frac{{{I_G}}}{{A\bar h}} + \bar h\)

Explanation:

A plate immersed vertically:

• The magnitude of hydrostatic force is given by F = ρgh̅A 
• Where ρ = Density of the fluid, h̅ = Depth of center of gravity of the surface from free liquid surface, A = Area of the surface

A plate immersed horizontally:

• The magnitude of hydrostatic force is given by F = ρgh̅A 
• Where ρ = Density of the fluid, h̅ = Depth of center of gravity of the surface from free liquid surface, A = Area of the surface
• Therefore, the hydrostatic force (total force) in both cases is same due to the same value of Depth of center of gravity of the surface from free liquid surface

Concept:

Hydrostatic Law:

It states that the rate of increase of pressure in a vertical direction is equal to the weight density of the fluid at that point.

\(\frac{{\partial P}}{{\partial Z}} = - pg\;\left[ {Going\;Upward} \right]\)

\(\frac{{\partial P}}{{\partial Z}} = + pg\;\left[ {Going\;Downward} \right]\)

P = ρgh

Calculation:

Given:

Converting all into equivalent head form

P_A + h_A⋅S_A – h_B⋅S₁ + h_B⋅S₃ – h_B S_B = P_B

P_A – P_B = h_BS₁ + h_BS_B – h_AS_A – h_BS₃

At base P_B = 750 mm of Hg

At top, P_A = 600 mm of Hg

∴ Pressure difference = 150 mm of Hg = 150 × 10^-3 × 13.6 × 103 × 9.81

(ΔP) = hρg

\( \Rightarrow \;h\; = \;\frac{{{\rm{\Delta }}P}}{{\rho g}}\; = \;\frac{{150 × 13.6 × {{10}^3} × {{10}^{ - 3}} × 9.81}}{{1 × 9.81}}\; = \;2040\;m\)

⇒ Height of mountain = 2040 m

Explanation:

Sensitivity:

Sensitivity is directly proportional to the length of fluid travel for a particular pressure difference.

\(\begin{array}{l} {\left( {Sensitivity} \right)_{U - tube}} \propto \;x\\ {\left( {sensitivity} \right)_{Inclined\;tube}} \propto \frac{x}{{\sin \theta }} \end{array}\)

\(\frac{(Sensitivity)_{Inclined\;tube}}{(Sensitivity)_{U-Tube}}=\frac{1}{\sin \theta}\)

Explanation:

When a body is either wholly or partially immersed in a fluid, a lift is generated due to the net vertical component of hydrostatic pressure forces experienced by the body. This lift is called the buoyant force and the phenomenon is called buoyancy.

The Archimedes principle states that the buoyant force on a submerged body is equal to the weight of the liquid displaced by the body and acts vertically upward through the centroid of the displaced volume.

Thus, the net weight of the submerged body, (the net vertical downward force experienced by it) is reduced from its actual weight by an amount that equals the buoyant force.

FB = ρghA = ρgV

F_B = f(V_displaced, ρ)

Explanation:

Types of equilibrium

Stable Equilibrium:

• If the body returns to its original position by retaining the originally vertical axis as vertical.

Unstable Equilibrium:

• If the body does not return to its original position but moves further from it.

Neutral Equilibrium:

• If the body neither returns to its original position nor increases its displacement further, it will simply adapt its new position.

Stability of Floating Bodies in Fluid:

When the body undergoes an angular displacement about a horizontal axis, the shape of the immersed volume changes and so the centre of buoyancy moves relative to the body.

Metacentre:

Meta Centre is defined as the point about which a body starts oscillating when the body is tilted by a small angle.

The meta-centre may also be defined as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement.

• For the body shown in the figure, M is above G, and the couple acting on the body in its displaced position is a restoring couple which tends to turn the body to its original position.
• If M were below G, the couple would be an overturning couple and the original equilibrium would have been unstable.
• When M coincides with G, the body will assume its new position without any further movement and thus will be in neutral equilibrium.

Hence the condition of stable equilibrium for a floating body can be expressed in terms of metacentric height as follows:

• GM > 0 (M is above G) ⇒ Stable equilibrium
• GM = 0 (M coinciding with G) ⇒ Neutral equilibrium
• GM < 0 (M is below G) ⇒ Unstable equilibrium