Fluid Mechanics Quiz

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Questions and Answers

What does the equation dP = -ρa dx represent in the context of a liquid in an accelerating beaker?

The change in pressure across a vertical segment of liquid

The force exerted by gravity on the liquid

The relationship between pressure change and horizontal acceleration (correct)

The relationship between liquid volume and temperature

In the scenario where a beaker is accelerated with components ax and ay, what happens to the pressure in the liquid?

Only the pressure along the y direction decreases

Pressure remains constant regardless of the acceleration

Pressure increases with both ax and ay

Pressure decreases along both x and y directions (correct)

Which of the following correctly simplifies the motion equation for a fluid element in an accelerated beaker?

dP/dx = ρa

dP/dy = ρ(g + ay)

dP = ρa dx (correct)

dP = -ρ(dx + dy)

What is the implication of the equation dP/dy = -ρ(g + ay) for a fluid element experiencing vertical acceleration?

Pressure decreases with the combined effects of gravity and vertical acceleration Signup and view all the answers

How does the acceleration of the beaker affect the pressure distribution within the liquid?

It results in pressure decreases in line with the direction of acceleration Signup and view all the answers

What is the mass term used in the equation of motion for the fluid element?

Aρ dy Signup and view all the answers

If the beaker is accelerating vertically with ay, what will dP/dy represent?

Pressure change influenced by vertical acceleration and gravity Signup and view all the answers

What characteristic behavior is observed in the pressure of liquid in a moving beaker?

Pressure decreases in the direction of the beaker's acceleration Signup and view all the answers

What is the relationship between the centre of gravity and the metacentre for a stable floating body?

The centre of gravity is below the metacentre. Signup and view all the answers

When a boat tilts, what happens to the centre of buoyancy?

It shifts relative to the centre of gravity. Signup and view all the answers

What factors determine the stability of a floating body?

The position of the centre of buoyancy and centre of gravity. Signup and view all the answers

What will happen if the centre of gravity is above the metacentre?

The body will be unstable and rotate away from equilibrium. Signup and view all the answers

In the context of a floating body, what does the term 'torque' refer to?

The rotational effect caused by the buoyant force. Signup and view all the answers

Which condition must be met for a floating body to achieve equilibrium?

The net torque about the centre of gravity must be zero. Signup and view all the answers

What is the specific gravity of a wooden plank if it has a weight density less than the water it displaces?

Less than 1 Signup and view all the answers

Which of the following best describes the effect of buoyancy on a floating object?

It counteracts the weight of the object. Signup and view all the answers

What does Bernoulli's equation state about an ideal fluid?

The sum of total energy per unit volume is constant. Signup and view all the answers

If the mass of a liquid is 2 kg and it is flowing with a velocity of 3 m/s, what is the kinetic energy per unit volume of the liquid?

9 J/m³ Signup and view all the answers

In the context of fluid dynamics, what is potential energy related to?

The height of the liquid from a reference point. Signup and view all the answers

What parameters are integrated in Bernoulli's equation?

Pressure, kinetic energy, and potential energy. Signup and view all the answers

When a liquid moves through a distance due to pressure, what is the relationship to pressure energy?

It is equal to the force exerted on the area times the distance moved. Signup and view all the answers

What is the correct formula for calculating pressure energy per unit volume of a liquid?

Pressure energy per unit volume = P Signup and view all the answers

If a liquid is at a height h, what is its potential energy per unit volume?

ρgh Signup and view all the answers

In the equation $P + \frac{1}{2} \rho v^2 + \rho gh = constant$, which term represents kinetic energy?

$\frac{1}{2} \rho v^2$ Signup and view all the answers

What is the definition of viscosity in the context of fluid mechanics?

The property of a liquid that opposes relative motion between its layers Signup and view all the answers

What does the negative sign in the viscous force equation indicate?

The force opposes the direction of relative velocity Signup and view all the answers

How does the coefficient of viscosity change with temperature for liquids?

It decreases with higher temperatures Signup and view all the answers

What is the SI unit of the coefficient of viscosity?

N-s/m² Signup and view all the answers

In the given liquid flow situation, what is the correct calculation of the velocity gradient?

2 m/s Signup and view all the answers

How is the tangential force calculated in the illustration provided?

Using the formula $F = -ηA * ∆v$ Signup and view all the answers

What happens to the relative motion between the layers of a liquid when an external force is applied?

The external force overcomes the backward drag Signup and view all the answers

What is the role of the viscous force in the motion of fluid layers?

It opposes the motion and tends to restore equilibrium Signup and view all the answers

What is the resistive force acting on a spherical body moving through a medium according to Stokes' Law?

$6 imes p imes u imes r imes V$ Signup and view all the answers

What happens to the terminal velocity of a spherical body when the density of the fluid is greater than that of the body?

It moves upward instead of falling Signup and view all the answers

In a scenario where two small raindrops with a terminal velocity of 1 m/s coalesce, what is the relationship between their radius and the new drop's radius?

The radius increases by a factor of 2^(1/3) Signup and view all the answers

What is the formula to express the terminal velocity of a spherical body in terms of its density and the fluid density?

$V = rac{4}{9} rac{(ρ - σ)g}{η}$ Signup and view all the answers

Given an air bubble rising at 0.35 cm/s in a solution of density 1750 kg/m³, how is the coefficient of viscosity represented in the context of the problem?

It opposes the movement of the bubble Signup and view all the answers

Using the information provided, what would be the terminal speed of a large drop formed from two smaller drops with terminal speed of 1 m/s?

1.587 m/s Signup and view all the answers

Which equation illustrates the relationship between the resistive force and terminal velocity for a falling body?

$W = F_t + 6 imes π imes r imes V$ Signup and view all the answers

What principle describes why an air bubble rises in water?

It is less dense than the surrounding fluid Signup and view all the answers

Study Notes

Equation of Motion for a Fluid Element

The equation of motion for a fluid element is derived from Newton's second law: Force = mass x acceleration
Considering a fluid element in a beaker accelerated in the y-direction, the pressure difference across the element causes a net force, balanced by the weight of the element and the inertial force due to acceleration.
This leads to the equation: dP/dy = -ρ(g + ay), where dP is the pressure difference, ρ is the density of the fluid, g is acceleration due to gravity, and ay is the acceleration in the y-direction.
Similarly, for acceleration in the x-direction, the equation is: dP/dx = -ρax.

Stability of a Floating Body

The stability of a floating body depends on the relative positions of its center of gravity (G) and the center of buoyancy (B).
The buoyant force acts at the center of buoyancy, which is the center of gravity of the displaced fluid.
When a floating body tilts, the center of buoyancy shifts.
The line of action of the buoyant force intersects the axis of the body at the metacenter (M).
A body is stable if the metacenter (M) is above the center of gravity (G), and unstable if G is above M.

Bernoulli's Equation

Bernoulli's equation relates the pressure, velocity, and height of a fluid in streamline motion.
It states that the sum of pressure energy per unit volume (P), kinetic energy per unit volume (1/2ρv^2), and potential energy per unit volume (ρgh) remains constant along a streamline.
This can be expressed as: P + 1/2ρv^2 + ρgh = constant.

Viscosity

Viscosity is the property of a fluid that resists the relative motion between its layers.
The viscous force is proportional to the area of contact and the velocity gradient (dv/dx).
The coefficient of viscosity (η) represents the fluid's resistance to flow. Its SI unit is Pascal-second (Pa·s).
In liquids, viscosity decreases with an increase in temperature as cohesive forces weaken.

Stokes' Law and Terminal Velocity

Stokes’ law states that the drag force on a spherical body moving through a viscous fluid is proportional to the body's radius, velocity, and the fluid's viscosity.
The force is given by: F = 6πηrV, where r is the radius, V is the velocity, and η is the coefficient of viscosity.
Terminal velocity is the constant speed reached by a falling object when the drag force equals the gravitational force.
For a spherical body, the terminal velocity is given by: V = 2r^2(ρ-σ)g/(9η), where ρ is the density of the body, σ is the density of the fluid, and g is acceleration due to gravity.
The terminal velocity is directly proportional to the difference in densities (ρ - σ). If σ > ρ, the terminal velocity is negative, indicating an upward motion (like air bubbles in water).

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Description

Test your understanding of fluid motion equations and the stability of floating bodies. This quiz covers the derivation of motion equations for fluid elements, as well as the principles governing the stability of floating objects. Enhance your knowledge in the field of fluid mechanics.