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What does the equation dP = -ρa dx represent in the context of a liquid in an accelerating beaker?
In the scenario where a beaker is accelerated with components ax and ay, what happens to the pressure in the liquid?
Which of the following correctly simplifies the motion equation for a fluid element in an accelerated beaker?
What is the implication of the equation dP/dy = -ρ(g + ay) for a fluid element experiencing vertical acceleration?
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How does the acceleration of the beaker affect the pressure distribution within the liquid?
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What is the mass term used in the equation of motion for the fluid element?
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If the beaker is accelerating vertically with ay, what will dP/dy represent?
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What characteristic behavior is observed in the pressure of liquid in a moving beaker?
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What is the relationship between the centre of gravity and the metacentre for a stable floating body?
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When a boat tilts, what happens to the centre of buoyancy?
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What factors determine the stability of a floating body?
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What will happen if the centre of gravity is above the metacentre?
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In the context of a floating body, what does the term 'torque' refer to?
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Which condition must be met for a floating body to achieve equilibrium?
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What is the specific gravity of a wooden plank if it has a weight density less than the water it displaces?
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Which of the following best describes the effect of buoyancy on a floating object?
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What does Bernoulli's equation state about an ideal fluid?
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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?
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In the context of fluid dynamics, what is potential energy related to?
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What parameters are integrated in Bernoulli's equation?
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When a liquid moves through a distance due to pressure, what is the relationship to pressure energy?
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What is the correct formula for calculating pressure energy per unit volume of a liquid?
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If a liquid is at a height h, what is its potential energy per unit volume?
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In the equation $P + \frac{1}{2} \rho v^2 + \rho gh = constant$, which term represents kinetic energy?
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What is the definition of viscosity in the context of fluid mechanics?
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What does the negative sign in the viscous force equation indicate?
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How does the coefficient of viscosity change with temperature for liquids?
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What is the SI unit of the coefficient of viscosity?
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In the given liquid flow situation, what is the correct calculation of the velocity gradient?
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How is the tangential force calculated in the illustration provided?
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What happens to the relative motion between the layers of a liquid when an external force is applied?
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What is the role of the viscous force in the motion of fluid layers?
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What is the resistive force acting on a spherical body moving through a medium according to Stokes' Law?
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What happens to the terminal velocity of a spherical body when the density of the fluid is greater than that of the body?
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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?
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What is the formula to express the terminal velocity of a spherical body in terms of its density and the fluid density?
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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?
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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?
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Which equation illustrates the relationship between the resistive force and terminal velocity for a falling body?
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What principle describes why an air bubble rises in water?
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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.