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Understand Bernoulli's Principle and its application in relating pressure, kinetic energy per unit volume, and potential energy per unit volume in streamline flow.
Calculate fluid dynamics using the Equation of Continuity, ensuring the product of cross-sectional area and velocity remains constant in incompressible fluid flow.
Apply Pascal's Law to determine pressure transmission in enclosed fluids and its applications in hydraulic systems.
Analyze the viscous drag force on spheres using Stokes' Law and its dependence on radius, velocity, and fluid viscosity.
Explore the concept of Surface Tension as a force per unit length at the liquid interface and its implications in various phenomena.
Investigate Capillary Action and the factors influencing the rise or fall of liquids in narrow tubes.
Derive the Pressure Variation with Depth formula and apply it to calculate pressure changes in fluids due to depth.
Examine the concept of Viscosity and its role in fluid resistance to deformation or flow.
Evaluate Dynamic Lift and the Magnus Effect in the context of lift forces on bodies moving through fluids.
Utilize Torricelli's Law to determine the speed of efflux of fluids under gravity from an orifice.
Define and calculate Pressure, Density, and Relative Density, and apply these concepts in numerical problems.
Differentiate between Atmospheric Pressure, Gauge Pressure, and use a Manometer for pressure-difference calculations.
Apply Pascal's Law in Hydraulic Machines to understand mechanical advantage and force transmission.
Distinguish between Streamline, Laminar, and Turbulent Flow, and understand the significance of critical speed.
Calculate Terminal Velocity using Stokes' Law and analyze the effects of buoyancy and density differences.
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Definition: Relates pressure, kinetic energy per unit volume, and potential energy per unit volume in a streamline flow, stating their sum remains constant.
Equation: P+21ρv2+ρgh=constant
Assumptions: Applies to incompressible, non-viscous fluids in steady flow.
Applications: Explains phenomena like lift on airplane wings and the functioning of carburetors.
Equation of Continuity
Definition: For incompressible fluid flow, the product of cross-sectional area and velocity remains constant along a streamline.
Equation: A1v1=A2v2
Conservation: Represents conservation of mass in fluid dynamics.
Pascal's Law
Statement: Pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.
Applications: Basis for hydraulic lifts and hydraulic brakes.
Stokes' Law
Definition: Describes the viscous drag force on a sphere moving through a fluid.
Equation: F=6πηav
Variables:
η: Viscosity of the fluid
a: Radius of the sphere
v: Velocity of the sphere
Surface Tension
Definition: The force per unit length acting at the interface between a liquid and another medium.
Equation: Surface tension S=2lF
Phenomena: Explains capillary action and the formation of droplets.
Capillary Action
Definition: The rise or fall of a liquid in a narrow tube due to surface tension and adhesive forces.
Equation: h=ρga2Scosθ
Variables:
h: Height of the liquid column
S: Surface tension
θ: Contact angle
ρ: Density of the liquid
a: Radius of the tube
Pressure Variation with Depth
Equation: P=Pa+ρgh
Explanation: Pressure in a fluid increases with depth due to the weight of the fluid above.
Viscosity
Definition: A measure of a fluid's resistance to deformation or flow.
Equation: η=AFvl
Units: Poiseuille (Pl), N s m⁻², or Pa s
Dynamic Lift and Magnus Effect
Dynamic Lift: Force on a body moving through a fluid due to pressure differences.
Magnus Effect: Lift force on a spinning object due to differences in velocity and pressure.
Torricelli's Law
Definition: Describes the speed of efflux of a fluid under gravity from an orifice.
Equation: v=2gh
Pressure, Density and Relative Density
Pressure: P=AF
Density: ρ=Vm
Relative Density: Ratio of the density of a substance to the density of a reference substance.
Atmospheric Pressure, Gauge Pressure and Manometer
Atmospheric Pressure: Pressure exerted by the weight of the atmosphere.
Gauge Pressure: Difference between absolute pressure and atmospheric pressure.
Manometer: Device for measuring pressure differences.
Hydraulic Machines
Principle: Based on Pascal's law.
Examples: Hydraulic lift and hydraulic brakes.
Streamline, Laminar and Turbulent Flow
Streamline Flow: Flow where each particle follows a smooth path.
Laminar Flow: Smooth, orderly fluid motion.
Turbulent Flow: Chaotic, irregular fluid motion.
Terminal Velocity
Definition: The constant velocity reached by a sphere falling through a viscous medium.
Equation: vt=9η2a2(ρ−σ)g
Variables:
vt: Terminal velocity
a: Radius of the sphere
ρ: Density of the sphere
σ: Density of the fluid
g: Acceleration due to gravity
η: Viscosity of the fluid
This chapter covers the mechanical properties of fluids, focusing on principles such as Bernoulli's principle, Pascal's law, and the equation of continuity, which are fundamental to understanding fluid dynamics and applications in real-world scenarios.
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The pressure on the swimmer is calculated using the formula P=Pa+ρgh. Substituting the given values: P=1.01×105 Pa+1000 kg/m3×9.8 m/s2×10 m=2.01×105 Pa.
Chapter Concept:
Pressure Variation with Depth
A.
The velocity decreases
B.
The velocity remains constant
C.
The velocity increases
D.
The velocity becomes zero
Correct Answer: C
Solution:
According to Bernoulli's principle and the equation of continuity, A1v1=A2v2, where A is the cross-sectional area and v is the velocity. When the fluid moves from a wider section to a narrower section, the area A decreases, causing the velocity v to increase to maintain the constant flow rate.
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True or False
Correct Answer: True
Solution:
Torricelli's Law states that the speed of efflux of a fluid from an orifice is the same as the speed of a freely falling body from the same height.
Chapter Concept :
Torricelli's Law
Correct Answer: False
Solution:
Bernoulli's equation is ideally applicable to incompressible, non-viscous fluids in steady flow. It does not hold for flows with significant viscosity or turbulence, where energy is lost to friction.
Chapter Concept :
Bernoulli's Principle
Correct Answer: False
Solution:
Streamlines cannot cross each other in a steady flow because if they did, a fluid particle at the intersection would have two different velocities, which is not possible.
Chapter Concept:
Stokes' Law
A.
Gravitational force and buoyant force
B.
Gravitational force and viscous force
C.
Buoyant force and surface tension
D.
Viscous force and surface tension
Correct Answer: B
Solution:
According to Stokes' Law, a raindrop reaches terminal velocity when the gravitational force is balanced by the viscous force and the buoyant force.
Chapter Concept:
Stokes' Law
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A.
F2=F1×A1A2
B.
F2=F1×A2A1
C.
F2=F1+A2
D.
F2=F1−A1
Correct Answer: A
Solution:
In a hydraulic lift, the force exerted on the larger piston is given by F2=F1×A1A2, demonstrating the mechanical advantage provided by the lift.
Chapter Concept:
Hydraulic Machines
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A.
The product of cross-sectional area and velocity remains constant.
B.
The pressure remains constant along a streamline.
C.
The density of the fluid changes with velocity.
D.
The flow rate is inversely proportional to the cross-sectional area.
Correct Answer: A
Solution:
The equation of continuity for incompressible fluid flow states that the product of cross-sectional area and velocity remains constant along a streamline.
Chapter Concept:
Equation of Continuity
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A.
5.15×106 Pa
B.
6.16×106 Pa
C.
5.16×106 Pa
D.
6.15×106 Pa
Correct Answer: C
Solution:
The absolute pressure is given by P=Pa+ρgh. Substituting Pa=1.01×105 Pa, ρ=1.03×103 kg/m3, g=9.8 m/s2, and h=500 m, we get P=1.01×105+1.03×103×9.8×500=5.16×106 Pa.
Chapter Concept:
Pressure Variation with Depth
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A.
50 Pa
B.
100 Pa
C.
200 Pa
D.
400 Pa
Correct Answer: C
Solution:
The pressure difference ΔP for a liquid drop is given by ΔP=r2S, where S is the surface tension and r is the radius. Substituting the given values, ΔP=0.001 m2×0.05 N/m=100 N/m2=100 Pa.
Chapter Concept:
Capillary Action
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A.
h=ρga2Scosθ
B.
h=ρgaScosθ
C.
h=ρga2Ssinθ
D.
h=ρgaSsinθ
Correct Answer: A
Solution:
The height h to which the liquid rises in a capillary tube is given by h=ρga2Scosθ, where S is the surface tension, θ is the contact angle, ρ is the density of the liquid, g is the acceleration due to gravity, and a is the radius of the tube.
Chapter Concept:
Viscosity
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A.
The ball's surface roughness increases air friction.
B.
The spin creates a pressure difference due to varying air velocities above and below the ball.
C.
The ball's weight changes due to spin.
D.
The ball's temperature affects its path.
Correct Answer: B
Solution:
The Magnus effect explains that the spin of the ball drags air with it, creating a pressure difference due to varying velocities of air above and below the ball, resulting in a lift force that deviates the ball's path.
Chapter Concept:
Dynamic Lift and Magnus Effect
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A.
The pressure at point A is higher than at point B
B.
The pressure at point A is lower than at point B
C.
The pressure at point A is equal to the pressure at point B
D.
The pressure at point A is zero
Correct Answer: B
Solution:
In a horizontal flow, according to Bernoulli's principle, an increase in velocity results in a decrease in pressure. Therefore, the pressure at point A is lower than at point B.
Chapter Concept:
Bernoulli's Principle
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Chapter Concept :
Streamline, Laminar and Turbulent Flow
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Correct Answer: True
Solution:
The Magnus effect describes how a spinning ball creates a pressure difference due to varying velocities of air around it, resulting in a lift force.
Chapter Concept :
Dynamic Lift and Magnus Effect
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Correct Answer: True
Solution:
Capillary action is indeed caused by surface tension and the adhesive forces between the liquid and the tube material, leading to the rise or fall of the liquid in a narrow tube.
Chapter Concept :
Capillary Action
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Correct Answer: True
Solution:
Viscosity quantifies the internal friction in a fluid, which resists flow and deformation.
Chapter Concept :
Viscosity
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Correct Answer: True
Solution:
This statement is true as it describes the equation of continuity for incompressible fluids, which ensures mass conservation.
Chapter Concept :
Equation of Continuity
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Correct Answer: True
Solution:
Surface tension minimizes the surface area for a given volume, resulting in a spherical shape for small liquid drops.
Chapter Concept :
Surface Tension
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Correct Answer: False
Solution:
The pressure inside a spherical drop is more than the pressure outside due to surface tension. This is because the surface tension causes a higher pressure on the concave side of the liquid-air interface.
Chapter Concept :
Pressure, Density and Relative Density
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Correct Answer: True
Solution:
Viscosity quantifies the internal friction in a fluid, indicating its resistance to flow or deformation.
