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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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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
A.
2gh
B.
21gh
C.
gh
D.
2gh
Correct Answer: A
Solution:
According to Torricelli's Law, the speed of efflux v=2gh for a fluid exiting a hole in a container open to the atmosphere.
Chapter Concept:
Bernoulli's Principle
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True or False
Correct Answer: True
Solution:
Viscosity quantifies the internal friction in a fluid, which resists flow and deformation.
Chapter Concept :
Viscosity
Correct Answer: False
Solution:
Bernoulli's equation is applicable to steady, incompressible flows with negligible viscosity. It does not hold for turbulent flows where velocity and pressure fluctuate.
Chapter Concept :
Bernoulli's Principle
Correct Answer: True
Solution:
The pressure in a fluid indeed increases with depth because of the weight of the fluid above, as described by the equation P=Pa+ρgh, where Pa is the atmospheric pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the depth.
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A.
12,500 N
B.
6,250 N
C.
25,000 N
D.
50,000 N
Correct Answer: A
Solution:
Using Pascal's Law, the pressure applied on the smaller piston is transmitted undiminished to the larger piston. The force exerted by the larger piston is given by F2=F1×A1A2, where A1 and A2 are the areas of the smaller and larger pistons, respectively. Therefore, F2=500×π(0.05)2π(0.25)2=12,500 N.
Chapter Concept:
Pascal's Law
A.
2 m/s
B.
4 m/s
C.
8 m/s
D.
16 m/s
Correct Answer: A
Solution:
According to the equation of continuity, A1v1=A2v2. If A1=21A2, then v2=A2A1v1=21×4=2 m/s.
Chapter Concept:
Bernoulli's Principle
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
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.
Chapter Concept:
Stokes' Law
A.
Viscosity
B.
Surface tension
C.
Density
D.
Pressure
Correct Answer: B
Solution:
Surface tension causes a liquid drop to acquire a spherical shape as it minimizes the surface area for a given volume.
Chapter Concept:
Surface Tension
A.
1000 N
B.
10000 N
C.
100 N
D.
100000 N
Correct Answer: B
Solution:
According to Pascal's law, the pressure applied to the smaller piston is transmitted equally to the larger piston. Thus, A1F1=A2F2. Solving for F2, we get F2=A1F1×A2=0.01100×1=10000 N.
Chapter Concept:
Hydraulic Machines
A.
9,800 Pa
B.
98,000 Pa
C.
1,013,000 Pa
D.
1,000 Pa
Correct Answer: B
Solution:
The pressure due to the water column is given by P=ρgh=1000×9.8×10=98,000 Pa.
Chapter Concept:
Pressure, Density and Relative Density
A.
P=Pa+ρgh
B.
P=Pa−ρgh
C.
P=Pa×ρgh
D.
P=Pa/ρgh
Correct Answer: A
Solution:
The pressure in a fluid increases with depth due to the weight of the fluid above, described by P=Pa+ρgh, where Pa is the atmospheric pressure, ρ is the fluid density, g is the acceleration due to gravity, and h is the depth.
Chapter Concept:
Pressure Variation with Depth
Chapter Concept :
Pressure Variation with Depth
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 :
Streamline, Laminar and Turbulent Flow
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
Correct Answer: False
Solution:
Stokes' Law states that the viscous drag force F on a sphere of radius a moving with velocity v through a fluid of viscosity η is given by F=6πηav. The force is proportional to the radius a, not the square of the radius.
Chapter Concept :
Stokes' Law
Correct Answer: True
Solution:
The pressure inside a spherical drop is greater than the pressure outside because the surface tension causes a pressure difference across the liquid-air interface, as described by the equation (Pi−Po)=r2S where S is the surface tension and r is the radius of the drop.
Chapter Concept :
Atmospheric Pressure, Gauge Pressure and Manometer
Correct Answer: True
Solution:
The Magnus effect occurs when a spinning ball drags air along with it, creating a pressure difference due to varying velocities of air above and below the ball, resulting in a net force that causes deviation.
Chapter Concept :
Dynamic Lift and Magnus Effect
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
Correct Answer: True
Solution:
Surface tension minimizes the surface area for a given volume, resulting in a spherical shape for small liquid drops.
