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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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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.
v=2gh
B.
v=gh
C.
v=2gh
D.
v=32gh
Correct Answer: A
Solution:
Torricelli's Law states that the speed of efflux, v, from an orifice is given by v=2gh, where g is the acceleration due to gravity and h is the height of the fluid column above the orifice.
Chapter Concept:
Torricelli's Law
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True or False
Correct Answer: True
Solution:
According to Pascal's law, a change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel. This principle is used in hydraulic lifts to exert a large force with a smaller force.
Chapter Concept :
Hydraulic Machines
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: True
A.
F=6πηav
B.
F=2πηav
C.
F=4πηav
D.
F=8πηav
Correct Answer: A
Solution:
Stokes' Law states that the viscous drag force F on a sphere is given by F=6πηav, where η is the viscosity, a is the radius, and v is the velocity.
Chapter Concept:
Stokes' Law
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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
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A.
1.0 m²
B.
0.2 m²
C.
0.1 m²
D.
0.5 m²
Correct Answer: A
Solution:
The volume displacement must be equal on both sides, so A1×h1=A2×h2. Solving for A2, we get A2=h2A1×h1=0.50.02×0.1=1.0 m2.
Chapter Concept:
Hydraulic Machines
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A.
Temperature
B.
Pressure
C.
Surface tension
D.
Density
Correct Answer: A
Solution:
The viscosity of a fluid is primarily affected by temperature. As temperature increases, the viscosity of liquids generally decreases, while the viscosity of gases increases.
Chapter Concept:
Viscosity
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A.
1.01×105 Pa
B.
2.01×105 Pa
C.
1.50×105 Pa
D.
3.01×105 Pa
Correct Answer: B
Solution:
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
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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
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A.
The height remains the same.
B.
The height doubles.
C.
The height halves.
D.
The height increases by a factor of four.
Correct Answer: B
Solution:
The height of the water column in a capillary tube is inversely proportional to the radius of the tube, h=ρga2Scosθ. Halving the radius doubles the height.
Chapter Concept:
Surface Tension
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A.
Increasing the radius of the sphere
B.
Decreasing the viscosity of the fluid
C.
Reducing the velocity of the sphere
D.
Decreasing the density of the sphere
Correct Answer: A
Solution:
According to Stokes' Law, the drag force F is directly proportional to the radius a, the viscosity η, and the velocity v. Increasing the radius of the sphere will increase the drag force.
Chapter Concept:
Stokes' Law
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Solution:
Pascal's Law indeed states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel.
Chapter Concept :
Pascal's Law
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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
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Correct Answer: False
Solution:
The pressure inside a spherical drop is more than the pressure outside due to surface tension, 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
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Correct Answer: True
Solution:
Pressure is indeed a scalar quantity. Although it is defined as force per unit area, the 'force' in this context refers to the component of force normal to the surface, not a vector quantity.
Chapter Concept :
Pressure, Density and Relative Density
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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: False
Solution:
Surface tension is defined as the force per unit length acting at the interface between a liquid and another medium, not per unit area.
Chapter Concept :
Surface Tension
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Correct Answer: True
Solution:
Stokes' law describes the viscous drag force on a sphere moving through a fluid as being proportional to the sphere's radius, velocity, and the fluid's viscosity.
Chapter Concept :
Stokes' Law
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Correct Answer: True
Solution:
Capillary action is a result of surface tension and adhesive forces between the liquid and the tube material, which creates a pressure difference across the curved liquid-air interface, causing the liquid to rise.
