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Questions and Answers
A solid object is submerged in a fluid. Which statement accurately describes the relationship between pressure and depth?
A solid object is submerged in a fluid. Which statement accurately describes the relationship between pressure and depth?
- Pressure increases with depth in the fluid. (correct)
- Pressure decreases with depth in the fluid.
- Pressure fluctuates erratically with changes in depth.
- Pressure is independent of depth within the fluid.
Under what condition will a submerged object neither rise nor sink but remain suspended at its current depth?
Under what condition will a submerged object neither rise nor sink but remain suspended at its current depth?
- When the buoyant force is less than the object's weight.
- When the buoyant force is greater than the object's weight.
- When the object displaces its own volume of the fluid.
- When the buoyant force equals the object's weight. (correct)
Why does buoyant force act in an upward direction on an object submerged in a fluid?
Why does buoyant force act in an upward direction on an object submerged in a fluid?
- It is due to the greater pressure at the lower part of the object compared to the top. (correct)
- It is caused by the displacement of the fluid's center of gravity.
- It is a result of the surface tension of the fluid.
- It is due to the fluid's viscosity resisting the object's downward motion.
According to Archimedes' principle, what determines the magnitude of the buoyant force acting on a submerged object?
According to Archimedes' principle, what determines the magnitude of the buoyant force acting on a submerged object?
A wooden block is floating in water. If additional weight is added to the top of the block, causing it to submerge further, what happens to the buoyant force acting on it?
A wooden block is floating in water. If additional weight is added to the top of the block, causing it to submerge further, what happens to the buoyant force acting on it?
A solid cube is submerged in a liquid. What can be said about the net horizontal hydrostatic pressure force acting on the cube?
A solid cube is submerged in a liquid. What can be said about the net horizontal hydrostatic pressure force acting on the cube?
How is the buoyant force, $F_B$, on an entirely submerged body mathematically expressed, considering fluid density $\rho_f$, gravitational acceleration $g$, and the submerged volume $V_{sub}$?
How is the buoyant force, $F_B$, on an entirely submerged body mathematically expressed, considering fluid density $\rho_f$, gravitational acceleration $g$, and the submerged volume $V_{sub}$?
When an object floats in a fluid, what must be true regarding the relationship between the average density of the object ($\rho_{avg,body}$), the fluid density ($\rho_f$), and the volumes of the object ($V_{body}$) and the submerged part ($V_{sub}$)?
When an object floats in a fluid, what must be true regarding the relationship between the average density of the object ($\rho_{avg,body}$), the fluid density ($\rho_f$), and the volumes of the object ($V_{body}$) and the submerged part ($V_{sub}$)?
Under what condition does a floating body become completely submerged regarding the density ratio (fraction of volume submerged)?
Under what condition does a floating body become completely submerged regarding the density ratio (fraction of volume submerged)?
A large cubic ice block is floating in seawater. The specific gravity of ice is 0.92, and seawater is 1.025. If a 25-cm portion of the ice block extends above the surface, what is the height of the ice block below the surface rounded to the nearest cm?
A large cubic ice block is floating in seawater. The specific gravity of ice is 0.92, and seawater is 1.025. If a 25-cm portion of the ice block extends above the surface, what is the height of the ice block below the surface rounded to the nearest cm?
In fluid mechanics, what is indicated by the term 'rigid-body motion'?
In fluid mechanics, what is indicated by the term 'rigid-body motion'?
What condition regarding strain and shear force is characteristic of fluid undergoing rigid-body motion?
What condition regarding strain and shear force is characteristic of fluid undergoing rigid-body motion?
A closed container partially filled with liquid is subjected to constant linear acceleration. How does this acceleration affect the liquid within the container?
A closed container partially filled with liquid is subjected to constant linear acceleration. How does this acceleration affect the liquid within the container?
When a tanker accelerates, the fluid rushes to the back. What is the nature of the free surface of the liquid, and how does each fluid particle behave?
When a tanker accelerates, the fluid rushes to the back. What is the nature of the free surface of the liquid, and how does each fluid particle behave?
When considering the characteristics of rigid-body motion, what can be said about the shear deformation between fluid elements?
When considering the characteristics of rigid-body motion, what can be said about the shear deformation between fluid elements?
What causes the fluid level to drop at the center and rise toward the edges when a vertical cylindrical container partially filled with water is rotated about its axis?
What causes the fluid level to drop at the center and rise toward the edges when a vertical cylindrical container partially filled with water is rotated about its axis?
In the context of pressure distribution within a fluid element, what do 'body forces' refer to?
In the context of pressure distribution within a fluid element, what do 'body forces' refer to?
When analyzing the forces acting on a fluid element, according to Newton's second law, what is the acceleration when the fluid element is in static equilibrium?
When analyzing the forces acting on a fluid element, according to Newton's second law, what is the acceleration when the fluid element is in static equilibrium?
How is the total surface force ($\delta F_s$) acting on the entire element expressed in vector form, incorporating pressure gradients in all three dimensions?
How is the total surface force ($\delta F_s$) acting on the entire element expressed in vector form, incorporating pressure gradients in all three dimensions?
Considering Newton's second law of motion for a fluid acting as a rigid body, what equation expresses the relationship between pressure gradient ($\nabla P$), gravitational force ($\rho g \vec{k}$), and acceleration ($\vec{a}$)?
Considering Newton's second law of motion for a fluid acting as a rigid body, what equation expresses the relationship between pressure gradient ($\nabla P$), gravitational force ($\rho g \vec{k}$), and acceleration ($\vec{a}$)?
For a fluid at rest, which of the following statements accurately describes the pressure variation?
For a fluid at rest, which of the following statements accurately describes the pressure variation?
In a two-dimensional case with the y-component of acceleration ($a_y$) equal to zero, a surface of constant pressure in a fluid:
In a two-dimensional case with the y-component of acceleration ($a_y$) equal to zero, a surface of constant pressure in a fluid:
What is the equation for the density as a function position $\rho(x,y,z)$ for a fluid in hydrostatic equilibrium?
What is the equation for the density as a function position $\rho(x,y,z)$ for a fluid in hydrostatic equilibrium?
Which parameter primarily determines fluid distribution, stability, and spill risk in designing a tank subjected to acceleration?
Which parameter primarily determines fluid distribution, stability, and spill risk in designing a tank subjected to acceleration?
In a scenario where a body is in free fall, what condition applies regarding its acceleration and the pressure within the body?
In a scenario where a body is in free fall, what condition applies regarding its acceleration and the pressure within the body?
A truck with a tank that has a cross section of 2 m x 0.6 m accelerates from 0 to 90 km/h in 10 seconds. What is the tangent of the angle that the new free surface makes with the original surface?
A truck with a tank that has a cross section of 2 m x 0.6 m accelerates from 0 to 90 km/h in 10 seconds. What is the tangent of the angle that the new free surface makes with the original surface?
A rectangular tank with a width of 1m, a length of 2m and a height of 1.5m contains gasoline to a depth of 1m. What is the maximum horizontal acceleration that can develop along the length of the tank before the gasoline begins to spill?
A rectangular tank with a width of 1m, a length of 2m and a height of 1.5m contains gasoline to a depth of 1m. What is the maximum horizontal acceleration that can develop along the length of the tank before the gasoline begins to spill?
Which of the following is a correct statement about buoyancy?
Which of the following is a correct statement about buoyancy?
Which of the following statements is true about pressure in a static fluid?
Which of the following statements is true about pressure in a static fluid?
What is the primary reason for the occurrence of buoyancy?
What is the primary reason for the occurrence of buoyancy?
When is the buoyant force on an object the greatest:
When is the buoyant force on an object the greatest:
Which of the following statements best describes the condition for an object to float in a fluid?
Which of the following statements best describes the condition for an object to float in a fluid?
What is another term used to describe volume-submerged ratio?
What is another term used to describe volume-submerged ratio?
A container containing a liquid is moved with a constant force. What will happen to the acceleration?
A container containing a liquid is moved with a constant force. What will happen to the acceleration?
Linear acceleration is due to what force?
Linear acceleration is due to what force?
Free surface of a liquid in a constant horizontal tank is also known as:
Free surface of a liquid in a constant horizontal tank is also known as:
For a two-dimensional view and the y component cancels, what equation is used to get the surface of the contant-pressure?
For a two-dimensional view and the y component cancels, what equation is used to get the surface of the contant-pressure?
In the design of tanks and the slope of the free surface are created, what parameter(s) are important to note?
In the design of tanks and the slope of the free surface are created, what parameter(s) are important to note?
Flashcards
What is Buoyancy?
What is Buoyancy?
Ability of an object to float in a fluid.
What is Buoyant Force?
What is Buoyant Force?
The upward force exerted by a fluid that opposes the weight of an immersed object.
When will an object float?
When will an object float?
A solid object will float is density is less than fluid's.
What is Archimedes' Principle?
What is Archimedes' Principle?
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Hydrostatic pressure forces.
Hydrostatic pressure forces.
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What is Buoyant Force
What is Buoyant Force
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Submerged Volume (Vsub)
Submerged Volume (Vsub)
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Floating Condition
Floating Condition
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When is body fully submerged.
When is body fully submerged.
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Rigid body motion.
Rigid body motion.
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Fluid behavior under Acceleration
Fluid behavior under Acceleration
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No Shear Strain
No Shear Strain
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No Velocity Gradient
No Velocity Gradient
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Linear Acceleration (Tanker example)
Linear Acceleration (Tanker example)
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Forces on Fluid Element
Forces on Fluid Element
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Body Force
Body Force
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Rigid Body Motion Equation
Rigid Body Motion Equation
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Case: Fluids at Rest
Case: Fluids at Rest
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Linear Acceleration
Linear Acceleration
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Spillage Limit Factors
Spillage Limit Factors
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Study Notes
- Fluid Mechanics PE 255 Lecture 2b covers fluid statics and surface forces
Class Objectives:
- Define buoyancy and its applications
- Derive the governing equation for fluids in rigid motion
Buoyancy and Stability: Archimedes' Principle
- A solid body in a fluid will float, remain at rest, or sink
- Pressure increases with depth in a fluid
- Upward force on the bottom of an object in a fluid exceeds downward force on top
- Buoyant force is the upward resultant net force
- If buoyant force overcomes object weight, the object floats
- If buoyant force is less than object weight, the object sinks
- If buoyant force equals object weight, the object remains suspended
- Buoyant force is always present
- Buoyant force direction is upward due to greater pressure at the lower part of the object
- Buoyant force counteracts the weight of the object
- Buoyant force equals the weight of the liquid displaced by the submerged body of volume ∀
- Archimedes' principle describes this phenomenon
- Bouyant force on submerged body equals the liquid weight displaced by the body
- It acts vertically upward through the centroid of displaced volume
Net Force
- Hydrostatic pressure forces act on the entire surface of an arbitrary shaped solid body completely submerged in a homogenous liquid
- The resultant horizontal force in any direction for a closed surface is zero
- The body is divided into elementary vertical prisms to calculate the vertical resultant force component
- The vertical forces on a prism with cross section dAz ends are dF1 and dF2
- Buoyant force represents a net vertically upward force on an elemental plasm
- Integrating this force over the entire submerged body defines the total buoyant force, FB
Submerged Volume
The volume of the body in the fluid is Vsub
- When an object is floating, the weight of the entire body must equal the buoyant force acting on it
- This ensures the object neither rises nor sinks, and remains at a constant depth
- For fully submerged objects, submerged body volume (Vsub) equals the object's volume
Density ratio
- When the density ratio (fraction of volume submerged) is one or greater, the floating body fully submerges
Problem
- Example determining the height of an ice block floating in sea water
Fluids Under Rigid-Body Motion
- Rigid-body motion encompasses fluid movement with uniform velocity and without deformation
- This motion category applies when the fluid behaves like a solid, with no strain, relative motion nor shear force
- A liquid container moving causes initial relative liquid particle movement
- After some time, there is no relative motion between liquid particles or container boundaries
- Acceleration causes the liquid to adopt a new position in its container
- The entire fluid mass shifts as a single unit
- Hydrostatic laws apply to determine liquid pressure, once the liquid attains a static position relative to the container
Characteristics of Rigid-Body Motion
- In rigid-body motion, there is no shear deformation because relative velocities in adjacent layers are zero
- Since every fluid particle has the same velocity, the velocity gradient is zero, generating no shear stress
- In an accelerating tanker, the fluid rushes to the back, causing initial splashing due to inertia
- A non-horizontal free surface appears; each particle accelerates alongside the tanker
- Since the body does not deform, there is no change in shape
- The equation for the free liquid surface can be derived by considering the equilibrium of a fluid element lying on the free surface.
- Rigid-body motion happens when a fluid is contained within a rotating body
- A vertical cylindrical container that is partially filled with water establishes rigid body motion when rotated about its axis
- The fluid level descends at the center and ascends towards the edges
- Centrifugal force causes the fluid at the center to push outward, resulting in the fluid level drop
Pressure Distribution
- Differential rectangular fluid element consideration with dx, dy and dz side lengths oriented in the x, y and z directions
- The z-axis is upward in the vertical direction
- It states that Newton's second law of motion can be expressed as δF = δm ⋅ a
- Body forces, like gravity, encompass the element's complete mass, contributing weight
- Surface forces as pressure directly act on element surfaces from surrounding fluids, normal to the unrestrained surface
Total Force
- Pressure at the center of the element (P), top and bottom surface pressures are expressed as (P + ∂P/∂z dz/2) and (P – ∂P/∂z dz/2)
- Net force yields -∂P/∂z dxdydz in the z-direction
- The net pressures in the x- and y- directions are -𝛿P/𝛿xdxdydz and -𝛿P/𝛿ydxdydz
- Surface force is expressed in vector form where ỉ, and kare length unit vectors
- Body force represents the element's weight in the negative z-direction
- Total force acting on the fluid element is: δF = δFs + δFB = −(∇P + ρgk)dxdydz
Rigid Body Motion
- The general fluid equation of motion, acting as a rigid body without shear stresses is
- With force due to pressure, 𝛻P + pgk = −pa, Equation for Rigid-body motion of fluids
- Vector relations are expanded into their components, this relation can be more explicitly expressed
- PRESSURE DISTRIBUTION IN EACH DIRECTION
Fluids at Rest
- Acceleration components are zero
- Pressure remains constant in any horizontal direction but varies only in the vertical
Linear Uniform Acceleration
- Angle of Inclination of Constant-pressure Surface is used for Linear Acceleration
- Horizontal acceleration generally results in tilted isobaric surfaces, and the unrestricted surface where p = patm (an explicit isobaric surface) tilts similarly with uniform acceleration
- The slope of the unrestrained surface or even the closed tank is: Tanθ = ax /g
- This slope dictates fluid distribution, stability, alongside spill risk, crucial when analysing design also safety
- As a/g is constant, so is Tan θ
Free Fall
- A freely falling body accelerates under the influence of gravity
- Negligible air resistance means that a body accelerates the acceleration of the body equates to gravitational acceleration, plus there is zero-acceleration in one horizontal direction
- The derived equation relates to motion of fluid becomes-zero
- In this case, pressure = constant
Problems
- Example problems determining acceleration and spill issues
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