This lesson explores the fundamental properties of fluids, including the molecular differences between solids, liquids, and gases, the concept of density, and the characteristics of ideal fluids. Students will learn to distinguish states of matter through atomic interactions and apply the density formula to various substances.
Pressure describes the concentration of force. It is defined as the magnitude of the perpendicular force component (\(F_{\perp}\)) exerted per unit area (\(A\)) over a given surface.
THE PRESSURE EQUATION
\[ P = \frac{F_{\perp}}{A} \]
Pressure is a Scalar Quantity
Unlike force (which is a vector), pressure is a scalar quantity. It has magnitude but no specific direction associated with the value itself; it acts in all directions within a fluid.
6. Particle Interactions
Where does this pressure come from? In a fluid, pressure is the result of the entirety of the interactions between the fluid's constituent particles (atoms/molecules) and the surface with which they interact.
Microscopic View
Every time a molecule bounces off a container wall, it exerts a tiny force. When we sum up billions of these tiny collisions over a specific area, we experience it as macroscopic pressure.
7. Absolute vs. Gauge Pressure
When measuring pressure, we often care about the difference between the pressure inside a system and the pressure outside (like the atmosphere).
Gauge Pressure (\(P_{gauge}\)): The pressure relative to atmospheric pressure. If your tire gauge reads 32 psi, that is the extra pressure above the atmosphere.
Absolute Pressure (\(P\)): The actual, total pressure. It is the sum of a reference pressure (\(P_0\)) and the gauge pressure.
ABSOLUTE PRESSURE EQUATION
\[ P = P_0 + P_{gauge} \]
Often \(P_0\) is atmospheric pressure (\(P_{atm}\))
8. Pressure in a Vertical Column
The deeper you go in a fluid, the higher the pressure. This is because there is more fluid above you pushing down. The gauge pressure of a vertical column of fluid is calculated using its density (\(\rho\)), gravity (\(g\)), and height (\(h\)).
\[ P_{gauge} = \rho gh \]
Reading Check: Pressure
3. Explain why pressure is considered a scalar quantity, even though force is a vector.
4. A diver is 10 meters underwater. Explain how the "absolute pressure" on the diver is calculated using both atmospheric pressure and gauge pressure.
5. What causes fluid pressure at a microscopic level?
"Fluid pressure is the macroscopic manifestation of microscopic particle-surface interactions."
Concept Check: Pressure Origins
3. Explain the relationship between particle collisions and macroscopic pressure.
4. Why must the force in the pressure equation be perpendicular to the surface?
7. Understanding Pressure Scales
We must distinguish between the Absolute total and the Gauge measurement.
Atmosphere
Gauge
Absolute Pressure (\(P\))
Absolute Zero (Vacuum)
\[ P = P_{atm} + P_{gauge} \]
Absolute: Total relative to vacuum.
Gauge: Reading relative to atmosphere.
Example: A Flat Tire
When a tire is "flat," its gauge pressure is 0. However, its absolute pressure is about \(1\text{ atm}\) (\(101,325\text{ Pa}\)) because it is still filled with air at the same pressure as the surrounding atmosphere.
8. Pressure in a Vertical Column
As you descend, pressure increases due to the weight overhead.
Vertical Gauge Pressure
\[ P_{gauge} = \rho gh \]
\[ P_{absolute} = P_{atm} + \rho gh \]
Comprehensive Review
5. Calculate pressure for a 500 N force acting perpendicularly on a 2 m² surface.
6. Scuba gauge reads 200 kPa. What is absolute pressure? (\(P_{atm} = 101.3\text{ kPa}\))
7. Using the microscopic view, explain why squeezing a balloon increases internal pressure.
8. How does the "Ideal Fluid" model treat density at extreme depths?
Concept Check: Pressure Origins
3. Explain the relationship between particle collisions and macroscopic pressure.
4. Why must the force in the pressure equation be perpendicular to the surface?
7. Understanding Pressure Scales
We must distinguish between the Absolute total and the Gauge measurement.
Atmosphere
Gauge Read
Absolute Pressure (\(P\))
Absolute Zero (Vacuum)
The Essential Sum
\[ P = P_{atm} + P_{gauge} \]
Absolute Pressure: Total pressure relative to absolute zero. Always positive.
Gauge Pressure: Measured relative to atmosphere. Zero if internal = external.
Example: A Flat Tire
When a tire is "flat," its gauge pressure is 0. However, its absolute pressure is about \(1\text{ atm}\) (\(101,325\text{ Pa}\)) because it is still filled with air at the same pressure as the surrounding atmosphere.
8. Pressure in a Vertical Column
As you descend, pressure increases due to the weight overhead. For an incompressible fluid, the gauge pressure depends on depth (\(h\)), density (\(\rho\)), and gravity (\(g\)).
Vertical Gauge Pressure
\[ P_{gauge} = \rho gh \]
Absolute Column Pressure
\[ P_{absolute} = P_{atm} + \rho gh \]
Comprehensive Review
5. Calculate the pressure exerted by a 500 N force acting perpendicularly on a 2 m² surface.
6. If a scuba diver's gauge reads 200 kPa, what is the absolute pressure on the diver? (\(P_{atm} = 101.3\text{ kPa}\))
7. Using the microscopic view, explain why squeezing a balloon (decreasing its volume) increases the internal pressure.
8. How does the "Ideal Fluid" model treat density when calculating vertical pressure at extreme depths?
Billions of Collisions
The Cumulative Effect
Every molecule collision exerts a tiny impulse. Billions of these combined create a measurable macroscopic pressure.
"Fluid pressure is the macroscopic manifestation of microscopic particle-surface interactions."
Concept Check: Pressure Origins
3. Explain the relationship between particle collisions and macroscopic pressure.
4. Why must the force in the pressure equation be perpendicular to the surface?
7. Understanding Pressure Scales
When measuring pressure, we must distinguish between Absolute total and Gauge measurement relative to the atmosphere.
Atmosphere
Gauge Read
Absolute (\\(P\\))
Absolute Zero (Vacuum)
The Essential Sum
\\[ P = P_{atm} + P_{gauge} \\]
Absolute Pressure: The actual total pressure relative to absolute zero. It is always positive.
Gauge Pressure: Pressure relative to the local atmospheric pressure. This is what most gauges read.
Example: A Flat Tire
When a tire is "flat," its gauge pressure is 0. However, its absolute pressure is about \\(1\text{ atm}\\) (\\(101,325\text{ Pa}\\)) because it is still filled with air at atmospheric pressure.
8. Pressure in a Vertical Column
As you descend, pressure increases due to the weight overhead. For an incompressible fluid, the gauge pressure depends on depth (\\(h\\)), density (\\(\rho\\)), and gravity (\\(g\\)).
Vertical Gauge Pressure
\\[ P_{gauge} = \rho gh \\]
Absolute Column Pressure
\\[ P_{absolute} = P_{atm} + \rho gh \\]
Comprehensive Review
5. Calculate the pressure exerted by a 500 N force acting perpendicularly on a 2 m² surface.
6. If a scuba diver's gauge reads 200 kPa, what is the absolute pressure on the diver? (\\(P_{atm} = 101.3\text{ kPa}\\))
7. Using the microscopic view, explain why squeezing a balloon (decreasing its volume) increases the internal pressure.
8. How does the "Ideal Fluid" model treat density when calculating vertical pressure at extreme depths?
The Cumulative Effect
Every molecule collision exerts a tiny impulse. When billions of these impacts occur, they create a steady macroscopic pressure.
"Fluid pressure is the macroscopic manifest of microscopic particle-surface interactions."
Concept Check: Pressure Origins
3. Explain the relationship between particle collisions and macroscopic pressure.
4. Why must the force in the pressure equation be perpendicular to the surface?
6. Absolute vs. Gauge Pressure
We must distinguish between the Absolute total pressure and the Gauge measurement relative to atmospheric levels.
ATM
GAUGE
Absolute (P)
Vacuum (0)
\[ P = P_{atm} + P_{gauge} \]
Absolute Pressure: The total pressure relative to a vacuum. Always positive.
Gauge Pressure: The difference between absolute and atmospheric pressure.
Example: A Flat Tire
When a tire is "flat," its gauge pressure is 0. However, its absolute pressure is about \(1\text{ atm}\) (\(101,325\text{ Pa}\)) because it is still filled with air at atmospheric pressure.
7. Pressure in a Vertical Column
As you descend into a fluid, pressure increases due to the weight overhead. For an incompressible fluid, the gauge pressure depends on depth (\(h\)), density (\(\rho\)), and gravity (\(g\)).
Vertical Gauge Pressure
\[ P_{gauge} = \rho gh \]
Absolute Column Pressure
\[ P_{absolute} = P_{atm} + \rho gh \]
Comprehensive Review
5. Calculate the pressure exerted by a 500 N force acting perpendicularly on a 2 m² surface.
6. If a scuba diver's gauge reads 200 kPa, what is the absolute pressure on the diver? (\(P_{atm} = 101.3\text{ kPa}\))
7. Using the microscopic view, explain why squeezing a balloon (decreasing its volume) increases the internal pressure.
8. How does the "Ideal Fluid" model treat density when calculating vertical pressure at extreme depths?
Non-Viscous
There is zero internal friction (viscosity). The fluid flows perfectly smoothly without losing energy as heat. Real water has a little viscosity; honey has a lot!
4. Defining Pressure: Concentrating Force
Pressure is often confused with force, but they are very different. While force is a push or pull, pressure describes how that push is distributed across a surface.
The Snowshoe Analogy
If you walk on deep snow in boots, you sink because your weight (force) is concentrated on a small area. If you wear wide snowshoes, your force is distributed over a massive area, resulting in lower pressure, allowing you to stay on top of the snow.
Contact Area (\(A\))
\(F_{\perp}\)
The Pressure Equation
\[ P = \frac{F_{\perp}}{A} \]
Unit: Pascal (Pa)
\(1\text{ Pa} = 1\text{ Newton / meter}^2\)
Perpendicular Force
Only the component of force that is 90 degrees to the surface creates pressure. Forces acting parallel to the surface (shear forces) do not count toward this calculation.
Pressure is a Scalar
Force has a direction, but pressure does not. In a static fluid, pressure acts equally in all directions. If you poke a hole in a water bottle, the water shoots out sideways, downwards, or even upwards if the hole is on the top.
5. The Microscopic Engine: Particle Impacts
When we feel pressure, we are actually feeling the collective momentum change of billions of particles. Every time a molecule strikes a wall and bounces off, it undergoes an impulse—it exerts a tiny force on the wall for a tiny fraction of a second.
Impulse = \(\Delta p\) / \(\Delta t\)
The Momentum Transfer
The force of a single collision is negligible. However, in a typical container, there are trillions of collisions per square millimeter every second. This cumulative force is so frequent and dense that we perceive it as a smooth, constant macroscopic pressure.
"Fluid pressure is the macroscopic manifestation of countless microscopic particle-surface interactions."
Effect of Temperature
As temperature increases, particles move faster. Faster particles strike walls with more momentum and more frequency, which is why heating a gas increases its pressure.
Effect of Concentration
Adding more particles to a fixed volume increases the frequency of collisions. More collisions per second directly leads to higher macroscopic pressure.
6. Understanding Pressure Scales
Imagine you have an empty tire. A pressure gauge reads "0 psi." But is the tire actually empty? No! It is full of air at atmospheric pressure. To solve physics problems, we must distinguish between two different starting points for our measurements.
Ref
\(P_0\)
Atmosphere
Added
\(P_{gauge}\)
Gauge Read
Absolute (P)
Absolute Zero (Vacuum)
The Absolute Relation
\[ P = P_0 + P_{gauge} \]
Absolute Pressure: The total, true pressure. It counts all collisions, starting from a perfect vacuum. It can never be negative.
Gauge Pressure: The "extra" pressure above local atmospheric levels. It's what your tires or scuba gear measure. If it's negative, it means a partial vacuum.
7. Pressure in a Vertical Column: Weight over Area
Why does pressure increase as you go deeper into a fluid? Because you are supporting the weight of everything above you. Imagine a vertical column of fluid with cross-sectional area (\(A\)) and height (\(h\)).
Notice that Area (\(A\)) cancels out! The pressure only depends on depth, density, and gravity.
Gauge Pressure Formula
\[ P_{gauge} = \rho gh \]
For Absolute Pressure at Depth:
\[ P = P_{atm} + \rho gh \]
Important: This formula only works for incompressible fluids (where \(\rho\) is constant). In a gas, the bottom layers are compressed by the layers above, making the density higher at the bottom and lower at the top!
Deep Understanding Check
Explain why an "ideal fluid" model is useful for engineering, even if real fluids have viscosity. Give a concrete example.
Derive the unit of Pascal (\(Pa\)) using the base SI units of mass (\(kg\)), length (\(m\)), and time (\(s\)). Hint: \(F = ma\).
A scuba diver is at a depth of 20 meters in seawater (\(\rho = 1025\text{ kg/m}^3\)). If atmospheric pressure is \(101,300\text{ Pa}\), calculate both the gauge and absolute pressure. Show all units.
Reflection Question
"If pressure is a scalar, how is it possible that it results from vector forces (particle impacts)? How does the macroscopic view 'average out' the microscopic directions?"
2. Mass = \(\rho\)V
3. Volume = Ah
4. P = Weight / A
\[ P_{gauge} = \rho gh \]
Incompressible: Density must be constant for this to work!
Max
Low
Depth (h)
The Ideal Fluid: Why We Use It
Incompressible
Density remains uniform. Volume never changes under pressure. Simplifies the conservation of mass.
Non-Viscous
Zero internal friction. Layers slide like perfect ice. Simplifies energy conservation (Bernoulli).
Simplifying Fluid Dynamics
Density Relationship
Force Relationship
Resulting Motion
\(\rho_{obj} > \rho_{fluid}\)
Weight > \(F_B\)
Sinks
\(\rho_{obj} < \rho_{fluid}\)
Weight < \(F_B\) (max)
Rises / Floats
\(\rho_{obj} = \rho_{fluid}\)
Weight = \(F_B\)
Neutral Buoyancy
4. The Physics of Floating and Sinking
Case 1: Sinking
An object sinks when its weight is greater than the maximum possible buoyant force the fluid can provide. This happens when the object is more dense than the fluid.
\(F_B\)
\(F_g = mg\)
NET FORCE IS DOWNWARD (\(F_{net} < 0\))
Case 2: Floating
An object floats when it reaches equilibrium. It sinks just deep enough to displace exactly enough fluid weight to match its own weight.
\(F_B\)
\(F_g = mg\)
NET FORCE IS ZERO (\(F_{net} = 0\))
Critical Explanation: The Floating Equilibrium
When an object is placed on water, if it is less dense than water, the buoyant force starts off larger than the weight. The object is pushed up until part of it is out of the water. As it leaves the water, the volume displaced decreases. This decreases the buoyant force until finally, \(F_B = F_g\). At this point, the object stops moving and floats steadily.
5. The "Eureka" Theory: A Deep Dive
To understand Archimedes' Principle fully, we must look at why the "Weight of the Displaced Fluid" matters. Imagine a container of water. If you replace a "cube" of that water with an identical cube of wood, the wood feels the exact same pressure forces from the surrounding water that the "cube" of water felt.
If that cube of water was just sitting there, it was in equilibrium. This means the water around it was pushing up with a force exactly equal to the weight of that water.
Theoretical Proof:
Upward force from fluid = Weight of water that would be there.
When you put an object in, the fluid doesn't know the difference.
It still pushes up with the same force: the weight of the fluid that was displaced.
Surrounding Pressure
Upward Support
Displaced
Volume (V)
Practical Application: Apparent Weight
When you weigh an object submerged in water, the scale shows a smaller number than in air. This is called the Apparent Weight (\(W_{app}\)).
\[ W_{app} = F_g - F_B \]
Apparent Weight = Actual Weight - Buoyant Force
Reading Mastery Check
1. Use Archimedes' Principle to explain why a heavy piece of steel (density \(\approx 7800\text{ kg/m}^3\)) can be used to make a ship that floats on water (density \(\approx 1000\text{ kg/m}^3\)).
2. Draw a Free Body Diagram for an object that is "neutrally buoyant" (it neither sinks nor rises).
Sketch Here
Hint: Consider the relative lengths of the \(F_B\) and \(F_g\) vectors and the resulting Net Force (\(F_{net}\)).
3. A stone has a mass of \(2\text{ kg}\) and a volume of \(0.0005\text{ m}^3\). If submerged in water (\(\rho = 1000\text{ kg/m}^3\)), what is the buoyant force acting on it? Does it sink or float?
Show all mathematical steps:
4. Summarize: How does the microscopic pressure difference between the top and bottom of an object lead to the macroscopic buoyant force?
5. The Microscopic Engine: Particle Impacts
At the molecular level, pressure is the collective momentum change of billions of particles striking a surface. Every time a molecule bounces off a wall, it exerts a tiny impulse.
Trillions of Collisions / Sec
Collective Impulse
While one collision is negligible, the cumulative force of billions of particles creates the smooth macroscopic pressure we measure with a gauge.
Checkpoint 2
3. Explain why heating a gas increases its internal pressure using the microscopic view.
4. Why must the force in the pressure formula be perpendicular to the surface?
6. Understanding Pressure Scales
Measurements can be relative or absolute. A tire gauge reads "0 psi" when a tire is flat, but the tire is still filled with air at atmospheric pressure (\(P_{atm}\)).
ATM
GAUGE
Scale starts at Vacuum (0)
\[ P = P_{atm} + P_{gauge} \]
Absolute Pressure: Total pressure from a perfect vacuum. Always positive.
Gauge Pressure: Measured relative to atmosphere. Zero means internal = external.
Example: The Scuba Diver
A diver's gauge might show 100 kPa of water pressure. To find the total pressure crushing the diver, you must add the 101.3 kPa of atmospheric pressure pressing down on the water's surface!
7. Pressure in a Vertical Column
As you descend, pressure increases linearly because you are supporting the weight of the fluid overhead.
The Derivation
1. Force (Weight) = \(m \cdot g\)
2. Mass = \(\rho \cdot V\)
3. Volume = \(A \cdot h\)
4. Pressure = \(F / A\)
Result: \(P = \rho gh\)
Area (A) cancels out, meaning pressure depends only on depth, density, and gravity.
Hydrostatic Formula
\[ P_{gauge} = \rho gh \]
Absolute at Depth:
\[ P = P_{atm} + \rho gh \]
Final Assessment
Calculate the pressure (in Pascals) exerted by a 800 N person standing on one foot with an area of \(0.02\text{ m}^2\).
A diver is at 30m in fresh water (\(\rho = 1000\text{ kg/m}^3\)). Calculate the absolute pressure in kPa. (\(P_{atm} = 101.3\text{ kPa}\), \(g=9.8\text{ m/s}^2\)).
Summarize the "Substitute Fluid" theory: Why can we replace an object with fluid to find the buoyant force? (Think about the pressure fields).
Reflection
"If density is a bulk property, why do we treat gases as compressible but liquids as incompressible in our ideal fluid model? What happens to the molecular spacing in each case?"
Net acceleration is downward.
NEUTRAL
Density Object = Density Fluid
\(F_g = F_B\)
Object hovers at any depth.
FLOATING
Density Object < Density Fluid
\(F_g = F_B\) (Equilibrium)
Volume sub < Volume object.
The Fraction Submerged
For a floating object, the percentage of the object that stays underwater depends entirely on the ratio of densities:
When an object is submerged, the buoyant force acts as an "upward assist," making the object seem lighter. This "measured" weight is called the Apparent Weight (\(F_{g,app}\)).
Fundamental Equation
\[ F_{g,app} = F_g - F_B \]
If \(F_{g,app}\) becomes 0, the object is floating!
SCALE (12 N)
The Theory Diagram: Fluid Replacement
Archimedes realized that a fluid provides the exact same upward support to an object that it would have provided to the fluid that used to be in that same space.
WATER
Weight = 10 N
OBJECT
\(F_B = 10\) N (Upward)
Mastering the Math: Practice Problem
The Problem
A solid gold crown has a mass of 1.5 kg. When it is fully submerged in water, it displaces \(0.00008\text{ m}^3\) of water. What is the apparent weight of the crown?
Problem Constants & Givens
Mass (\(m\)) = 1.5 kg
Volume Displaced (\(V\)) = 0.00008 m³
Water Density (\(\rho\)) = 1000 kg/m³
Gravity (\(g\)) = 9.8 m/s²
Show your mathematical procedure and final reasoning below:
Final Result:
N
Reading Mastery Check
1. A ship is made of steel, which is denser than water, yet it floats. Explain this using the "Weight of Displaced Fluid" concept.
2. Draw the FBD for a submarine that is hovering at a constant depth of 50m.
Sketch Here
Include labels for \(F_B\) and \(F_g\). What is the relationship between their magnitudes?
3. A beach ball is held underwater. When released, it accelerates toward the surface. Draw the FBD at the moment it is released (while still fully submerged) and explain why it moves upward.
4. Summarize Archimedes' Principle in exactly one sentence using the terms "buoyant force" and "fluid weight."
Simplifying Fluid Dynamics
\[ F_{net} = 0 \implies F_B = F_g \]
CASE: SINKING
Occurs when \(F_g > F_B\). This happens when the object's average density is higher than the fluid's density.
RESULT: NEGATIVE NET FORCE (DOWN)
4. Why Shapes Matter: Steel Ships
A solid block of steel sinks because its density (\(7,800\text{ kg/m}^3\)) is far greater than water (\(1,000\text{ kg/m}^3\)). However, if we reshape that steel into a hollow boat, we increase the displaced volume without increasing the mass.
Master Concept: Effective Density
"By including a large volume of air inside the hull, the ship's average density (\(m_{total} / V_{total}\)) becomes lower than the density of water. The ship floats because it can displace a weight of water equal to its own weight before it is fully submerged."
AIR
Increased Volume = Increased Max \(F_B\)
Technical Review
1. Use the pressure gradient theory to explain why the buoyant force is always directed upward.
2. If an object is "Neutrally Buoyant," what can you conclude about the relationship between its density and the fluid's density?
3. Why does the side-to-side (horizontal) pressure not result in a net horizontal force on a submerged sphere?
4. Synthesis: A piece of aluminum (\(2.7\text{ g/cm}^3\)) and a piece of lead (\(11.3\text{ g/cm}^3\)) both have a volume of exactly \(100\text{ cm}^3\). Which one experiences a greater buoyant force when fully submerged in water? Explain.
Density remains uniform. Volume never changes under pressure.
Non-Viscous
Zero internal friction. Layers slide perfectly without energy loss.
Simplifying Fluid Dynamics
Deep Reflection
"If you could remove the Earth's gravity, would an object still experience a buoyant force in a fluid? Why or why not?"
\(F_B = W_{parcel}\)
1. Static Equilibrium
A parcel of fluid is supported by its neighbors with a force exactly equal to its own weight.
SOLID
2. The "Solid" Replacement
Replace the parcel with a solid object of the same volume. The surrounding fluid still pushes with the same force.
BUOYANT FORCE
\(F_B = W_{displaced}\)
3. The Archimedes Identity
The solid object experiences an upward force equal to the weight of the fluid that would have been there.
Key Insight: The buoyant force is a property of the fluid and the volume displaced. It does not matter if the object is gold, wood, or lead; if the volume is the same, the buoyant force is the same.
4. Net Force and Equilibrium
The behavior of an object (rising, sinking, or floating) depends on the Net Force (\(F_{net}\)), which is the vectorial sum of Gravity and Buoyancy.
Case A: Sinking (\(\rho_{obj} > \rho_{f}\))
The object's weight is greater than the maximum possible buoyant force. Acceleration is downward.
\(F_g > F_B\)
Case B: Floating (\(\rho_{obj} < \rho_{f}\))
The object rises until it breaks the surface. It then settles at a depth where \(F_B\) perfectly equals \(F_g\).
\(F_g = F_B\)
Technical Synthesis
1. Use the diagram on Page 1 to explain why an object's weight has no effect on the buoyant force it experiences.
2. An iron anchor and a giant hollow iron ship both have the same mass. Why does the anchor sink while the ship floats? Use the terms Effective Density and Displaced Volume.
BUOYANCY
Identical Volume = Identical Upward Push
Net acceleration is downward.
NEUTRAL
Density Object = Density Fluid
\(F_g = F_B\)
Object hovers at any depth.
FLOATING
Density Object < Density Fluid
\(F_g = F_B\) (Equilibrium)
Volume sub < Volume object.
The Fraction Submerged
For a floating object, the percentage of the object that stays underwater depends entirely on the ratio of densities:
When an object is submerged, the buoyant force acts as an "upward assist," making the object seem lighter. This "measured" weight is called the Apparent Weight (\(F_{g,app}\)).
Fundamental Equation
\[ F_{g,app} = F_g - F_B \]
If \(F_{g,app}\) becomes 0, the object is floating!
SCALE (12 N)
The Theory Diagram: Fluid Replacement
Archimedes realized that a fluid provides the exact same upward support to an object that it would have provided to the fluid that used to be in that same space.
WATER
Weight = 10 N
OBJECT
\(F_B = 10\) N (Upward)
Mastering the Math: Practice Problem
The Problem
A solid gold crown has a mass of 1.5 kg. When it is fully submerged in water, it displaces \(0.00008\text{ m}^3\) of water. What is the apparent weight of the crown?
Problem Constants & Givens
Mass (\(m\)) = 1.5 kg
Volume Displaced (\(V\)) = 0.00008 m³
Water Density (\(\rho\)) = 1000 kg/m³
Gravity (\(g\)) = 9.8 m/s²
Show your mathematical procedure and final reasoning below:
Final Result:
N
Reading Mastery Check
1. A ship is made of steel, which is denser than water, yet it floats. Explain this using the "Weight of Displaced Fluid" concept.
2. Draw the FBD for a submarine that is hovering at a constant depth of 50m.
Sketch Here
Include labels for \(F_B\) and \(F_g\). What is the relationship between their magnitudes?
3. A beach ball is held underwater. When released, it accelerates toward the surface. Draw the FBD at the moment it is released (while still fully submerged) and explain why it moves upward.
4. Summarize Archimedes' Principle in exactly one sentence using the terms "buoyant force" and "fluid weight."
Net acceleration is downward.
NEUTRAL
Density Object = Density Fluid
\(F_g = F_B\)
Object hovers at any depth.
FLOATING
Density Object < Density Fluid
\(F_g = F_B\) (Equilibrium)
Volume sub < Volume object.
The Fraction Submerged
For a floating object, the percentage of the object that stays underwater depends entirely on the ratio of densities:
When an object is submerged, the buoyant force acts as an "upward assist," making the object seem lighter. This "measured" weight is called the Apparent Weight (\(F_{g,app}\)).
Fundamental Equation
\[ F_{g,app} = F_g - F_B \]
If \(F_{g,app}\) becomes 0, the object is floating!
SCALE (12 N)
The Theory Diagram: Fluid Replacement
Archimedes realized that a fluid provides the exact same upward support to an object that it would have provided to the fluid that used to be in that same space.
WATER
Weight = 10 N
OBJECT
\(F_B = 10\) N (Upward)
Mastering the Math: Practice Problem
The Challenge
A solid gold crown has a mass of 1.5 kg. When it is fully submerged in water, it displaces 0.00008 m³ of water.
Using your knowledge of Archimedes' Principle and the concept of apparent weight, determine the apparent weight of the crown.
Mathematical Analysis & Procedure:
Final Result:
N
Reading Mastery Check
1. A ship is made of steel, which is denser than water, yet it floats. Explain this using the "Weight of Displaced Fluid" concept.
2. Draw the FBD for a submarine that is hovering at a constant depth of 50m.
Sketch Here
Include labels for \(F_B\) and \(F_g\). What is the relationship between their magnitudes?
3. A beach ball is held underwater. When released, it accelerates toward the surface. Draw the FBD at the moment it is released (while still fully submerged) and explain why it moves upward.
4. Summarize Archimedes' Principle in exactly one sentence using the terms "buoyant force" and "fluid weight."
