University Physics Notes: Density, Archimedes' Principle, and Flotation

Fundamental Definitions and Mathematical Representation of Density

• Definition of Density: The density ($\rho$) of a substance or a material sample is defined as the mass ($m$) per unit volume ($V$) of that material. The governing formula is:     $\text{Density} = \frac{\text{Mass of material}}{\text{Volume of material}}$

• Units and Dimensions:

  • The SI unit for density is $kg\,m^{-3}$.
  • Physical dimensions are expressed as $ML^{-3}$.
  • Standard density of water: $1\,g\,cm^{-3}$ or $1000\,kg\,m^{-3}$.

• Theoretical vs. Practical Considerations:

  • At a specific point in a fluid, density is defined as the limit of the ratio of a small mass element ($\Delta m$) to its corresponding volume element ($\Delta V$) as the volume approaches zero:         $\rho = \frac{\Delta m}{\Delta V}$
  • In practical physics and engineering, fluids are assumed to be large relative to atomic scales, allowing for the assumption of uniform density throughout the sample. This yields the standard equation:         $\rho = \frac{m}{V}$

• Compressibility and Phase Differences:

  • The density of a gas varies significantly based on pressure, as gases are readily compressible.
  • Liquids are considered incompressible, meaning their density does not vary significantly with changes in pressure.
  • Compactness: Density reflects the degree of compactness of matter. Loosely packed molecules results in low density. Consequently, a substance's density is lowest in its gaseous phase and highest in its solid phase (with liquids usually being intermediate).

High vs. Low Density and Porosity

• High Density: Occurs when particles are packed together tightly with minimal space between them. Objects with high density, such as an iron nail, will sink easily in water.

• Low Density: Occurs when particles are loosely packed with more space between them. Objects with low density, such as wood, float more easily.

• Comparative Density in Soil (Bulk vs. Particle Density):

  • Bulk Density: Calculated using a sample that contains both solids and pore spaces (voids). For example, a soil sample of $1.0\,cm^3$ weighing $1.32\,g$ has a bulk density of:         $\text{Bulk Density} = \frac{1.32\,g}{1.0\,cm^3} = 1.32\,g\,cm^{-3}$
  • Particle Density: Calculated based only on the volume of the solids (excluding pores). If those solids are compressed into a volume of $0.5\,cm^3$, the density becomes:         $\text{Particle Density} = \frac{1.32\,g}{0.5\,cm^3} = 2.64\,g\,cm^{-3}$

• Porosity ($\Phi$): Defined as the pore (cavity) volume per unit volume of the sample. It represents the fraction of the total (bulk) volume occupied by voids.

• Example 1: Rock with Cavity Analysis:

  • Given: Cube side = $5.0\,cm$; Mass = $306\,g$; Particle density = $2.55\,g\,cm^{-3}$.
  • Bulk volume = $5.0^{3} = 125\,cm^3$.
  • True volume of rock material = $\frac{306}{2.55} = 120\,cm^3$.
  • Volume of the cavity ($V_c$) = $\text{Bulk Volume} - \text{True Volume} = 5.0\,cm^3$.
  • Analysis: Bulk density = $2.45\,g\,cm^{-3}$; Porosity = $\frac{5}{125} = 0.04$ (or $4\%$).

Relative Density (Specific Gravity)

• Definition: Relative density (RD) is a dimensionless measure comparing the density of a substance to the density of a reference substance, typically pure water. It is a ratio of masses for equal volumes.

• Characteristics:

  • RD has no units.
  • The numerical value of density in $g\,cm^{-3}$ is identical to the relative density.
  • Example: Mercury density = $13.6\,g\,cm^{-3}$, therefore its relative density is $13.6$. This implies mercury is $13.6$ times heavier than an equal volume of water.

• Density Tower Examples (Materials in descending order of density):

  • Solids/Liquids found in layering: Bolt, Honey, Popcorn kernel, Corn syrup, 100% Maple syrup, Die, Milk, Cherry tomato, Beads, Dish soap, Water, Soda cap, Vegetable oil, Rubbing alcohol, Ping pong ball, Oil, Lamp oil.

• Example 2: Density Bottle Problem:

  • Bottle with olive oil weights $90\,g$.
  • Bottle with pure water weights $100\,g$.
  • Relative density of olive oil = $0.75$.
  • (Finding the mass of the empty bottle involves the comparison: $\frac{90 - m_e}{100 - m_e} = 0.75$).

Properties and Behavior of Fluids

• Fluid Definition: A substance characterized by its tendency to flow, encompassing both liquids and gases.

• Mechanical Properties:

  • Fluids assume the shape of their containers because they cannot sustain or maintain tangential (shearing) forces.
  • Fluids can exert force in a direction perpendicular to their surface.
  • Molecular behavior: Molecules are in constant random motion, exerting pressure on container walls.
  • Constraint: Fluids have no definite shape and (for gases) no definite volume.

Archimedes’ Principle and Buoyancy

• Statement of the Principle: When a body is wholly or partially immersed in a fluid, it experiences an upthrust (buoyant force) equal to the weight of the fluid displaced by the body.

• The Buoyant Force ($F_b$):

  • Magnitude: $F_b = m_f g$, where $m_f$ is the mass of the displaced fluid.
  • Formula using density: $F_b = \rho_f V_f g$.
  • Origin: The net upward force exists because the pressure in a fluid increases with depth.
  • Apparent Loss in Weight: When submerged, an object appears lighter; this loss in weight is exactly equal to the buoyant force.

• Geometric Applications for Buoyant Force:

  • For a cylindrical body: $F_b = \pi r^2 h \rho_f g$ (where $r$ is radius and $h$ is height).
  • For a spherical body: Calculated using the volume of displaced fluid based on the radius of the sphere ($V = \frac{4}{3} \pi r^3$).

• Net Resultant Forces ($F_R$):

  • For a submerged wooden cube (rising): $F_R = F_b - F_g = (\rho_f - \rho_o) V_o g$.
  • For a submerged metal cube (sinking): $F_R = F_g - F_b = (\rho_m - \rho_f) V_m g$.

Practical Applications of Archimedes' Principle

• Measuring Specific Gravity of a Solid:

  • Calculated as the ratio: $\frac{\text{Weight in Air (True Weight)}}{\text{Upthrust (Loss of weight in water)}}$.
  • Formula for measured weight ($W_s$) when submerged in fluid ($\rho_f$): $W_s = Vg(\rho - \rho_f)$.

• Measuring Density of a Liquid: Required variables include:

  1. Weight of solid in air ($W_A$).
  2. Apparent weight when immersed in the liquid ($W_L$).
  3. Apparent weight when immersed in water ($W_w$).
  • Upthrust in liquid ($U_L$) = $W_A - W_L$.
  • Upthrust in water ($U_w$) = $W_A - W_w$.

• Worked Metal Cube Example:

  • Weight in air = $10\,N$; Weight in water = $8\,N$.
  • Upthrust ($U_w$) = $2\,N$.
  • Mass of water displaced = $\frac{2}{9.8} = 0.204\,kg$.
  • Volume of cube = $\frac{0.204\,kg}{1000\,kg\,m^{-3}} = 2.04 \times 10^{-4}\,m^3$.
  • Density of metal = $\frac{1.02}{2.04 \times 10^{-4}} = 5000\,kg\,m^{-3}$.

Principle of Flotation

• Statement: A floating body displaces its own weight of the fluid in which it floats.

• Conditions for Flotation:

  • Static Equilibrium: occurs when the upward buoyant force ($F_b$) equals the downward gravitational force ($F_g$).
  • $\text{Apparent Weight} = \text{True Weight} - \text{Buoyant Force} = 0$.
  • Example: A boat weighing $1000\,kg$ is less dense than the water it displaces as long as it contains air; it displaces a mass of water equal to its own mass ($1000\,kg$) rather than sinking.

• Calculations for Submerged Fractions:

  • Submerged volume ($V_s$) is correlated to the ratio of object density to fluid density.
  • Example: Hollow sphere (inner radius $8.0\,cm$, outer radius $9.0\,cm$) floating half-submerged in liquid ($800\,kg\,m^{-3}$).
    • Inner Radius ($r_i$) = $0.08\,m$, Outer Radius ($r_o$) = $0.09\,m$.
    • Mass ($m$) = $\rho \times V_s = 800 \times \frac{1}{2} \times (\frac{4}{3} \pi r_o^3) = 1.22\,kg$.
    • Density of sphere material ($\rho_m$) = $\frac{m}{V_m} = \frac{1.22}{\frac{4}{3} \pi (r_o^{3} - r_i^{3})} = 1.3 \times 10^{3}\,kg\,m^{-3}$.

Hydrometers

• Definition: A scientific instrument designed for measuring the density or relative density of liquids.

• Anatomy and Design:

  • Narrow hollow glass stem: Sensitivity is improved with a narrower stem.
  • Wide bulb: Provides the necessary volume to displace enough liquid for flotation.
  • Graduated scale: Etched on the stem. Crucially, the density values decrease upward and the scale is non-uniform (non-equal divisions).
  • Loaded end: Positioned below the bulb, containing lead-shots to ensure the instrument remains vertical in the liquid.

• Operating Principle:

  • Operates on the Principle of Flotation. The instrument sinks until the weight of the displaced liquid equals the instrument's weight. It sinks deeper in less dense liquids.

• Primary Uses:

  • Testing dilute sulfuric acid in accumulators (batteries).
  • Measuring the strength of spirits and alcohols.
  • Determining the richness of milk and other beverage products.