Physics

Physics · modern-physics

Problem

In general, we can describe the quadratic drag on an object by the following force law: FD=12CDρAv2F_{D}=\frac{1}{2} C_{D} \rho A v^{2} where AA is the cross-sectional area of the object exposed to the airflow, vv is the speed of the object in a fluid, and CDC_{D} is the drag coefficient, a dimensionless quantity that varies based on shape. Another useful quantity to know is the Reynold's number, a dimensionless quantity that helps predict fluid flow patterns. It is given by the formula: Re⁡=ρvLμ\operatorname{Re}=\frac{\rho v L}{\mu} where ρ\rho is the density of the surrounding fluid, μ\mu is the dynamic viscosity of the fluid, and LL is a reference length parameter that varies based on each object. For a smooth 1{ }^{1} sphere traveling in a fluid, its diameter serves as the reference length parameter. <image_1> A logarithmic graph of CDC_{D} vs Re of a sphere from the NASA Glenn Research Center. The relationship between the drag coefficient and the Reynold's number holds significant importance. Due to the complexity of fluid dynamics, empirical data is commonly used, as depicted in the figure provided above. Notably, the figure indicates a significant decrease in the drag coefficient around Re⁡≈4×105\operatorname{Re} \approx 4 \times 10^{5}. This phenomenon, known as the drag crisis, occurs when a sphere transitions from laminar to turbulent flow, resulting in a broad wake and high drag. Let's consider a smooth ball with a radius of 0.2 m0.2 \mathrm{~m} and a mass of 0.1 kg0.1 \mathrm{~kg} dropped in air with a constant density of ρ=1.255 kg/m3\rho=1.255 \mathrm{~kg} / \mathrm{m}^{3}. It is found that at velocity 5 m/s5 \mathrm{~m} / \mathrm{s}, the Reynold's number of the ball is 3.41⋅1053.41 \cdot 10^{5}. If the ball is dropped from rest, it approaches a stable terminal velocity v1v_{1}. If the ball is thrown downwards with enough velocity, it will experience turbulence, and approach a stable terminal velocity v2v_{2}. Find Δv=v2−v1\Delta v=v_{2}-v_{1}. Ignore any terminal velocities found for Reynold numbers less than an order of magnitude 10−110^{-1}. Note: This problem is highly idealized as it assumes the atmosphere has air of constant density and temperature. In reality, this is not true! Given constants: - Proton mass, mp=1.67⋅10−27 kgm_{p}=1.67 \cdot 10^{-27} \mathrm{~kg} - Neutron mass, mn=1.67⋅10−27 kgm_{n}=1.67 \cdot 10^{-27} \mathrm{~kg} - Electron mass, me=9.11⋅10−31 kgm_{e}=9.11 \cdot 10^{-31} \mathrm{~kg} - Avogadro's constant, N0=6.02⋅1023 mol−1N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1} - Universal gas constant, R=8.31 J/(mol⋅K)R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K}) - Boltzmann's constant, kB=1.38⋅10−23 J/Kk_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K} - Electron charge magnitude, e=1.60⋅10−19Ce=1.60 \cdot 10^{-19} \mathrm{C} - 1 electron volt, 1eV=1.60⋅10−19 J1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J} - Speed of light, c=3.00⋅108 m/sc=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s} - Universal Gravitational constant, G=6.67⋅10−11( N⋅m2)/kg2G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2} - Solar Mass M⊙=1.988⋅1030 kgM_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg} - Acceleration due to gravity, g=9.8 m/s2g=9.8 \mathrm{~m} / \mathrm{s}^{2} - 1 unified atomic mass unit, 1u=1.66⋅10−27 kg=931MeV/c21 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2} - Planck's constant, h=6.63⋅10−34 J⋅s=4.41⋅10−15eV⋅sh=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s} - Permittivity of free space, ϵ0=8.85⋅10−12C2/(N⋅m2)\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right) - Coulomb's law constant, k=14πϵ0=8.99⋅109( N⋅m2)/C2k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2} - Permeability of free space, μ0=4π⋅10−7 T⋅m/A\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A} - Magnetic constant, μ04π=1⋅10−7( T⋅m)/A\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A} - 1 atmospheric pressure, 1 atm=1.01⋅105 N/m2=1.01⋅105 Pa1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa} - Wien's displacement constant, b=2.9b=2.9. 10−3 m⋅K10^{-3} \mathrm{~m} \cdot \mathrm{K} - Stefan-Boltzmann constant, σ=5.67⋅10−8 W/m2/K4\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}

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