🔬 科学与工程Science & Engineering （第 2 页）(page 2)
导线截面积计算器（按压降选线，铜/铝，单相/三相，AWG）Wire Size Calculator (Voltage Drop, Copper/Aluminium, Single/Three Phase, AWG)
按允许电压降计算所需导线截面积：输入电流、单程长度、电压和允许压降百分比，选择铜或铝、直流/单相或三相，得到最小截面积、推荐的标准规格（mm²）和近似 AWG，以及选用该规格时的实际压降。载流量还取决于敷设方式，请同时核对规范。本地处理。Find the wire cross-section needed for an allowed voltage drop: enter current, one-way length, voltage and allowed drop %, pick copper or aluminium and DC/single phase or three phase, and get the minimum area, the recommended standard size (mm²), the nearest AWG and the actual drop with that size. Ampacity also depends on installation, so check your wiring code too. Processed locally.
弹性势能计算器（E = ½kx²，可反求 k 或 x，弹力）Elastic Potential Energy Calculator (E = ½kx², Solve for k or x, Spring Force)
胡克弹簧的弹性势能 E = ½kx²：由劲度系数和形变量求能量，或由能量反求劲度系数或形变量；同时给出弹力 F = kx。形变量可用 m、cm 或 mm。本地处理。Elastic potential energy of a Hookean spring, E = ½kx²: get the energy from the spring constant and displacement, or solve for the spring constant or displacement from the energy. Also gives the spring force F = kx. Displacement in m, cm or mm. Processed locally.
溶液浓度示意Concentration Calculator
c = n/V（mol/L）或质量浓度 m/V。本地处理。可设置「模式」、「溶质（mol 或 g）」、「溶液体积 L」。c = n/V (mol/L) or mass concentration m/V. Processed locally. Settings: Mode, Solute (mol or g), Solution volume L.
超焦距计算器（焦距、光圈、弥散圆）Hyperfocal Distance Calculator (Focal Length, f-Number, CoC)
H ≈ f²/(N·c) + f，其中 f 为焦距(mm)、N 为光圈系数、c 为弥散圆直径(mm)。再给出米制结果。本地处理。H ≈ f²/(N·c) + f with focal length f (mm), f-number N and circle of confusion c (mm), plus metres. Processed locally.
氢离子浓度 / pH 计算器Hydrogen Ion Concentration / pH Calculator
由 pH 求 [H+]=10^(-pH)（并示意 25C 的 [OH-]），或由 [H+] 求 pH。本地处理。可设置「已知」、「pH」、「[H⁺] mol/L」。From pH get [H+]=10^(-pH) (with a 25C [OH-] demo), or from [H+] get pH. Processed locally. Settings: Known, pH, [H⁺] mol/L.
稀释倍数计算器Dilution Factor Calculator
按体积比 DF = V_final / V_stock，或按浓度比 DF = C1/C2。本地处理。可设置「模式」、「原液体积」、「终体积」、「C₁」、「C₂」。By volumes DF = V_final / V_stock, or by concentrations DF = C1/C2. Processed locally. Settings: Mode, Stock volume, Final volume, C₁, C₂.
离子强度计算器Ionic Strength Calculator
I = 1/2 Sum c_i z_i^2。每行填写：浓度(mol/L) 与电荷数 z。本地处理。I = 1/2 Sum c_i z_i^2. Each line: concentration (mol/L) and charge z. Processed locally.
摩尔比计算器Molar Ratio Calculator
由两份物质的量 n1、n2 给出比值 n1/n2 与未约分的 n1 : n2。二者须大于 0。本地处理。From amounts n1 and n2 get the quotient n1/n2 and the unreduced ratio n1 : n2. Both must be greater than 0. Processed locally.
皂化值估算器（示意）Saponification Value Estimator (Demo)
示意 SV ≈ 56100 / 平均摩尔质量(g/mol)，单位 mg KOH / g 油脂。非实验滴定。本地处理。Demo SV ≈ 56100 / average molar mass (g/mol), in mg KOH per g fat. Not a lab titration. Processed locally.
真空速估算器（示意）True Airspeed Estimator (Demo)
TAS ≈ IAS × √(ρ₀/ρ) 密度比示意（简化）。本地处理。可设置「指示空速」、「密度比 ρ/ρ₀」。TAS ≈ IAS × √(ρ₀/ρ) density-ratio demo (simplified). Processed locally. Settings: Indicated airspeed, Density ratio ρ/ρ₀.
螺栓圆坐标Bolt Circle Calculator
在直径 D 的圆上均布 n 个孔的坐标（从角 0° 起）。本地处理。可设置「直径」、「孔数」、「起始角 °」。Coordinates of n holes on a circle of diameter D from angle 0°. Processed locally. Settings: Diameter, Holes, Start angle °.
酒精稀释Alcohol Dilution
C₁V₁ = C₂V₂：由浓酒兑水到目标酒精度。本地处理。可设置「原酒精度 %」、「原体积」、「目标酒精度 %」。C₁V₁ = C₂V₂: dilute strong spirit to a target ABV. Processed locally. Settings: Source ABV %, Source volume, Target ABV %.
截止频率（RC）Cutoff Frequency (RC)
f_c = 1 / (2π R C)。本地处理。可设置「电阻 Ω」、「电容 F」。f_c = 1 / (2π R C). Processed locally. Settings: Resistance Ω, Capacitance F.
树木年龄示意Tree Age (demo)
年龄 ≈ 胸径(英寸) × 生长因子。本地处理。可设置「胸径 英寸」、「生长因子」。Age ≈ diameter (inches) × growth factor. Processed locally. Settings: DBH inches, Growth factor.
切削速度与进给Speeds and Feeds
主轴转速 n = 1000 V / (π D)；进给 F = n × f × z。本地处理。可设置「切削速度 m/min」、「刀具直径 mm」、「每齿进给 mm」、「齿数」。Spindle n = 1000 V / (π D); feed F = n × f × z. Processed locally. Settings: Cutting speed m/min, Tool diameter mm, Feed per tooth mm, Flutes.
卷材长度Roll Length
长度 ≈ π × 层数 × (外径+内径)/2（示意）。本地处理。可设置「外径」、「内径」、「厚度」。Length ≈ π × layers × (OD+ID)/2 (demo). Processed locally. Settings: Outer diameter, Inner diameter, Thickness.
有效值电压RMS Voltage
正弦波 V_rms = V_peak / √2；也可由峰峰值。本地处理。可设置「已知」、「电压」。Sine V_rms = V_peak / √2; or from peak-to-peak. Processed locally. Settings: Given, Voltage.
惠斯通电桥Wheatstone Bridge
平衡：R₁/R₂ = R₃/R₄；未知 Rₓ = R₂×R₃/R₁。本地处理。可设置「R₁」、「R₂」、「R₃」。Balance R₁/R₂ = R₃/R₄; unknown Rₓ = R₂×R₃/R₁. Processed locally. Settings: R₁, R₂, R₃.
LC 谐振频率LC Resonant Frequency
f₀ = 1 / (2π √(L C))。本地处理。可设置「电感 H」、「电容 F」。f₀ = 1 / (2π √(L C)). Processed locally. Settings: Inductance H, Capacitance F.
分频点（示意）Crossover Frequency (demo)
一阶 RC：f_c = 1/(2π R C)；也可由目标 f 反推 C。本地处理。可设置「计算」、「电阻 Ω」、「电容 F」、「目标频率 Hz」。First-order RC: f_c = 1/(2π R C); or solve C from target f. Processed locally. Settings: Solve, Resistance Ω, Capacitance F, Target Hz.
压缩比Compression Ratio
CR = V_max / V_min。本地处理。可设置「最大容积」、「最小容积」。CR = V_max / V_min. Processed locally. Settings: Max volume, Min volume.
瓷砖填缝用量Grout Calculator
填缝体积示意 ≈ 缝长×缝宽×缝深。本地处理。可设置「缝总长」、「缝宽」、「缝深」。Grout volume demo ≈ length × width × depth of joints. Processed locally. Settings: Joint length, Joint width, Joint depth.
简支梁均布荷载挠度示意Simply Supported Beam Deflection (demo)
δ_max = 5 w L⁴ / (384 E I)。本地处理。可设置「均布荷载 w」、「跨度 L」、「弹性模量 E」、「惯性矩 I」。δ_max = 5 w L⁴ / (384 E I). Processed locally. Settings: UDL w, Span L, Modulus E, Inertia I.
比尔-朗伯定律Beer-Lambert Law
A = ε c l。本地处理。可设置「ε」、「浓度 c」、「光程 l」。A = ε c l. Processed locally. Settings: ε, Concentration c, Path length l.
皮带周长示意Belt Length (demo)
开口皮带 L ≈ 2C + π(D+d)/2 + (D-d)²/(4C)。本地处理。可设置「中心距 C」、「大轮直径 D」、「小轮直径 d」。Open belt L ≈ 2C + π(D+d)/2 + (D-d)²/(4C). Processed locally. Settings: Center distance C, Large pulley D, Small pulley d.
沸点海拔修正示意Boiling Point vs Altitude (demo)
示意：沸点 ≈ 100 - 海拔(m)/300（粗估）。本地处理。可设置「海拔 m」。Demo: BP ≈ 100 - altitude(m)/300 (rough). Processed locally. Settings: Altitude m.
沸点升高Boiling Point Elevation
ΔT_b = i · K_b · m。本地处理。可设置「范特霍夫 i」、「K_b」、「质量摩尔浓度 m」。ΔT_b = i · K_b · m. Processed locally. Settings: van ʼt Hoff i, K_b, Molality m.
螺栓扭矩计算器Bolt Torque Calculator
按 T = K × F × d 估算螺栓拧紧扭矩，可选力、直径和扭矩的单位（N/kN/lbf、mm/in、N·m/ft·lbf）。本地处理。Estimate bolt tightening torque with T = K × F × d, with selectable units for force, diameter and torque (N/kN/lbf, mm/in, N·m/ft·lbf). Processed locally.
相机裁切系数与等效焦距Crop Factor Calculator
裁切系数 = 全画幅对角线 43.27 mm / 传感器对角线；等效焦距 = 焦距 × 系数，等效光圈 = 光圈 × 系数。本地处理。Crop factor = 43.27 mm full-frame diagonal / sensor diagonal; equivalent focal length = focal × factor, equivalent aperture = f-number × factor. Processed locally.
德布罗意波长De Broglie Wavelength
λ = h / (m v)，h = 6.62607015×10⁻³⁴ J·s。本地处理。可设置「质量 kg」、「速度 m/s」。λ = h / (m v), with h = 6.62607015×10⁻³⁴ J·s. Processed locally. Settings: Mass kg, Velocity m/s.
多普勒效应（声波）Doppler Effect Calculator
f′ = f (v + v₀) / (v - vₛ)：观察者靠近 v₀ 取正，声源靠近 vₛ 取正（远离为负）。本地处理。f′ = f (v + v₀) / (v - vₛ): observer speed v₀ positive when approaching, source speed vₛ positive when approaching (negative when receding). Processed locally.
质能方程 E = mc²E = mc² Calculator
E = m c²，c = 299,792,458 m/s；同时换算为千瓦时和 TNT 当量。本地处理。可设置「质量 kg」。E = m c² with c = 299,792,458 m/s, also in kWh and TNT equivalent. Processed locally. Settings: Mass kg.
发动机排量Engine Displacement Calculator
排量 = π/4 × 缸径² × 行程 × 缸数。本地处理。可设置「缸径 mm」、「行程 mm」、「缸数」。Displacement = π/4 × bore² × stroke × cylinders. Processed locally. Settings: Bore mm, Stroke mm, Cylinders.
凝固点降低Freezing Point Depression Calculator
ΔT = i × Kf × m（i 范特霍夫因子，Kf 溶剂常数，m 质量摩尔浓度 mol/kg）。水的 Kf = 1.86 °C·kg/mol。本地处理。ΔT = i × Kf × m (i dissociation (van-t-Hoff) factor, Kf solvent constant, m molality in mol/kg). Water Kf = 1.86 °C·kg/mol. Processed locally.
建筑热损失Heat Loss Calculator
Q = U × A × ΔT（W）：U 为传热系数 W/(m²·K)，A 面积，ΔT 室内外温差；并估算每天耗能。本地处理。Q = U × A × ΔT (W): U-value W/(m²·K), area A and indoor-outdoor temperature difference; plus daily energy. Processed locally.
导热传热（傅里叶定律）Heat Transfer Calculator (Conduction)
Q = k × A × ΔT / d：k 导热系数 W/(m·K)，A 面积，ΔT 两侧温差，d 厚度；并给出热阻 R = d/(kA)。本地处理。Q = k × A × ΔT / d: conductivity k W/(m·K), area A, temperature difference ΔT and thickness d; plus thermal resistance R = d/(kA). Processed locally.
亨德森-哈塞尔巴尔赫方程（缓冲液 pH）Henderson-Hasselbalch Calculator
pH = pKa + log₁₀([A⁻]/[HA])；也可由目标 pH 反求所需的共轭碱/酸浓度比。本地处理。可设置「pKa」、「共轭碱 [A⁻] mol/L」、「弱酸 [HA] mol/L」、「目标 pH（可选）」。pH = pKa + log₁₀([A⁻]/[HA]); also gives the base/acid ratio needed for a target pH. Processed locally. Settings: pKa, Conjugate base [A⁻] mol/L, Weak acid [HA] mol/L, Target pH (optional).
斜面受力（含摩擦）Inclined Plane Calculator
质量 m 的物体在倾角 θ 的斜面上：沿斜面分力 mg sinθ，法向力 mg cosθ，摩擦力 μ mg cosθ，净加速度 a = g(sinθ - μ cosθ)。本地处理。For mass m on a slope of angle θ: parallel force mg sinθ, normal force mg cosθ, friction μ mg cosθ and acceleration a = g(sinθ - μ cosθ). Processed locally.
感抗 XL = 2πfLInductive Reactance Calculator
交流电路中电感的感抗 XL = 2πfL（欧姆），同时给出电流（若填电压）。本地处理。可设置「频率 Hz」、「电感 H」、「电压 V（可选）」。Inductive reactance in an AC circuit, XL = 2πfL (ohms), plus current if a voltage is given. Processed locally. Settings: Frequency Hz, Inductance H, Voltage V (optional).
kVA 视在功率kVA Calculator
单相 kVA = V × I / 1000，三相 kVA = √3 × V（线电压）× I / 1000；给出功率因数时同时算 kW。本地处理。Single-phase kVA = V × I / 1000; three-phase kVA = √3 × V (line) × I / 1000; with a power factor, kW too. Processed locally.
机械效益（理想 / 实际 / 效率）Mechanical Advantage Calculator
实际机械效益 AMA = 负载力 ÷ 施力，理想机械效益 IMA = 施力移动距离 ÷ 负载移动距离，效率 = AMA ÷ IMA。适用于杠杆、滑轮、斜面等。本地处理。Actual mechanical advantage AMA = load ÷ effort, ideal IMA = effort distance ÷ load distance, efficiency = AMA ÷ IMA. Works for levers, pulleys, ramps and more. Processed locally.
当量浓度（N）Normality Calculator
当量浓度 N = 溶质质量 ÷ 当量质量 ÷ 溶液体积，当量质量 = 摩尔质量 ÷ 当量数（酸的可电离 H⁺ 数、碱的 OH⁻ 数等）。同时给出摩尔浓度。本地处理。Normality N = solute mass ÷ equivalent weight ÷ solution volume, where equivalent weight = molar mass ÷ equivalents (H⁺ per acid, OH⁻ per base, etc.). Molarity is shown too. Processed locally.
渗透压（范特霍夫方程）Osmotic Pressure Calculator
π = i M R T：i 为解离因子，M 为摩尔浓度 mol/L，R = 0.082057 L·atm/(mol·K)，T 为绝对温度。本地处理。π = i M R T: i dissociation factor, M molarity in mol/L, R = 0.082057 L·atm/(mol·K) and T the absolute temperature. Processed locally.
PCB 走线宽度（IPC-2221）PCB Trace Width Calculator (IPC-2221)
IPC-2221：I = k × ΔT^0.44 × A^0.725（A 为截面积 mil²，外层 k = 0.048，内层 k = 0.024），由电流、允许温升和铜厚求最小线宽。本地处理。IPC-2221: I = k × ΔT^0.44 × A^0.725 (A cross-section in mil², k = 0.048 external, 0.024 internal) gives the minimum width from current, allowed temperature rise and copper weight. Processed locally.
管道流量与流速Pipe Flow Calculator
流量 Q = 截面积 × 流速，截面积 = π d² / 4；同时给出雷诺数（按 20 °C 水）判断层流或湍流。本地处理。Flow Q = cross-section × velocity, with area = π d² / 4; plus the Reynolds number (water at 20 °C) to tell laminar from turbulent. Processed locally.
串联电阻Resistors in Series Calculator
串联总电阻 R = R₁ + R₂ + …；填写电源电压时同时给出电流和每个电阻上的分压与功率。本地处理。Total series resistance R = R1 + R2 + ...; with a supply voltage, also the current and each resistor's voltage drop and power. Processed locally.
钢材重量Steel Weight Calculator
按形状和尺寸算钢材重量：钢板 = 长 × 宽 × 厚，圆钢 = π d²/4 × 长，钢管 = π (D² - d²)/4 × 长；密度默认 7850 kg/m³（不锈钢约 7930）。本地处理。Steel weight by shape: plate = length × width × thickness, round bar = π d²/4 × length, tube = π (D² - d²)/4 × length; density 7,850 kg/m³ by default (stainless ≈ 7,930). Processed locally.
刹车距离（反应 + 制动）Stopping Distance Calculator
停车距离 = 反应距离（车速 × 反应时间）+ 制动距离 v² / (2 μ g)。干燥沥青 μ ≈ 0.7，湿滑 ≈ 0.4，冰面 ≈ 0.1。本地处理。Stopping distance = reaction distance (speed × reaction time) + braking distance v² / (2 μ g). Dry asphalt μ ≈ 0.7, wet ≈ 0.4, ice ≈ 0.1. Processed locally.
功与功率Work and Power Calculator
功 W = F × d × cos θ（焦耳），功率 P = W ÷ t（瓦特），并换算为马力。本地处理。可设置「力 N」、「位移 m」、「力与位移夹角 °」、「时间 s」。Work W = F × d × cos θ (joules) and power P = W ÷ t (watts), also in horsepower. Processed locally. Settings: Force N, Displacement m, Angle between force and motion °, Time s.
角分辨率（瑞利判据）Angular Resolution Calculator
θ = 1.22 λ / D（弧度），λ 为波长、D 为口径；换算成角秒，并给出 Dawes 极限 116/D(mm) 角秒作为参考。本地处理。θ = 1.22 λ / D (radians) for wavelength λ and aperture D, in arcseconds too, with the Dawes limit 116 / D(mm) arcsec for reference. Processed locally.
望远镜口径面积与集光力Aperture Area Calculator
口径面积 = π D² / 4；集光力与面积成正比，和人眼瞳孔（默认 7 mm）比较；有中央遮挡时扣除副镜面积。本地处理。Aperture area = π D² / 4; light grasp scales with area, compared with a dark-adapted pupil (7 mm by default); a central obstruction is subtracted. Processed locally.
空气阻力（阻力方程）Drag Equation Calculator
F = ½ ρ v² C_d A：ρ 流体密度（空气 1.225 kg/m³），v 速度，C_d 阻力系数，A 迎风面积；并给出克服阻力所需功率 P = F v。本地处理。F = ½ ρ v² C_d A: fluid density ρ (air 1.225 kg/m³), speed v, drag coefficient C_d and frontal area A; plus the power to overcome it, P = F v. Processed locally.
弗劳德数Froude Number Calculator
Fr = v / √(g L)：v 流速，L 特征长度（明渠为水深）。Fr < 1 缓流（亚临界），= 1 临界，> 1 急流（超临界）。本地处理。Fr = v / √(g L): flow speed v and characteristic length L (water depth for open channels). Fr < 1 subcritical, = 1 critical, > 1 supercritical. Processed locally.
传热系数 hHeat Transfer Coefficient Calculator
对流传热系数 h = Q ÷ (A × ΔT)（W/(m²·K)）；也可由 h 反算热流。同时给出对应的热阻 1/(hA)。本地处理。Convective heat transfer coefficient h = Q ÷ (A × ΔT) in W/(m²·K), with the matching thermal resistance 1/(hA). Processed locally.
液压压力与输出力（帕斯卡原理）Hydraulic Pressure Calculator
压力 P = F ÷ A（A = π d²/4）；按帕斯卡原理，大活塞输出力 F₂ = P × A₂ = F₁ × (d₂/d₁)²。本地处理。Pressure P = F ÷ A (A = π d²/4); by Pascal's principle the output force is F₂ = P × A₂ = F₁ × (d₂/d₁)². Processed locally.
液体静压力Hydrostatic Pressure Calculator
P = ρ g h：液体密度 ρ（淡水 1000，海水约 1025 kg/m³）、深度 h；同时给出加上大气压后的绝对压力。本地处理。P = ρ g h for fluid density ρ (fresh water 1000, seawater about 1025 kg/m³) and depth h; plus the absolute pressure including the atmosphere. Processed locally.
努森数Knudsen Number Calculator
Kn = λ / L：分子平均自由程 λ 与特征长度 L 之比。Kn < 0.01 连续介质，0.01-0.1 滑移流，0.1-10 过渡区，> 10 自由分子流。本地处理。Kn = λ / L: mean free path λ over characteristic length L. Kn < 0.01 continuum, 0.01-0.1 slip, 0.1-10 transition, > 10 free-molecular. Processed locally.
长度收缩（相对论）Length Contraction Calculator
运动物体沿运动方向的长度 L = L₀ √(1 - v²/c²)，L₀ 为固有长度；同时给出洛伦兹因子 γ。本地处理。Length along the motion L = L₀ √(1 - v²/c²) for rest length L₀; the Lorentz factor γ is shown too. Processed locally.
雷电距离（声光时差）Lightning Distance Calculator
看到闪电到听到雷声的秒数 × 声速 ≈ 距离。声速用 331.3 + 0.606 × T(°C) m/s，或近似"秒数 ÷ 3 ≈ 公里"。本地处理。Seconds between flash and thunder × speed of sound ≈ distance. Sound speed is 331.3 + 0.606 × T(°C) m/s, or roughly "seconds ÷ 3 ≈ kilometres". Processed locally.
马赫数Mach Number Calculator
马赫数 Ma = v / a，当地声速 a = √(γ R T)，空气 γ = 1.4、R = 287.05 J/(kg·K)。Ma < 0.8 亚音速，0.8-1.2 跨音速，> 1 超音速，> 5 高超音速。本地处理。Mach number Ma = v / a with local speed of sound a = √(γ R T); for air γ = 1.4 and R = 287.05 J/(kg·K). Ma < 0.8 subsonic, 0.8-1.2 transonic, > 1 supersonic, > 5 hypersonic. Processed locally.