SOR
202 lines
8.2 KiB
Plaintext
202 lines
8.2 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# 杨氏模量计算器 (光杠杆 & 逐差法)\n",
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"\n",
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"本 Notebook 用于通过光杠杆镜尺法测量金属丝在拉伸过程中的微小伸长量，利用**逐差法**计算金属丝的杨氏弹性模量 $E$，并进行完整的各分量不确定度传递与规范修约。\n",
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"\n",
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"---\n",
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"### 实验原理与数学公式\n",
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"\n",
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"#### 1. 杨氏模量计算公式\n",
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"根据胡克定律与光杠杆光学放大原理：\n",
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"$$E = \\frac{F/S}{\\Delta L / L} = \\frac{4 F L}{\\pi d^2 \\Delta L}$$\n",
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"设单个砝码质量为 $m$，增重过程每加 4 个砝码（通过逐差法将 8 个加重、减重平均刻度值 $n_i$ 隔 4 项逐差）产生的刻度尺平均位移为 $\\Delta n_{\\mathrm{avg}}$，光杠杆后足长为 $b$，反射镜至标尺距离为 $D$，则伸长量 $\\Delta L = \\frac{b \\Delta n_{\\mathrm{avg}}}{2 D}$，代入得：\n",
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"$$E = \\frac{32 m g L D}{\\pi d^2 b \\Delta n_{\\mathrm{avg}}}$$\n",
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"* $L$：金属丝原长\n",
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"* $D$：光杠杆镜面到刻度尺标尺距离\n",
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"* $d$：金属丝直径（千分尺测量）\n",
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"* $b$：光杠杆常数（刀口到后足支点的垂直距离）\n",
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"* $m$：单组砝码质量；$g$：重力加速度\n",
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"* $\\Delta n_{\\mathrm{avg}}$：4 个砝码引起的标尺平均位移\n",
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"\n",
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"#### 2. 不确定度传递公式\n",
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"* 相对不确定度平方和：\n",
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"$$u_r(E) = \\frac{u_E}{E} = \\sqrt{ \\left(\\frac{u_L}{L}\\right)^2 + \\left(\\frac{u_D}{D}\\right)^2 + \\left(2\\frac{u_d}{d}\\right)^2 + \\left(\\frac{u_b}{b}\\right)^2 + \\left(\\frac{u_{\\Delta n}}{\\Delta n_{\\mathrm{avg}}}\\right)^2 }$$\n",
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"* 各直接测量量的不确定度按 A 类与仪器 B 类合成：\n",
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"$$u_X = \\sqrt{u_{A,X}^2 + u_{B,X}^2}, \\quad u_{A,X} = \\frac{s_X}{\\sqrt{n}}, \\quad u_{B,X} = \\frac{\\Delta_{\\mathrm{inst}}}{\\sqrt{3}}$$"
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],
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"id": "5763889c1dbb6480"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import math\n",
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"from decimal import Decimal\n",
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"from python.utils import scientific_round, calculate_stats\n",
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"\n",
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"print(\"杨氏模量模块加载完成。\")"
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],
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"id": "a70d2b4077726fc0"
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 1. 实验参数、几何尺寸与标尺读数输入\n",
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"> **提示**：可直接在下方单元格中修改各几何长度的多次测量列表以及加减砝码的刻度尺读数矩阵。"
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],
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"id": "b3fe9624ea56f3d0"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# --- 常量设置 ---\n",
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"M_WEIGHT = Decimal(\"0.350\") # 单个砝码质量 350g = 0.350kg\n",
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"G = Decimal(\"9.80\") # 重力加速度 m/s^2\n",
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"PI = Decimal(str(math.pi))\n",
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"\n",
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"# --- 几何长度测量值 (单位: mm，各6次) ---\n",
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"L_vals = [Decimal(\"905.0\"), Decimal(\"905.5\"), Decimal(\"904.5\"), Decimal(\"905.0\"), Decimal(\"905.2\"), Decimal(\"904.8\")]\n",
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"D_vals = [Decimal(\"1502.0\"), Decimal(\"1501.5\"), Decimal(\"1502.5\"), Decimal(\"1502.0\"), Decimal(\"1501.8\"), Decimal(\"1502.2\")]\n",
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"d_vals = [Decimal(\"0.702\"), Decimal(\"0.704\"), Decimal(\"0.701\"), Decimal(\"0.703\"), Decimal(\"0.702\"), Decimal(\"0.703\")]\n",
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"b_vals = [Decimal(\"68.20\"), Decimal(\"68.22\"), Decimal(\"68.18\"), Decimal(\"68.20\"), Decimal(\"68.24\"), Decimal(\"68.20\")]\n",
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"\n",
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"# --- 仪器误差限 Δ_inst (mm) ---\n",
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"delta_L = Decimal(\"1.0\")\n",
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"delta_D = Decimal(\"1.0\")\n",
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"delta_d = Decimal(\"0.005\")\n",
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"delta_b = Decimal(\"0.02\")\n",
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"delta_n_inst = Decimal(\"0.5\")\n",
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"\n",
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"# --- 标尺读数 n (0~7 个砝码，单位: mm；每项为 [加重读数, 减重读数]) ---\n",
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"n_readings = [\n",
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" [Decimal(\"15.2\"), Decimal(\"15.4\")], # 0个\n",
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" [Decimal(\"22.1\"), Decimal(\"22.3\")], # 1个\n",
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" [Decimal(\"29.0\"), Decimal(\"29.1\")], # 2个\n",
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" [Decimal(\"35.8\"), Decimal(\"36.0\")], # 3个\n",
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" [Decimal(\"42.7\"), Decimal(\"42.9\")], # 4个\n",
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" [Decimal(\"49.5\"), Decimal(\"49.7\")], # 5个\n",
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" [Decimal(\"56.3\"), Decimal(\"56.5\")], # 6个\n",
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" [Decimal(\"63.1\"), Decimal(\"63.2\")], # 7个\n",
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"]\n",
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"\n",
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"print(\"实验数据加载完成。\")"
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],
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"id": "b6816ecdb2083f36"
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 2. 逐差法处理标尺读数"
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],
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"id": "7cecf972d1d0806c"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 计算各加重减重的平均读数 n_i\n",
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"n_means = [(row[0] + row[1]) / Decimal(\"2\") for row in n_readings]\n",
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"\n",
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"# 逐差法计算 Δn (隔 4 项相减，相当于 4 个砝码的位移)\n",
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"dn_list = []\n",
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"print(\"--- 逐差过程 (Δn_j = n_{j+4} - n_j) ---\")\n",
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"for j in range(4):\n",
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" diff = n_means[j + 4] - n_means[j]\n",
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" dn_list.append(diff)\n",
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" print(f\"Δn_{j} = n_{j+4} - n_{j} = {diff:.2f} mm\")\n",
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"\n",
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"dn_mean = sum(dn_list) / Decimal(\"4\")\n",
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"variance_dn = sum((x - dn_mean)**2 for x in dn_list) / Decimal(\"3\")\n",
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"s_dn = Decimal(str(math.sqrt(float(variance_dn))))\n",
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"u_a_dn = s_dn / Decimal(str(math.sqrt(4)))\n",
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"u_b_dn = delta_n_inst / Decimal(str(math.sqrt(3)))\n",
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"dn_u = Decimal(str(math.sqrt(float(u_a_dn**2 + u_b_dn**2))))\n",
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"\n",
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"print(f\"\\nΔn 平均值 : {dn_mean:.3f} ± {dn_u:.3f} mm\")"
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],
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"id": "67a1199d24b1f978"
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### 3. 杨氏模量计算与不确定度评定"
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],
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"id": "66279721f812b579"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"# 几何参数的均值与不确定度\n",
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"L_mean, L_u = calculate_stats(L_vals, delta_L)[:2]\n",
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"D_mean, D_u = calculate_stats(D_vals, delta_D)[:2]\n",
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"d_mean, d_u = calculate_stats(d_vals, delta_d)[:2]\n",
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"b_mean, b_u = calculate_stats(b_vals, delta_b)[:2]\n",
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"\n",
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"# 转换为国际单位制 (m)\n",
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"L_m = L_mean / Decimal(\"1000\")\n",
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"D_m = D_mean / Decimal(\"1000\")\n",
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"d_m = d_mean / Decimal(\"1000\")\n",
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"b_m = b_mean / Decimal(\"1000\")\n",
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"dn_m = dn_mean / Decimal(\"1000\")\n",
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"\n",
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"# E = (32 * m * g * L * D) / (pi * d^2 * b * Δn)\n",
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"numerator = Decimal(\"32\") * M_WEIGHT * G * L_m * D_m\n",
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"denominator = PI * (d_m**2) * b_m * dn_m\n",
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"E_val = numerator / denominator\n",
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"\n",
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"# 不确定度合成传递\n",
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"rel_u_sq = (L_u/L_mean)**2 + (D_u/D_mean)**2 + (Decimal(\"2\")*d_u/d_mean)**2 + (b_u/b_mean)**2 + (dn_u/dn_mean)**2\n",
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"rel_u = Decimal(str(math.sqrt(float(rel_u_sq))))\n",
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"E_u = E_val * rel_u\n",
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"\n",
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"# 科学修约 (以 10^11 Pa 数量级表示)\n",
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"E_11 = E_val / Decimal(\"1e11\")\n",
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"E_u_11 = E_u / Decimal(\"1e11\")\n",
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"E_final, u_final = scientific_round(E_11, E_u_11)\n",
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"\n",
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"print(\"=\" * 45)\n",
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"print(\" 杨 氏 模 量 计 算 结 果 \")\n",
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"print(\"=\" * 45)\n",
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"print(f\"金属丝长度 L : {L_mean:.2f} ± {L_u:.2f} mm\")\n",
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"print(f\"镜面到尺距离 D: {D_mean:.2f} ± {D_u:.2f} mm\")\n",
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"print(f\"金属丝直径 d : {d_mean:.4f} ± {d_u:.4f} mm\")\n",
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"print(f\"光杠杆臂长 b : {b_mean:.3f} ± {b_u:.3f} mm\")\n",
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"print(f\"标尺位移 Δn : {dn_mean:.3f} ± {dn_u:.3f} mm (4个砝码对应位移)\")\n",
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"print(\"-\" * 45)\n",
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"print(f\"杨氏模量 E (原始): {E_val:.4e} Pa\")\n",
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"print(f\"相对不确定度 u_r : {rel_u * 100:.2f}%\")\n",
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"print(f\"绝对不确定度 u_E : {E_u:.4e} Pa\")\n",
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"print(\"-\" * 45)\n",
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"print(f\"★ 最终结果表示 : E = {E_final} ± {u_final} × 10^11 Pa\")\n",
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"print(\"=\" * 45)"
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],
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"id": "d4864c1fd01816d6"
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}
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],
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"metadata": {
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"language_info": {
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"name": "python"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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