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"source": [
"from pathlib import Path\n",
"import sys\n",
"ROOT=next(p for p in (Path.cwd(),*Path.cwd().parents) if (p/'pyproject.toml').exists())\n",
"sys.path.insert(0,str(ROOT/'src'))\n",
"import numpy as np\n",
"import pandas as pd\n",
"import matplotlib.pyplot as plt\n",
"from IPython.display import display,Markdown,HTML\n",
"from physicslab.common.theme import EXPORT_CSS\n",
"from physicslab.common.manuscripts import table_display,metrics_display,display_plotly,series_plot\n",
"from physicslab.experiments import prism_apex as adapter\n",
"EXP=ROOT/'experiments/course-04/prism-apex'\n",
"result=adapter.analyze(EXP)\n",
"import json\n",
"labels=json.loads((EXP/'authoring.json').read_text(encoding='utf-8')).get('metric_labels',{})\n",
"for key,value in result['metrics'].items():value.setdefault('label',labels.get(key,key))\n",
"display(HTML(EXPORT_CSS))\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-001",
"metadata": {},
"source": [
"# 用分光计测三棱镜的顶角\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-002",
"metadata": {},
"source": [
"## 实验名称\n",
"\n",
"实验二 用分光计测三棱镜的顶角\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-003",
"metadata": {},
"source": [
"## 实验目的\n",
"\n",
"\n",
"\n",
"1. 进一步熟悉分光计的调节与双游标读数方法。\n",
"2. 掌握反射法测量三棱镜顶角的原理和操作。\n",
"3. 分析反射像对准、棱边姿态和仪器读数对测量结果的影响。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-004",
"metadata": {},
"source": [
"## 实验原理\n",
"\n",
"\n",
"\n",
"平行光分别照射三棱镜的两个折射面时,两表面的反射光方向之间夹角为棱镜顶角的两倍。用望远镜先后对准两个反射像,设对应的两组游标读数为 $\\left( \\theta_{1},\\theta_{2} \\right)$ 和 $\\left( \\theta_{3},\\theta_{4} \\right)$,则\n",
"\n",
"$$2A = \\frac{1}{2}\\left( \\left\\vert \\theta_{3} - \\theta_{1} \\right\\vert + \\left\\vert \\theta_{4} - \\theta_{2} \\right\\vert \\right),$$\n",
"\n",
"因此\n",
"\n",
"$$A = \\frac{1}{4}\\left( \\left\\vert \\theta_{3} - \\theta_{1} \\right\\vert + \\left\\vert \\theta_{4} - \\theta_{2} \\right\\vert \\right).$$\n",
"\n",
"两个游标相差约 $180^{\\circ}$,对其角度改变量取平均,可减小刻度盘偏心误差。测量时应使三棱镜棱边与仪器转轴平行,并使两个反射像都清晰地落在望远镜视场内。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-005",
"metadata": {},
"source": [
"## 主要仪器设备\n",
"\n",
"JYY 型分光计、钠灯或汞灯、三棱镜、平面反射镜。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-006",
"metadata": {},
"source": [
"## 实验步骤\n",
"\n",
"\n",
"\n",
"1. 调节目镜、用平面镜完成望远镜自准直,使平行光管出射平行光,并调平载物台。\n",
"2. 将三棱镜置于载物台中央,使顶角朝向平行光管,调节棱镜使其棱边近似平行于分光计转轴。\n",
"3. 转动望远镜找到一个折射面的反射狭缝像,使叉丝竖线与其重合,记录 $\\theta_{1},\\theta_{2}$。\n",
"4. 转动望远镜寻找另一折射面的反射像,记录 $\\theta_{3},\\theta_{4}$。\n",
"5. 改变载物台起始方位,重复测量 5 次。\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-007",
"metadata": {},
"source": [
"## 原始数据记录\n",
"\n",
"\n",
"\n",
"测量方法采用反射法。两游标读数分别记为 $\\theta_1,\\theta_2$;望远镜转到另一反射面后,两游标读数记为 $\\theta_3,\\theta_4$。\n",
"\n",
"\n",
"\n",
"表 1 反射法测顶角原始读数\n",
"\n",
"\n",
"\n",
"| 次数 | $\\theta_1$(左) | $\\theta_2$(右) | $\\theta_3$(左) | $\\theta_4$(右) |\n",
"| --- | --- | --- | --- | --- |\n",
"| 1 | $15^{\\circ}20^{\\prime}$ | $195^{\\circ}18^{\\prime}$ | $135^{\\circ}25^{\\prime}$ | $315^{\\circ}23^{\\prime}$ |\n",
"| 2 | $86^{\\circ}45^{\\prime}$ | $266^{\\circ}43^{\\prime}$ | $206^{\\circ}48^{\\prime}$ | $26^{\\circ}46^{\\prime}$ |\n",
"| 3 | $92^{\\circ}18^{\\prime}$ | $272^{\\circ}16^{\\prime}$ | $212^{\\circ}20^{\\prime}$ | $32^{\\circ}18^{\\prime}$ |\n",
"| 4 | $121^{\\circ}16^{\\prime}$ | $301^{\\circ}14^{\\prime}$ | $241^{\\circ}19^{\\prime}$ | $61^{\\circ}17^{\\prime}$ |\n",
"| 5 | $139^{\\circ}52^{\\prime}$ | $319^{\\circ}54^{\\prime}$ | $20^{\\circ}02^{\\prime}$ | $200^{\\circ}04^{\\prime}$ |\n",
"\n",
"\n"
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"id": "prism-apex-008",
"metadata": {},
"source": [
"## 数据处理\n",
"\n",
"每组先计算两游标的角度改变量。遇到读数跨越 $0^\\circ$ 时,以圆周模 $360^\\circ$ 的最短角差计算;不将跨零读数直接相减。计算保留半分等未修约信息,最后才修约显示。\n",
"\n",
"\n",
"\n",
"表 2 顶角逐次计算(角度以小数度列出)\n",
"\n",
"\n",
"\n",
"| 次数 | 左游标角差 / ° | 右游标角差 / ° | 顶角 / ° |\n",
"| --- | --- | --- | --- |\n",
"| 1 | 120.08333 | 120.08333 | 60.041667 |\n",
"| 2 | 120.05 | 120.05 | 60.025 |\n",
"| 3 | 120.03333 | 120.03333 | 60.016667 |\n",
"| 4 | 120.05 | 120.05 | 60.025 |\n",
"| 5 | 119.83333 | 119.83333 | 59.916667 |\n",
"\n",
"未修约均值为 $\\bar A=60.00500000^\\circ$,即 $60^\\circ0.3000^\\prime$。A 类分量由五次结果计算:\n",
"\n",
"$$\n",
"u_A=\\sqrt{\\frac{\\sum_i(x_i-\\bar x)^2}{n(n-1)}}=1.34722^\\prime.\n",
"$$\n",
"\n",
"沿用记录的角度误差限 $\\Delta=1^\\prime$,将其作为最终角度估计的均匀分布限值,得到 $u_B=\\Delta/\\sqrt3=0.57735^\\prime$,两项独立合成为 $u_c=1.46572^\\prime$。该处理采用指定误差限的约定,未另行包含分光计校准与其他系统分量。\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"图 1 顶角重复测量\n",
"\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-009",
"metadata": {},
"source": [
"#### 逐组计算与结果核验\n",
"\n",
"读取观测并逐组计算,再形成最终指标;修约只作用于显示。"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "prism-apex-010",
"metadata": {
"execution": {
"iopub.execute_input": "2026-10-15T17:06:44.546021Z",
"iopub.status.busy": "2026-10-15T17:06:44.545760Z",
"iopub.status.idle": "2026-10-15T17:06:44.549621Z",
"shell.execute_reply": "2026-10-15T17:06:44.549131Z"
},
"jupyter": {
"source_hidden": true
}
},
"outputs": [
{
"data": {
"text/markdown": [
"表 计算结果\n",
"\n",
"| 指标 | 数值 | 单位 |\n",
"| --- | --- | --- |\n",
"| 顶角均值 | 60.005 | ${}^\\circ$ |\n",
"| 角度均值A类分量 | 1.3472194 | ${}^{\\prime}$ |\n",
"| 指定误差限B类分量 | 0.57735027 | ${}^{\\prime}$ |\n",
"| 两分量合成标准不确定度 | 1.4657194 | ${}^{\\prime}$ |"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import physicslab.common.manuscripts as _presentation\n",
"_presentation._table_captioned=False\n",
"metrics_display(result)"
]
},
{
"cell_type": "markdown",
"id": "prism-apex-011",
"metadata": {},
"source": [
"## 实验结果与讨论\n",
"\n",
"反射法顶角未修约均值为 $60.00500000^\\circ$,约 $60^\\circ00^\\prime$。重复性分量为 $1.3472^\\prime$,指定误差限下两项合成值为 $1.4657^\\prime$;给出的是标准不确定度,不能将其任意修约为 $1^\\prime$。\n",
"\n",
"反射像对准、游标估读和棱边未与转轴平行是主要影响因素。五组原始数据全部保留,最后一组得到约 $59^\\circ55^\\prime$,对均值离散有明显贡献;没有独立依据将其排除。\n"
]
}
],
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